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Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Effective mass signatures in multiphoton pair production

Christian Kohlf¨ urst, Holger Gies, Reinhard Alkofer

University of Graz Institute of Physics

Non-Perturbative Methods in Quantum Field Theory Balatonf¨ ured , October 8, 2014

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Outline

Motivation Theoretical Considerations Model for the Field Numerical Results Concepts Particle Distribution and Spectra Summary & Outlook

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

QED Vacuum

Cite: G. Dunne, PIF 2013, July 2013

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

External Field

• Strong electric field → charge separation
• Particles become measurable

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Dirac Sea Picture

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

Motivation Theoretical Considerations Model for the Field 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 Model for the Field 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 Model for the Field Numerical Results Summary & Outlook

Multiphoton Absorption

• Absorption of multiple photons
• C. Kohlfurst et al.: Phys. Rev. Lett. 112(050402), 2014

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Effective Mass

• Particle in background field
• Apparent mass of a particle in response to a perturbation
• Reduction of various interactions into effective mass m∗
• Treatment of m∗ as it were a free particle

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Considerations

Goal

Describe e− and e+ in a homogeneous electric field in mean field approximation

Requirement

• Describe dynamical pair creation
• Particle statistics

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Quantum Vlasov Equation

Integro-differential equation

∂tF (q,t) = W (q,t)

t

tVac

dt′W

• q,t′

1−F

• q,t′

cos

• 2θ
• q,t,t′

(1)

W (q,t) = eE (t)ε⊥ (q,t) ω2 (q,t) , ε2

⊥ (q,t) = m2 +q⊥2,

p = q−eA θ

• q,t,t′

=

t

t′ ω

• q,t′′

dt′′, ω2 (q,t) = ε2

⊥ (q,t)+(q3−eA(t))2

• Non-Markovian integral equation, Rapidly oscillating term
• Fermions and Pauli statistics
• S. A. Smolyansky et al. hep-ph/9712377 GSI-97-72, 1997
• S. Schmidt et al.: Int.J.Mod.Phys. E7 709-722, 1998

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Differential Equation

Auxiliary functions

G =

t

tVac

dt′W

• q,t′

1−F

• q,t′

cos

• 2θ
• q,t,t′

(2a) H =

t

tVac

dt′W

• q,t′

1−F

• q,t′

sin

• 2θ
• q,t,t′

(2b)

Coupled differential equation

   ˙ F ˙ G ˙ H    =   W −W −2ω 2ω     F G H  +   W   (3)

• J. C. R. Bloch et al.: Phys. Rev. D 60(116011), 1999

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Pros and Cons of QKT

Positive Aspects

• Works for arbitrary time-dependent fields
• Insight into time evolution of system
• Gives particle momentum spectra
• Particle density easily calculable

Negative Aspects

• Works only for spatial homogeneous fields
• No magnetic field present
• Mean field approximation

N.B.: Back-reaction and particle collisions can be included

• J. C. R. Bloch et al.: Phys. Rev. D 60(116011), 1999

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Outline

Motivation Theoretical Considerations Model for the Field Numerical Results Concepts Particle Distribution and Spectra Summary & Outlook

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Characteristic Scales

Pair Production

• Ecr ∼ 1016V/cm
• Icr ∼ 1029W/cm2
• τ ∼ 10−21s
• ω ∼ 1MeV/c2

Laser Parameters(Accessible region)

• Epeak ∼ 0.001Ecr
• τ ∼ 10−15s
• ωγ ∼ 0.05m
• ωe ∼ 50GeV

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Experimental Setup

• Two colliding laser fields
• Time-dependent electric field in interaction region
• M. Marklund: Nature Photonics 4, 72-74 2010

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Model for the Field

• Electric field: E (t) = εEcrexp
• −t2/(2τ2)
• cos(ωt)
• Photon energy: ω
• Field strength: ε
• Pulse duration: τ

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Above Threshold Pair Production

• Absorption of photons beyond the “ionization” threshold
• Particles produced with non-vanishing momenta
• Photon number is intensity dependent ∝ Iγ
• Likelyhood of creating particles increases with In

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

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Above Threshold Pair Production

• Photon absorption beyond the threshold → results in

higher net momentum

• Two-, three- and higher n-photon absorption processes

possible

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

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Effective Mass

• Electric field acts as background field
• e−e+ interact with background
• “Ionization” energy depends on laser field

EKin = nω −m∗

• Effective mass m∗ = m
• 1+ε2/(2ω2)
• AT peak position

nω

2

2 = m2

∗ +qn2

P . Agostini et al.: Phys. Rev. A 36, 4111, 1987

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Particle Distribution

• 0.0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1014 1012 1010 108 106 104 Parallel Momentum qm Fq ,

Parameters: τ = 300[1/m], ε = 0.2, ω = 0.3[m]

• Above-Threshold peaks
• Peak position predictable via effective mass concept

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Pulse Intensity

Blue: Peak height at q = 0 Blue: Peak height at first AT peak

• Power law behavior ∼ ε2n
• Configuration: ω = 2/3[m],τ = 60[1/m]

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Pulse Frequency

0.0 0.2 0.4 0.6 0.8 1.0 1019 1016 1013 1010 107 Momentum qm Distribution

Parameters: τ = 100[1/m], ε = 0.1, ω = 0.3305[m] Parameters: τ = 100[1/m], ε = 0.1, ω = 0.34[m]

• Increased photon energy → higher particle net momentum

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Particle Yield

• 0.285

0.4 0.667 2 1013 1010 107 104 0.1 0.333 0.5 1 Field Frequency Ωm Particle Y ield 1 ΛC

Parameters: τ = 100[1/m], ε = 0.1

• Resonant at n-photon frequencies
• Peak position at ωn = 2m∗/n

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Influence of Electric Field

0.28 0.29 0.30 0.31 0.32 0.33 0.01 0.02 0.05 0.10 0.20 0.50 1.00 Field Frequency Ωm Normalized Yielda.u .

0.20 m 2 0.15 m 2 0.10 m 2 0.05 m 2 e 0.02 m 2

Parameters: τ = 100[1/m], n = 7, LinLog-Plot

• Influence of field parameters on particle yield
• Peak position at ωn = 2m∗/7 = 2/7
• 1+ε2/(2ω2

n)

• Normalized via ε2n

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Effective Mass

• 0.00

0.05 0.10 0.15 0.20 0.25 0.30 1.00 1.05 1.10 1.15 Field Strength em 2 Effective Mass m m

m Model

• QKT, n 5

m Model

• QKT, n 7

Parameters: τ = 100[1/m],n = 7 Parameters: τ = 100[1/m],n = 5

• Comparison between numerical simulation and m∗ model
• Effective mass m∗ = m
• 1+ε2/(2ω2)

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Channel Closing

• 0.00

0.05 0.10 0.15 0.20 0.25 0.0 0.1 0.2 0.3 0.4 0.5 Field Strength em 2 Peak Position qm

m Model

• QKT

Parameters: τ = 300[1/m] ω = 0.322[m]

• AT peak position in distribution function
• The higher ε the closer the peak comes to the threshold
• Resonance: Peak at threshold(q = 0)

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Summary

• Quantum Kinetic Theory is capable of describing

multiphoton processes in the non-perturbative threshold domain

• Various consequences of field dependent threshold
• Above-threshold pair production
• Concept of an effective mass

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Outlook

• Understand full particle spectrum
• Perform calculations with more realistic parameters
• Include back-reaction(effect of internal electric field)
• Having a look at spatial inhomogeneities
• Investigate influence of magnetic fields

Motivation Theoretical Considerations Model for the Field Numerical Results Summary & Outlook

Thank you!

supported by FWF Doctoral Program on Hadrons in Vacuum, Nuclei and Stars (FWF DK W1203-N16)

Appendix

Problem Statement

Quantum electrodynamic

Only QED-Lagrangian and Dirac equation is known L = ¯ ψ(iγµDµ −m)ψ − 1 4FµνF µν (4) iγµ∂µψ −mψ = eγµAµψ (5) Covariant derivative Dµ = ∂µ +ieAµ Field strength tensor Fµν = ∂µAν −∂νAµ

Open question

How to obtain statistical quantity F (q,t) from particle description?

Appendix

Quantization

Vector Potential

Spatial-independent vector potential in one direction Aµ = (0,A(t)e3) (6)

• Very strong fields → regarded as classical
• Mean field approximation(background field)
• S. Schmidt et al.: Int. J. Mod. Phys. E 7(709), 1998
• F. Hebenstreit: Dissertation, 2011

Appendix

Quantization

Dirac Field

Fully quantized Ψ ∼ χ+a(q)+ χ−b†(q) (7) Equation of motion

• ∂ 2

t +m2 +(q3 −eA(t))2 +ieE(t)

• χ± = 0

(8)

Creation/Annihilation Operators

• Operators a(q), b†(q) hold information about particle

statistics

• Fermions → anti-commutation relations

Appendix

Operator Transformation

Hamiltonian

• Non-vanishing off-diagonal elements
• Diagonalization by Bogoliubov transformation
• Switching to quasi-particle picture

A(q,t) = α (q,t)a(q)−β ∗ (q,t)b† (−q) (9a) B† (−q,t) = β (q,t)a(q)+α∗ (q,t)b† (−q) (9b) Bogoliubov coefficients have to fulfill |α (q,t)|2 +|β (q,t)|2 = 1

Appendix

Particle Distribution

One-particle distribution function

F (q,t) = A† (q,t)A(q,t) (10) Fulfills equation of motion ∂tF (q,t) = S (q,t) (11)

• Gives distribution in momentum space
• Time-dependent quantity
• Interpretation as electron/positron distribution for t → ±∞
• nly

Appendix

Magnetic Fields

Pure Magnetic Fields

• Particle creation not possible

Collinear EM-Fields

• Space-dependent vector potential Aµ (x,t) = (0,0,xB,A(t))
• Additional constraints compared to pure electric case
• Quantised particle distribution

ε2

⊥ → m2 +eB (2n +1+(−1)s)

• N. Tanji: Ann. Phys. 8(324), 2009