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


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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.


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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.


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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+) .


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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.).


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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|

!

.


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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.


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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 .


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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).


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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.


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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 .


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



Conclusion


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


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



Conclusion


Parameters

Aµ = a0 ǫµ cos(φ), kµ = (k0, kx , 0, 0), ǫµ = (0, 0, 0, 1)

a0 = 20

ω = 1018 s−1


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



Conclusion


Parameters

Two charges hit by a short laser pulse.


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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.


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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).


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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.


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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

.


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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.


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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

.


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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

.


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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

.


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



Conclusion

Grid-free simulation code: Growth rate

Comment

Exponential growth with time.


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



Conclusion

Grid-free simulation code: Cascading

Comment

Photons cannot propagate far.


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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

.


Fields


Munich

People involved

Introduction

Radiation reaction

LAD equations

LAD and LL

LF in a plane wave

LL in a plane wave

Remarks



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.

h. ruhl-simulation of qed-cascading in ultra-intense laser fields

Documents