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“Frontiers in Intense Laser Physics”, KITP, Santa Barbara, CA, Aug 4, 2014
Pair creation in external
electric and magnetic fields
Q. Charles Su
Intense Laser Physics Theory Unit
Department of Physics, Illinois State University
Support:
National Science Foundation, NSFC
Acknowledgement: Illinois State University
Prof. Grobe
Gospodarczyk
Vikartofsky
Dr. Wagner
Alexander
Graybeal
Dr. Cheng
Rogers
Dr. Krekora
Ware
Shields
Lamb
Acknowledgement: China
Beijing
Dr. Y.T. Li
Dr. X. Lu
Dr. Y.J. Li
Dr. M. Jiang
W. Su
Q.Z. Lv
Y. Liu
Shanghai
W.Y. Wu
Dr. F. He
Dr. Z.M. Sheng
Dr. J. Zhang
ultra-intense laser development
Vacuum physics
Vacuum breakdown
e+
E = mc2
e–
E
Pair creation
Extreme Light Infrastructure
I ～ 1025 W/cm2
Int. Zetta-Exawatt Sci. & Tech.
I > 1027 W/cm2
“Zeptotechnology is just around the corner”
The Economist, page 77 Feb 28 2004
Schwinger limit
IZEST
ELI
CUOS
Relevant experiments
heavy ion
1980s，Argonne, GSI
pairs observed
nuclear triggered
laser-electron 1997， Stanford,
SLAC
electron-laser collision
indirect
pure laser
ELI，IZEST, …
pure light —> matter
Theoretical exploration
goals
new concepts
threshold estimation
possible experiments ?
theory
Europe (Sweden, ELI, …)
US (CA, CT, …)
simulation
US (ILP)
Europe (Heidelberg)
Asia (CAS)
Progress obtained with related numerical models
Relativistic quantum mechanics (1996–2003)
numerical solution to Dirac equation
motion in electric and magnetic fields
superluminal in barrier tunneling
harmonic generation
retardation effect
resonances in cycloatoms
Computational quantum field theory (2003–)
resolution of Klein paradox
electron-electron correlation
transition to negative Dirac sea
Zitterbewegung
entanglement
supercritical bound states
interference in charge density
pair creation in varying forces
Quantum fermion-boson interaction (2008–)
numerical solution to Yukawa model
virtual boson formation
transfer of fermion source correlation to bosons
pair creation with Klein-Gordon field
Processes leading to pair creation
connection with ionization?
mc2
–mc2
pair creation =
Schwinger
tunneling
or
photon
resonance
dived
or
or
transition enhancement bound state
Outline of my talk

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


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
Introduction to the subject ✔
Introduction to numerical approach
Application to the Klein paradox
Time dependent field
Field-induced bound states
Magnetic field influence
Boson versus fermion
Dived bound states
Outlook
QM Dirac equation
¶
ih f = hf
¶t
h = ca(p - qA/c) + mc 2b + qV
when A=V=0
E=± m2c 4 + p 2c 2
mc2
0
–mc2
but <(t)|(t)>=1
How to describe creation ?
Quantum field description
ˆ = å bˆ l
Y
l
l
QF operator
¶ ˆ
ˆ ,H
ˆ]
ih Y = [ Y
¶t
QM state
¶ ˆ
ˆ
ih Y = hY
¶t
ˆ =Y
ˆ †h Y
ˆ
H
ˆ (t) = å bˆ (t) l = å bˆ
Y
l
l
l
l
l(t)
From wave functions to operator solutions
bˆ l (t) = å bˆ a l a(t)
a
operator
time involution of |a>
¶
ih a = h a
¶t
Ŷ e– (t) = å b̂ l (t) l
J. Braun, Q. Su, R. Grobe, PRA 59, 604 (1999)
l
Now we know the electron’s quantum field
Apply
ˆ to pair creation
Y
ˆ =Y
ˆ + CY
ˆ
Y
e–
e+
Number of created pairs
N(t) = <<vac||
ˆ † (t) Y
ˆ (t) ||vac>>
Y
e–
e–
= åå < p | u(t) | n >
n
p
2
|p>
ih
¶
n(t) = h n(t)
¶t
|n>
n(t) = u(t) n
How to get pairs
step1: WF start with: |n>
the Dirac Sea
¶
step2: solve ih
n(t) = h n(t) to get |n(t)> picture
¶t
step3: project |n(t)> on |p>, call it <p|n(t)>
step4: repeat steps 1 to 3, …
Number of the created pairs
step5: add up results N(t) =
å < p | n(t) >
2
p,n
|p>
+
|n>
|p>
+
|n>
|p>
+
|n>
|p>
…… =
|n>
The space-time resolved pair creation
energy
e–
Phys. Rev. Lett. 92, 040406 (2004)
e+
Klein paradox
Oskar Klein
(1894-1977)
Z Physik 53, 157 (1929)
“Normal” potential height (V << 2mc2 )
Ekin
potential height V
Classical mechanics predicts:
if Ekin< V  ball cannot roll up
Traditional quantum mechanics
if Ekin< V
 bounces back
Movie: Courtesy of Bernd Thaller
http://www.kfunigraz.ac.at/imawww/thaller/
“Abnormal” potential height (V > 2mc2 )
?
height V
Single-particle quantum mechanics predicts:
even if Ekin< V  some of the “particle” rolls up
Wave packet evolution under single particle Dirac equation
V > 2 mc2
Ekin < V
Interpretation of mysterious transmitted portion unclear
Braun, Su and Grobe, PRA 59, 604 (1999)
The Klein paradox
electron
e–
no e–
pair-creation ????
e–
e+
before 2004:
simplification:
no e–- e+
affects
approach:
Quantum
mechanics
e–
e–
e+
Quantum field
theory
2004: Resolution of the Klein-paradox
Ekin < V
V > 2 mc2
• Electron cannot move into the barrier
• Klein’s state is the amount of suppressed positrons
P. Krekora, Q. Su and R. Grobe, Phys. Rev. Lett. 92, 040406 (2004).
ways to create pairs
super-critical, large force
N(t)
N
V0>2c2
0.3
V0 >2c2
3
2
1
V0=2c
0.2
V0<2c
0.1
0
0
1
2
1
EuroPhys. Lett. 86, 13001 (2009)
2
3´10-3 a.u.
0
under-critical,
time dependent force
2
2
3 t [10-4a.u.]
multi-field combination
Lowering the field threshold with
2 subcritical constant fields ?
2 subcritical fields
displaced by space D
V1 =V2=1.5c2, dash is for V=2.5c2
M. Jiang, et al, Phys. Rev. A 83, 053402 (2011)
Lowering the field threshold with
alternating field + constant field ?
Different mechanisms for pair creation:
h = E / cw
1. Multi-photon transition h <<1
for an alternating field
Key parameter:
field frequency
Threshold: ωcr > 2c 2
2. Schwinger tunneling
for a constant field
Key parameter:
field strength
Threshold: Vcr > 2c 2
h >>1
[with W ~1/c while E0 =V0/(2W )]
M. Jiang, et al, Phys. Rev. A 85, 033408 (2012)
Pair creation enhancement due to
combined external fields
Motivation:
Only alternating field:
· suppression due to Pauli blocking
Possible solution:
· pull out the created pairs
Model:
F1= V1S(z) sin(ωt)
F2= V2S(z)
with V1=1.5c2 and
with V2=2.5c2.
PRA 85, 033408 (2012)
Pair creation enhanced in combined fields !
Blue lines: both added
Red lines: only alternating
Black line: only static
Time evolution of the spatial probability
F1: creates particles
F2: drags the created particles out.
For k>0: accelerated (green balls)
For k<0: decelerated then accelerated (red balls)
Multiphoton or Schwinger tunneling ??
Perturbation theory
Two-level model
N(t) for F1 only and F1+F2
[W = 5/c ]
 For F1 only (black line), sudden rise at ω = 2c2,
suggests the start of single photon transition
 By adding F2 (red line), the region ω < 2c 2 is no longer forbidden,
due to single photon transition and
Schwinger tunneling
Above Threshold Pair-creation
M. Jiang, et al, Phys. Rev. A 85, 033408 (2012)
(A)
(B)
(C)
Pair production vs energy,
for F1 only and with six field strengths.
[W = 3/c, ω = 2.5c2, T = 0.002]
> location of the peaks: independent of V1.
peak (A): one-photon transition Einitial(EA) + ω = Efinal(EA);
peak (B) and (C): two and three-photo transitions.
Field-induced bound states enhance pair creation ?
Model: a time dependent potential well
V (z,t ) = V0 [ S(z–D/2)–S(z+D/2) ] sin(ωt )
Quantum coherent effect
Transition from the bound states
W
V0
ω
D
Key parameters:
V0-depth D-width
W-field width ω-frequency
PRA 87, 042503 (2013)
Pair creation non-monotonic with D
D = 2/c, 4/c, 6/c, 8/c and
∞ (red dashed line).
[V0 = 1.5c2, W = 0.5/c and ω = 2.5c2]
> Supercritical frequency, particles are produced continuously
> The slope varies with the well width D, and has an oscillatory behavior
Perturbation theory works
The coherent term
N
0.02
num. = pert.
N
0
0
6
10
20
Dc
0.5/c=W
4
1/c
2
3/c
0
0
5
10
15
Dc
The N-D plot, at time t=0.004. [V0=1.5c2, ω=3c2].
The inset [V0=0.1c2, W=0.5/c, ω=2.5c2, t=0.002]
TD = p / p while p : n, E = w / 2 = c c 2 + p2
Momentum distribution at t=0.002
[ V0=1.5c2, W=1/c, ω=4c2, and for
D=5/c, D=10/c, and D=82.2/c (∞) ]
Tp = p / D
New peaks in the electron energy spectrum (D/c ~ 2π/ω)
D →∞
D=3/c
The time evolution of the energy distribution from t=0 to 0.005
(a) D →∞, and (b) D=3/c. [ω=c2, V0=1.5c2, W=0.5/c]
E2a~1.2c2, E2b~1.8c2, E2c=E2a+ω and E2d=E2b+ω
New pathways due to field induced bound states
2
E/c 2
2b
2a
1
3
0
1
2
2
1
Instantaneous eigenvalues of the
potential well in one temporal
period [D=3/c, V0=1.5c2, W=0.5/c]
3
-1
-2
14.00
14.25
14.50
14.75
t/T
15.00
The excitations of the three
electron (positron) bound states
14
14.50
15
E+ 2ω =-0.8c2+2c2=1.2c2
E+ ω = 0.8c2+c2=1.8c2
Magnetic field influence

J. Schwinger (1951）
considered pure E-field
creation threshold:
Bz(x)
Ex(x)
Þ Ec ~ 1.32 ´1016 V / cm
 magnetic field can not be neglected
 model from laser plasma physics
 static fields, both change along x
 uniform along y, py conserved
Suppression of pair creation
due to a steady magnetic field
Model:
E(x) = (Ex (x),0,0) = (V0 / 2WE sech 2 (x / WE ),0,0)
B(x) = (0,0, Bz (x))= (0,0, M 0 / 2WB sech 2 (x / WB ))
Yellow line:
V0=2.5c2, M0=0
Black solid line:
V1=2.5c2, M1=0.6c2
[WE=WB=0.1/c]
Black dashed line:
V2=√(V12-M12), M2=0
PRA 86, 013422 (2012)
Pair creation in unequal width, while WB>WE
Time evolution of N for different WB
Phys. Rev. Lett. 109, 253202 (2012)
Energy spectrum of the total h vs magnetic field width WB
When WB > 1.25/c, continua begin to separate, the system become subcritical.
In the gap area new discrete energy levels emerge.
Oscillations in N(t): transition between field dressed Landau levels
Why might boson creation work better ?
• Traditional:
e-
e+
• New idea: force p, p+ to return to creation zone
B Field
B Field
p-
p+
Dream: stimulated emission amplifies p, p+ creation
# fermions (time)
5
N(t) ~ a t
# bosons (time)
N(t)
W=¥
W=6.15 l
4
c
N (t)
104
~ e kt
103
¥
W=4.55 l
2
W=6.15 l
N (t)
1
1
W=4.55 l
0
N(t)
0
c
101
c
c
model
W
D = 2.5 l
W
D= 5l
W
D = 10 l
W
D = 20 l
102
N(t) = 2g t
100
4
8 time [10-3 a.u.]
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
time [in units of l/c]
Pauli exclusion principle
Stimulated emission
Pair production stops
Pair production increases
P. Krekora, K. Cooley, Q. Su and R. Grobe, Phys. Rev. Lett. 95, 070403 (2005)
R.E. Wagner, M.R. Ware, Q. Su and R. Grobe, Phys. Rev. A 81, 052104 (2010)
Speculation
Number of pairs in a magnetic field
e–, e+ attenuation
p, p+ amplification
Exponential creation confirmed
250
100
N (t)
10
150
1
0
N(t)
slope = 598.37
0.002 t [a.u.] 0.006
100
WB = 1.10
(a)
50
0
WB
(b)
1.15
1.25
(c)
B=0
0
1.30
(d)
0 .0 0 2
0 .0 0 4
t [a.u .]
0 .0 0 6
• Four different widths WB of the magnetic field
• Inset: log scale shows exponential growth for WB=1.25
V0=2.5c2, WE=0.3=0.0022 a.u., B=0.6c3 and WB=1.25=0.0091 a.u.
suppression for very large widths
25
W c = 1 .2 5
20
B
1 .3 0
1 .3 5
N(t)
15
1 .3 7 5
10
2 .0
5
0
3 .0
0
0 .0 0 1
0 .0 0 2
t [a.u .]
0 .0 0 3
V0=2.5c2, WE=0.3, B=0.6c3
Multi-channel non-competing mechanism
ways lead to pair creation:
• Dirac +/– Sea overlap
• bound states dive in the
Dirac Sea
general belief：
strength of interaction
• when multiple states dive in,
dominant channel matters
101
4
N(t)
n(t)
(B)
10-1
(A)
3
10
(A)
-3
0
0.0006
0.0012
t [a.u.]
2 states dive in
2
(B)
1
0
0
1 state dive in
0.002
0.004
0.006
t [a.u.]
E = Er – i Ei
N(t) = N [ 1 – exp(–Gt) ]
G = Ei1 + Ei2 + …
PRL, 111, 183204 (2013)
PRA 90, 013405 (2014)
Next step: extend our model
Dirac field model
particle
external
field
antiparticle
neglected：（1）inter-particle “forces”
（2）particle reaction on external field
Maxwell-Dirac field model
particle
Maxwell field
antiparticle
Topics and goals
boson
fermion
light field correction
matter correction
Maxwell-Dirac
field model
vacuum correction
polarized charge
dynamics
？
static
new algorithm
2D/3D（supercomputer）
Initial attempts
correction on pairs
before (no force):
N(t)
with force
external field ＝》 Dirac field
1
now (yes force):
0.5
Maxwell field 《＝》 Dirac field
without force
0
0
0.4
time [10
-20
sec]
1.6
new formulas：
correction on width
0.6
Dz
analytical
Maxwell field ＝》Dirac field
i ∂y(r,t)/∂t = h(V,A) y(r,t)
with force
0.4
0.2
without force
0
0
0.3
time c [a.u.]
0.9
Dirac field ＝》 Maxwell field
(c–2∂2/∂t2 –∇2) V(r,t) = 4 π ρ(y)
(c–2∂2/∂t2 –∇2) A(r,t) = (4π/c) j(y)
Thank you for your attention!
Thank you to the organizers!