H. Ruhl-Simulation of QED-Cascading in Ultra-Intense Laser Fields
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Transcript of H. Ruhl-Simulation of QED-Cascading in Ultra-Intense Laser Fields
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Simulation of QED-Cascading inUltra-Intense Laser Fields
Hartmut Ruhl and Nina Elkina, LMU Munich
ELI Meeting
Prague, April 27, 2010
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numerical model: The classical realm
The numerical model: Quantum regime
Conclusion
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
People involved
Nina Elkina, Sergey Rykovanov, Karl-Ulrich Bamberg,LMU Munich.
Radiative MD, Radiative AMR-PIC, Perturbative QED.
Alexander Fedotov, Moscow Physics Insitute.
Investigations of QED processes in strong laser fields.
Kai Germaschewski, University of New Hampshire.
Implementation of AMR-PIC on distributed GPUs.
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Introduction
Consistent model for classical radiation reaction.
Grid-free electromagnetic simulation model.
Simulation of e+e− cascading in intense laser fields.
Exponential growth of pair production over a wide range ofparameters.
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Self-fields
To understand the origin of self-fields we consider thepotential of a charge
Aµ(y) = −e
ǫ0c
∫
d4z θ(y0 − z0) δ[(y − z)2] jµ(z) ,
where
jµ(z) = −ec∫
dτ xµ(τ) δ4[y − x(τ)] .
Explicitly this reads
Aµ
ret(y) =e
2ǫ0
xµ(τ+)
x(τ+) · [y − x(τ+)],
where
[y − x(τ+)]2 = 0 , y0 > x0(τ+) .
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Self-fields
Introducing
~r = ~y − ~x(τ+) , ~v =d~xdt
(τ+)
one obtains
A0ret(y) =
e2ǫ0
1r −~r · ~v
, ~Aret(y) =e
2ǫ0
~vr −~r · ~v
.
In simple words: The field contribution of those fields atthe world line of the particle is the self-field. However, thefield on the world line is diverging. It is problematic tointroduce point particles into a continuum theory as isClassical Electomagnetism (see Gralla et al.).
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
When are self-field effects important?
Radiation force:
∼e4E2
0 γ2
ǫ0m2ec4
Acceleration force:∼ eE0
Radiation reaction large when both forces are equal:
eE0 =e4E2
0 γ2
ǫ0m2ec4
, γ ≈ a =eE0
me ω c
This implies:
a3 =λ
re→ a ≈ 100
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
What are the LAD equations?
We consider a single electron exposed to an externalfield and its self-field:
me uµ= −
e
cFµν uν , ∂
α∂αAµ
=1
ǫ0c2jµ
jµ(x) = −ecZ
dτ uµ(τ) δ
4[x − x(τ)]
Aµself = −
e
ǫ0c
Z
dτ uµ(τ) G+[x − x(τ)]
mr uµ= −
e
cFµν
ext uν +2e2
3ǫ0c3
»
uµ+
1
c2uη uηuµ
–
,
mr = me
1 +e2
2ǫ0c3
Z
dαδ[α]
|α|
!
.
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
LAD and Landau Lifshitz equations
me uµ= −
e
cFµν uν +
2e2
3ǫ0c3
»
uµ+
1
c2uη uηuµ
–
uµ=
d
dτ
„
−e
mecFµν uν
«
= −e
mec
`
∂ηFµν uν uη+ Fµν uν
´
= −e
mec
„
∂ηFµν uν uη −e
mecFµν Fη
ν uη
«
Fµself = −
2e3
3ǫ0mec4
„
∂ηFµν uν uη −e
mecFµν Fη
ν uη
«
+2e4
3ǫ0m2ec7
Fην Fηδuν uδuµ
Valid if ω γ2 a ≪ 1.
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Lorentz force solution in a plane wave
We consider a single electron in an external plane wave:
me uµ= −
e
cFµν uν , Fµν
= ∂µAν − ∂
ν Aµ,
Aµ= ǫ
µf (φ), φ = k · x, k2= ǫ · k = 0, ǫ
2= −1 ,
Fµν=
`
kµǫν − kν
ǫµ´ f
′(φ) ,
0 = kµFµν.
d(k · u)
dτ= k · u = 0 .
The phase as a function of proper time can be obtained:
k · u(τ) = k · u(0), φ = k · x = k · u(0) τ .
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Lorentz force solution in a plane wave
The solution in an external field becomes:
duµ
dφ= −
e
mecf′(φ)
kµ ǫ · u
k · u(0)− ǫ
µ
!
,
ǫ · u(φ) = ǫ · u(0) −e
mec(f (φ) − f (0)) ,
uµ(φ) = uµ
(0) −e
mec(f (φ) − f (0))
kµ ǫ · u(0)
k · u(0)− ǫ
µ
!
+e2
m2ec2
(f (φ) − f (0))2 kµ
k · u(0).
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Solution of Landau Lifshitz in a plane wave
0 = kµ Fµν →d(k · u)
dφ=
2e4
3ǫ0m3ec5
(k · u)2 A2
d(k · u)
(k · u)2(φ) = C A2
(φ)dφ , C =2e2
3ǫ0mec3
e2
m2ec2
(k · u)(φ) =(k · u)(0)
1 − C (k · u)(0)R φ
0 dz A2(z)
Aµ(φ) = ǫ
µ a0 cos(φ) , φ = ωt − kx x , ǫµ
= (0, 0, 0, 1)
A2(φ) = −a2
0 cos2(φ) ,
Z φ
0dz cos2
(z) =φ
2+
sin(2φ)
4
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Solution of Landau Lifshitz in a plane wave
1
ω(k · u)(φ) =
1
1 + Cωa20
“
φ2 +
sin(2φ)4
” =
s
1 +u2
c2−
ux
c
dφ
dτ= (k · u)(φ)
φ(τ) ≈1
Cωa20
„
q
1 + 8(ωτ) (Cωa20) − 1
«
Cωa20 = 1.48 · 10−8 q2
e2
me
m
ω
ωLa2
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Remarks
Radiation reaction is large if a large number of electronscan be accelerated coherently or a few charges are athigh energies prior to interacting with the intense laserfield or the laser field is very intense or a combination ofall.
The Landau-Lifshitz approach becomes invalid andquantum vacuum effects enter as soon as self-field effectsbecome significant.
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Grid-free simulation code
d~up
dt=
em
~F ext + ~F self
p +∑
i 6=p
~F LWi
d~rp
dt=
~up
γ
~F ext is the external force.~F self
p = ~Eself + ~vp × ~Bself is the self-force.
~Eself = 2e γ4(
~β + 3γ2(~β · ~β)~β)
/3.
~Bself = ~β × ~Eself .
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Grid-free simulation code
The electromagnetic force ~F LW is obtained fromretarded fields:
~ELW=
e
4π
0
@
~n − ~β
γ2(1 −~n · ~β)2 R2+
1
c
~n × [(~n − ~β) × ~β]
(1 −~n · ~β)3 R
1
A
~BLW= ~n × ~ELW
~R = ~r(t) −~r′
, ~n =~r −~r
′
|~r −~r ′ |
c (t − t′) =
q
(~r −~r ′ )2
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Grid-free simulation code
Parameters
Aµ = a0 ǫµ cos(φ), kµ = (k0, kx , 0, 0), ǫµ = (0, 0, 0, 1)
a0 = 20
ω = 1018 s−1
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Grid-free simulation code
Parameters
Two charges hit by a short laser pulse.
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Grid-free simulation code
Parameters
Two charges hit by a short laser pulse.
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Grid-free simulation code
Parameters
Two charges hit by a short laser pulse.
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Grid-free simulation code: Vacuum stability
If photons reach energies high enough to overcome thepair creation threshold the quantum regime of radiation isreached.
The acceleration of an electron in an external magneticfield is used to estimate the peak of the observableradiation spectrum based on self-radiation.
The peak of the synchrotron radiation spectrum scales as~ ωc γ3, where ωc is the cyclotron frequency. Hence, itfollows that m γ c2 ≈ ~ ω γ3.
This relation implies a ≈ γ ≈√
m c2/~ ω ≈ 103, whichmeans I ≈ 1024 W/cm2.
For I > 1024 W/cm2 quantum vacuum effects, like paircreation, may become important, since sufficiently manyhard photons are produced.
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Grid-free simulation code: Assumptions
Transition rates will be functions of the externalelectromagnetic field.
Energy-momentum transfer from the external field will beneglected.
Dressed transition rates must be computed (see Landau& Lifshitz and Nikishov & Ritus and Reiss, PRL 26, 1971).
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Grid-free simulation code: Pair creation
Perturbative result for the pair creation rate in anelectromagnetic field:
W =α
3
Z
d4q θ(q2 − 4m2)h
|E(q)|2 − |B(q)|2i
1 −4m2
q2
!1/2
1 +2m2
q2
!
Pair creation is an electric effect, since|E(q)|2 − |B(q)|2 ≥ 0.
To make pair creation viable in laser fields external fieldconfigurations must be found, for which the electric field islarge or the magnetic field disappears.
Transition rates should be computed for those fieldtopologies in a non-perturbative way. This is not possiblein the general case. One is forced to make sacrifices.
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Grid-free simulation code: Essentialparameters
Dimensionless laser amplitude:
a =e a0
me c≫ 1
.
Formation length for pair creation:
l ≈me c2
e E0≪ λ
.
Field can be considered as locally constant.
Invariant intensity parameters (Es = m2e c3/e ~):
f = −Fµν Fµν
2 E2s
=~E2 − c2~B2
E2s
≪ 1 , g =ǫµνλκFµν Fλκ
8 E2s
=c ~E · ~B
E2s
≪ 1
.
The external field can be considered as crossed.
We make the approximation:Aµ
= ǫµ a0 k · x
.
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Grid-free simulation code: Fundamentalprocesses
Photon emission process (Ritus & Nikishov, Reiss) e±(ǫ) → e±(ǫ − ω) + γ(ω):
dW~E,~Brad
dω(ω, ǫ) = −
α m2
ǫ2
„Z
∞
xdξ Ai(ξ) +
»
2
x+ κ
√x–
dAi
dx(x)
«
x =
κ
χ(χ − κ)
!2/3
, 0 ≤ κ < χ
.
Pair creation process (Ritus & Nikishov, Reiss) γ(ω) → e+(ǫ) + e−(ǫ − ω):
dW~E,~Bpair
dǫ(ω, ǫ) = −
α m2
ω2
Z
∞
ydξ Ai(ξ) +
»
2
y− κ
√y–
dAi
dx(y)
!
y =
κ
χ(κ − χ)
!2/3
, 0 ≤ χ < κ
.
Pair creation threshhold is reached for κ → 1.
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Grid-free simulation code: Efficiencyparameters
Proper acceleration in units of Schwinger field Es :
χ =|pµ|e Es
=e ~
m3 c4
r
−“
Fµν pνin
”2=
Eproper frame
Es
κ =e ~
2
m3 c4
r
−“
Fµν kνin
”2
.
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Grid-free simulation code: Efficiencyparameters
Let us consider an electric field that is rotating:
There is an electron initally at rest. The equation of motion is:
d~p(t)
dt= e ~E(t)
.
For a rotation electric field we find:~p 6‖ ~E
.
It holds:
θ =ωt
2, E⊥ ≈ E0 θ =
E0ωt
2.
It holds:
χ(t) ∼E⊥ ǫ
Es me≈
E0ωt
2Es
eE0t
me
.
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Grid-free simulation code: Scales
We can solve for the time. For the creation time (χ ≈ 1) it holds that
t ∼Es me
e E20 ω
∼E2
s
E20 me ω
.
This time is much smaller than 1/ω if:
E0 ≫r
ω
mEs ≈ 10−3 Es = 1015 V
m≪ Es
.
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Grid-free simulation code: Growth rate
Comment
Exponential growth with time.
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Grid-free simulation code: Cascading
Comment
Photons cannot propagate far.
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Grid-free simulation code: Extrapolation tolarger fields
For the number of e±-pairs obtained in the laser field withone initial slow electron it holds:
µ I[
Wcm2
]
N±
0.1 3 · 1023 non1.0 3 · 1025 ≈ 104
2.0 1.2 · 1026 ≈ 1010
.
Simulation ofQED-Cascading inUltra-Intense Laser
Fields
Hartmut Ruhl andNina Elkina, LMU
Munich
People involved
Introduction
Radiation reaction
LAD equations
LAD and LL
LF in a plane wave
LL in a plane wave
Remarks
The numericalmodel: Theclassical realm
The numericalmodel: Quantumregime
Conclusion
Conclusion
The quantum radiation regime is probably reached wellbelow the Schwinger field.
With a first cold electron present cascading sets in assoon as the intensity threshold is reached and growth isexpontential.
Codes that can handle the physics represent a newgeneration of simulation codes because of self-fields,rapidly rising particle numbers and dynamic scales.