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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
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Outline
Motivation
Theoretical ConsiderationsPreliminary ThoughtsFormalism
Numerical ResultsFew-Cycle PulseMany-Cycle Pulse
Summary & Outlook
Motivation Theoretical Considerations Numerical Results Summary & Outlook
QED Vacuum
Cite: G. Dunne, PIF 2013, July 2013
Motivation Theoretical Considerations Numerical Results Summary & Outlook
External Field
• Strong electric field→ charge separation• Particles become measurable
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Dirac Sea Picture
• Blue: positron band, Red: electron band• Measurement: Overcome band gap
Motivation Theoretical Considerations Numerical Results Summary & Outlook
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
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Photon Absorption
• Photon absorption P ≈(
eEτ
2m
)4mτ
• Relies on photon energy
N. Narozhnyi: Sov. J. Nucl. Phys. 11(596), 1970
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Multi-Photon Absorption
• Simultaneous absorption of multiple photons• Production rate is given by the n-th power of the intensity
Motivation Theoretical Considerations Numerical Results Summary & Outlook
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
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Dynamically Assisted Schwinger Effect
• Photon absorption→ virtual electron state• Subsequent particle tunneling
R. Schutzhold et al.: Phys. Rev. Lett. 101(130404), 2008
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Outline
Motivation
Theoretical ConsiderationsPreliminary ThoughtsFormalism
Numerical ResultsFew-Cycle PulseMany-Cycle Pulse
Summary & Outlook
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Experimental Setup
• Two colliding laser fields• Model for the electromagnetic field in interaction region
M. Marklund: Nature Photonics 4, 72-74 2010
Motivation Theoretical Considerations Numerical Results Summary & Outlook
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
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Standing-Wave Approximation
E = Ez+ + Ez− = 2cos(ωt)cos(ωx) (1)B = By+ + By− = 2sin(ωt)sin(ωx) (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
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Standing-Wave Approximation
E ∼ 2cos(ωt) (1)B ∼ 0 (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
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Standing-Wave Approximation
Ex = g (x)h (t)cos(ωt) (1)
• Model laser pulse as plane wave• Collision→ standing wave• Investigating pair production at x ∼ 0• Toy model in order to study spatially inhomogeneous
background
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Considerations
GoalDescribe e−e+ pair production in an electric field
Requirement
• Describe dynamical pair creation• Inhomogeneous background field• Particle statistics
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Quasi-probability Distribution
Wigner operator
W (x ,p) =12
∫d4y eip·y U(Aµ ,x ,y)
[ψ(x− y
2),ψ(x +
y2
)]
(2)
• 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
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Quasi-probability DistributionWigner operator
W (x ,p) =12
∫d4y eip·y U(Aµ ,x ,y)
[ψ(x− y
2),ψ(x +
y2
)]
(2)
• 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
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Dirac-Heisenberg-Wigner Formalism
Equation of motion(Dt1+ ∂xA + 2pB
)w = Mw (3)
Wigner vector w =(s, v‖, v⊥, v0
)T
Matrices 1, A, B and M
Pseudo-differential operator
Dt = ∂t + e∫
dξE (x + iξ ∂p, t) ·∂p
F. Hebenstreit: Dissertation, 2011
Motivation Theoretical Considerations Numerical Results Summary & Outlook
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)
Motivation Theoretical Considerations Numerical Results Summary & Outlook
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
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Lorentz Force
ddt
(γv) = eE(x , t) (7)
• Simple and descriptive• Easy to use→ particle trajectory• Explains aspects of distribution of created particles
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Outline
Motivation
Theoretical ConsiderationsPreliminary ThoughtsFormalism
Numerical ResultsFew-Cycle PulseMany-Cycle Pulse
Summary & Outlook
Motivation Theoretical Considerations Numerical Results 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
Motivation Theoretical Considerations Numerical Results Summary & Outlook
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
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Self-bunching
Homogeneous field
• Particle position irrelevant• Particles created at same point in time→ acquire same
momentum
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Self-bunching
Inhomogeneous field
• Particles accelerated out of strong background field• Particles bunched into smaller phase space volume
Motivation Theoretical Considerations Numerical Results Summary & Outlook
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
• Spatial extent: λ
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Particle Density
Parameters: τ = 5[1/m] ε = 0.5
Large spatial extent
• Interference pattern• Similarity to double slit in time
E. Akkermans et al.: Phys. Rev. Lett. 108, 030401 (2012)
CK, in preparation
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Particle Density
Parameters: τ = 5[1/m] ε = 0.5
Small spatial extent
• Peak center shifted to lower p• Vanishing interference pattern• Double peak structure
CK, in preparation
Motivation Theoretical Considerations Numerical Results Summary & Outlook
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
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Interference Pattern
Path I
• Particle measured at x with momentum p• Possibility, that it was created at second peak
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Interference Pattern
Path II
• Particle measured at x with momentum p• Equally possible, that it was created at first peak
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Interference Pattern
Small spatial extent
• Particles created at first peak do not interact with secondpeak
• Particles with negative and positive momentum
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Outline
Motivation
Theoretical ConsiderationsPreliminary ThoughtsFormalism
Numerical ResultsFew-Cycle PulseMany-Cycle Pulse
Summary & Outlook
Motivation Theoretical Considerations Numerical Results 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
• Spatial extent: λ
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Particle Yield
Parameters: τ = 1000[1/m] ε = 0.5
Overview
• Calculation for spatially homogeneous field• Different mechanism dominate in different regions
CK, in preparation
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Particle Yield
Parameters: τ = 1000[1/m] ε = 0.5
Schwinger dominance
• Schwinger dominance at ω = 0• Inhomogeneous background→ self-bunching
F. Hebenstreit et al.: Phys. Rev. Lett. 107, 180403 (2011)
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Particle Yield
Parameters: τ = 1000[1/m] ε = 0.5
CEP region
• Tunneling and absorption processes• Carrier Envelope Phase influences particle distribution
CK, in preparation
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Particle Yield
Parameters: τ = 1000[1/m] ε = 0.5
Multiphoton region
• Above-Threshold peaks, effective mass concept• Inhomogeneous background→ particle deflection
CK et al.: Phys. Rev. Lett. 112, 050402 (2014)
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Particle Yield
Parameters: τ = 1000[1/m] ε = 0.5
High energy region
• Every energy package is capable of creating particles
CK, in preparation
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Ponderomotive Force
• Force on particle in inhomogeneous oscillating field• F ∼− 1
ω2 ∇E2
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Particle Density - CEP Region
Parameters: τ = 100[1/m] ε = 0.5 ω = 0.3[m] φ = 0
• Ponderomotive force results in additional acceleration ofparticles
• Particles created at main peak with px ∼ 0[m] areaccelerated to px > 0[m]
CK, in preparation
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Particle Density - CEP Region
Parameters: τ = 100[1/m] ε = 0.5 ω = 0.3[m] φ = π/2
• Particle accelerated in both directions• No offset in particle momentum
CK, in preparation
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Particle Density - Multiphoton Region
Parameters: τ = 100[1/m] ε = 0.5 ω = 0.7[m]
• Ponderomotive force accelerates all particles• Symmetry in px remains
CK, in preparation
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Normalized Particle Yield - Multiphoton Region
Parameters: τ = 100[1/m] ε = 0.5 ω = 0.7[m]
• Shows non-monotonic behaviour
CK, in preparation
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Summary
Dirac-Heisenberg-Wigner formalism
• Pair production process in the non-perturbative thresholddomain
• All mechanism of pair production(Schwinger, Multiphoton, ...)
• Spatially inhomogeneous electric field
Spatially inhomogeneous background
• Self-bunching• Vanishing of interference pattern• Ponderomotive force
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Outlook
• Perform calculations with more realistic parameters• Include back-reaction(effect of internal electric field)• Further investigate effects stemming from inhomogeneous
background• Calculation for QED3+1 including magnetic fields
Motivation Theoretical Considerations Numerical Results Summary & Outlook
Thank you!
supported by FWF Doctoral Program on Hadrons in Vacuum, Nuclei and Stars (FWFDK W1203-N16)