Electron-Positron Pair Production in Spatially...

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Motivation Theoretical Considerations Numerical Results Summary & Outlook

Electron-Positron Pair Production inSpatially Inhomogeneous Electric Fields

Christian KohlfurstPhD-Advisor: Reinhard Alkofer

University of GrazInstitute of Physics

PhD SeminarGraz, April 22, 2015

Outline

Motivation

Theoretical ConsiderationsPreliminary ThoughtsFormalism

Numerical ResultsFew-Cycle PulseMany-Cycle Pulse

Summary & Outlook

QED Vacuum

Cite: G. Dunne, PIF 2013, July 2013

External Field

• Strong electric field→ charge separation• Particles become measurable

Dirac Sea Picture

• Blue: positron band, Red: electron band• Measurement: Overcome band gap

Schwinger Effect

• Electron tunneling P ≈ exp(−πm2/eE)

• Relies on field strength Ecr = 1.3 ·1018V/m

F. Sauter: Z. Phys. 69(742), 1931

J. S. Schwinger: Phys. Rev. 82(664), 1951

Photon Absorption

• Photon absorption P ≈(

• Relies on photon energy

N. Narozhnyi: Sov. J. Nucl. Phys. 11(596), 1970

Multi-Photon Absorption

• Simultaneous absorption of multiple photons• Production rate is given by the n-th power of the intensity

Above-Threshold Pair Production

• Absorption of additional photons beyond the threshold• Produced particles have non-vanishing momentum

P. Agostini et al.: Phys. Rev. Lett. 42(1127-1130), 1979

Dynamically Assisted Schwinger Effect

• Photon absorption→ virtual electron state• Subsequent particle tunneling

R. Schutzhold et al.: Phys. Rev. Lett. 101(130404), 2008

Outline

Motivation

Summary & Outlook

Experimental Setup

• Two colliding laser fields• Model for the electromagnetic field in interaction region

M. Marklund: Nature Photonics 4, 72-74 2010

Standing-Wave Approximation

Ez± = cos(ω (t±x)) (1)By± =±cos(ω (t±x)) (2)

• Model laser pulse as plane wave• Collision→ standing wave• Investigating pair production at x ∼ 0• Toy model in order to study spatially inhomogeneous

background

E = Ez+ + Ez− = 2cos(ωt)cos(ωx) (1)B = By+ + By− = 2sin(ωt)sin(ωx) (2)

background

E ∼ 2cos(ωt) (1)B ∼ 0 (2)

background

Ex = g (x)h (t)cos(ωt) (1)

background

Considerations

GoalDescribe e−e+ pair production in an electric field

Requirement

• Describe dynamical pair creation• Inhomogeneous background field• Particle statistics

Quasi-probability Distribution

Wigner operator

W (x ,p) =12

∫d4y eip·y U(Aµ ,x ,y)

[ψ(x− y

2),ψ(x +

• Aµ in mean field approach

• W (x ,p) is gauge invariant

Equal-time ApproachW(x,p, t) =

∫ dp0

2πW (x ,p)

D. Vasak et al.: Annals of Physics 173(462-492), 1987

I. Bialynicki-Birula et al.: Phys. Rev. D 44(6), 1991

Quasi-probability DistributionWigner operator

W (x ,p) =12

∫d4y eip·y U(Aµ ,x ,y)

[ψ(x− y

2),ψ(x +

• Aµ in mean field approach

• W (x ,p) is gauge invariant

Equal-time Approach

W(x,p, t) =∫ dp0

2πW (x ,p) =

14(s+ iγ5p+ γ

µvµ + γµ

γ5aµ + σµνtµν

)D. Vasak et al.: Annals of Physics 173(462-492), 1987

I. Bialynicki-Birula et al.: Phys. Rev. D 44(6), 1991

Dirac-Heisenberg-Wigner Formalism

Equation of motion(Dt1+ ∂xA + 2pB

)w = Mw (3)

Wigner vector w =(s, v‖, v⊥, v0

Matrices 1, A, B and M

Pseudo-differential operator

Dt = ∂t + e∫

dξE (x + iξ ∂p, t) ·∂p

F. Hebenstreit: Dissertation, 2011

Observables

Particle Density

N(t → ∞) =∫

n(p, t → ∞) dp, (4)

n(p, t) =∫

dxs(x ,p, t) + p · v‖ (x ,p, t)

ω(p)(5)

with one-particle energy ω(p) =√

1 + p2

Charge Density

Q(t) =∫

dx dp v0 (x ,p, t) (6)

Pros and Cons

Positive Aspects

• Works for spatially inhomogeneous and time-dependentelectric fields

• Insight into time evolution of system• Gives particle spectra

Negative Aspects

• Mean field approximation• No back-reaction or particle collisions• ∇ ·A 6= 0

Lorentz Force

(γv) = eE(x , t) (7)

• Simple and descriptive• Easy to use→ particle trajectory• Explains aspects of distribution of created particles

Outline

Motivation

Summary & Outlook

Model for the Field I

• Electric field: E(x , t) = εEcr sech2 ( tτ

)exp(− x2

2λ 2 )

• Field strength: ε

• Temporal scale: τ

• Spatial extent: λ

Particle Density

Parameters: τ = 10[1/m] ε = 0.75

• Self-Bunching• λ → 0: Total field energy vanishes

F. Hebenstreit et al.: Phys. Rev. Lett. 107, 180403 (2011)

CK, in preparation

Self-bunching

Homogeneous field

• Particle position irrelevant• Particles created at same point in time→ acquire same

momentum

Self-bunching

Inhomogeneous field

• Particles accelerated out of strong background field• Particles bunched into smaller phase space volume

Model for the Field II

• Electric field:E(x , t) = εEcr

(sech2 ( t

τ−1)− sech2 ( t

τ+ 1))

exp(− x2

2λ 2 )

• Field strength: ε

• Temporal scale: τ

Particle Density

Large spatial extent

• Interference pattern• Similarity to double slit in time

E. Akkermans et al.: Phys. Rev. Lett. 108, 030401 (2012)

CK, in preparation

Particle Density

Small spatial extent

• Peak center shifted to lower p• Vanishing interference pattern• Double peak structure

CK, in preparation

Interference Pattern

Particle trajectory

• Particle created at peak field strength of first peak• Accelerated due to presence of first peak in electric field• Second peak accelerates particle in opposite direction

Path I

• Particle measured at x with momentum p• Possibility, that it was created at second peak

Path II

• Particle measured at x with momentum p• Equally possible, that it was created at first peak

Small spatial extent

• Particles created at first peak do not interact with secondpeak

• Particles with negative and positive momentum

Outline

Motivation

Summary & Outlook

Model for the Field III

• Electric field:E(x , t) = εEcr cos4 (t/τ)cos(ωt + φ)exp(− x2

2λ 2 )

• Field strength: ε, Photon energy ω

• Temporal scale: τ, Phase φ

Particle Yield