schwinger pair production in strong electric fields€¦ · · 2009-08-04qed vacuum schwinger...

QED Vacuum Schwinger Effect for E(x, t) = E(t) Results Schwinger Effect for E(x, t) Summary

Schwinger Pair Production inStrong Electric Fields

Florian Hebenstreit

Advisers: R. Alkofer (KFU Graz), H. Gies (FSU Jena)

Seminar des GraduiertenkollegsFSU Jena

23.06.2009


Historical overview: Does the nothingness exist?

Does the nothingness exist?What is the meaning of the vacuum?

• Democritus, Leucippus: ∼ 450BCVacuum necessary for the motion of particles

• Aristotle: ∼ 350BCNature abhors a vacuum→ horror vacui

• Torricelli, Pascal, von Guericke: ∼ 1650ADExperiments (diluted gases)→ vacuum exists

• Quantum Field Theory: ∼ 1950ADVacuum fluctuations, virtual particles, vacuum energy...


Outline

QED Vacuum

Schwinger Effect for E(x , t) = E(t)

Results

Schwinger Effect for E(x , t)

Summary


The Vacuum of QuantumElectroDynamics

Vacuum of QFT: State in which no real particles are presentBUT

Vacuum fluctuations: Virtual particles do exist

QuantumMechanics

→ Quantum FieldTheory

← SpecialRelativity

↓ l ↓UncertaintyPrinciple:

∆E ·∆t & ~

→Vacuum

Fluctuations:Virtual Particles

←Mass-EnergyEquivalence:

E = mc2

Virtual particles exist for ∆t ≈ ~

mc2 ≈ 10−21 s


Feynman Diagrams for QED

• QED Lagrangian: L = −14FµνFµν + ψ(i /∂ − e /A + m)ψ

• Amplitudes→ Cross sections, Decay rates...• Feynman Diagrams: Graphical representation

Electron Line Photon Line Elementary VertexNLO (1-Loop) Diagrams↔ QED Corrections

Electron Self Energy Vacuum Polarization Vertex Function


Effects due to the Non-Trivial Vacuum of QED

QED effects in atomic physics:

• Anomalous Magnetic Moment: Deviation from g = 2

• Lamb Shift: Lift of degenerate energy levels in the H-atom

QED effects in pure vacuum:

• Casimir Effect: Attractive force between two parallel plates

QED effects in perturbed vacuum:

• Non-Linear Compton Scattering: → T.Heinzl (21.4.)

• Vacuum Birefringence: Linear→ elliptic polarization

• Schwinger Effect: Spontaneous production of e+e− pairs


QED effects in atomic physics

Anomalous Magnetic Moment

• Electron spin↔ Magnetic moment• Magnetic moment: ~µ = −gµB~s/~• Relativistic QM: g = 2

LO (Tree level)↔ Relativistic QM

NLO↔ QED Corrections

g − 2 = α/π → QED precision test



Lamb Shift

Lift of degenerate energy levels in the H-atom

Non-relativistic QM

2p (n = 2, l = 1)

2s (n = 2, l = 0)

1s (n = 1, l = 0)

Relativistic QM QED Corrections

Most important: Emission and re-absorption of virtual photons



Lamb Shift


Non-relativistic QM

2p (n = 2, l = 1)

2s (n = 2, l = 0)HHAAAA

Relativistic QM

2p3/2 (j = 3/2, l = 1)

2p1/2 (j = 1/2, l = 1)

2s1/2 (j = 1/2, l = 0)

QED Corrections




Lamb Shift


Non-relativistic QM

2p (n = 2, l = 1)

2s (n = 2, l = 0) AAAA

Relativistic QM

2p1/2 (j = 1/2, l = 1)

2s1/2 (j = 1/2, l = 0)

@@

QED Corrections

2s1/2 (j = 1/2, l = 0)

2p1/2 (j = 1/2, l = 1)



QED effects in pure vacuum

Casimir effect

• Outside: Fluctuations with all frequencies/wavelengths• Conducting plates→ Boundary condition: E‖ = 0• Inside: Possible frequencies/wavelengths restricted

ǫvacoutside > ǫvac

inside

FC

A= − ~cπ2

240d4

A. Lambrecht, Physik in unserer Zeit 36 (2005)

Attractive force between the plates


QED effects in perturbed vacuum

Vacuum Birefringence

• Strong background field polarizes the vacuum• Vacuum polarization→ refractive index n• Different n for different polarization states

n± = 1 +α

45π

(

EEcr

)2

(11±3)

T. Heinzl and A. Ilderton, arXiv:0809.3348 (2008)

Linear polarized probe beam→ eliptically polarized



Schwinger Effect: The Analogy to Atomic Physics

• Sub-barrier Tunnelling: Ionization of H-atom by E-fieldJ. Oppenheimer, Phys. Rev. 31 (1928)

• Ground state: Electron bound with Eb = me4

2~2 = −13.6 eV• Perturbation: Constant electric field→ ΦE ∼ −E z

r @a.uD

-13.6

VHrL @eVD

P ∼ exp(

−23

m2e5

E~4

)

∼ exp

(

−43

√2mE3/2

b

eE~

)



Schwinger Effect

• Sub-barrier Tunnelling: Production of e+e− pairs by E-fieldF. Sauter, Z. Phys. 69, 742 (1931)

W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1935)

• Vacuum state: Dirac sea picture→ Eb = 2mc2

• Perturbation: Constant electric field→ ΦE ∼ −E z

pz z

+mc2

-mc2

P ∼ exp(

−πm2c3

eE~

)

∼ exp

(

−π4

√2mE3/2

b

eE~

)



Yet another Analogy: Oscillatory InhomogeneityL. Keldysh, Sov. Phys. JETP. 20 (1965)

• 1st (laser) time scale ω: E(t) = E cos(ω t)

• 2nd (tunnelling) time scale ωT : ωT ∼ vL ∼√

Eb/2mEb/eE = eE√

2mEb

• Keldysh adiabaticity parameter γ: γ = ωωT

=ω√

2mEbeE

Non-perturbative regime: ω ≪ ωT ↔ γ ≪ 1Low frequency/Strong fields↔ ’Instantaneous’; Tunneling

P ∼ exp

(

− 43

√2mE3/2

beE~

)

Perturbative regime: ω ≫ ωT ↔ γ ≫ 1High frequency/Weak fields↔ ’No time to tunnel’; Multi-Photon

P ∼(

eE2ω

√2mEb

)2Eb/~ω



Schwinger Effect with Oscillatory Inhomogeneity

E. Brezin and C. Itzykson, Phys. Rev. D. 2 (1970)

• 1st (laser) time scale ω: E(t) = E cos(ωt)• 2nd (tunnelling) time scale ωT : ωT ∼ v

L ∼ cmc2/eE = eE

mc

• Keldysh adiabaticity parameter γ: γ = ωωT

= ωmceE

Non-perturbative regime: ω ≪ ωT ↔ γ ≪ 1Low frequency/Strong fields↔ ’Instantaneous’; Tunneling

P ∼ exp(

−πm2c3

eE~

)

Perturbative regime: ω ≫ ωT ↔ γ ≫ 1High frequency/Weak fields↔ ’No time to tunnel’; Multi-Photon

P ∼(

eE2ωmc

)4mc2/~ω



Atomic Physics Schwinger Effect

Effect: Ionization of Atoms Vacuum Pair Production

Keldysh γ: γ = ω√

2mEbeE γ = ωmc

eE

γ ≪ 1: P ∼ exp

(

− 43

√2mE3/2

beE~

)

P ∼ exp(

−πm2c3

eE~

)

γ ≫ 1: P ∼(

eE2ω

√2mEb

)2Eb/~ωP ∼

(

eE2ωmc

)4mc2/~ω


Outline

QED Vacuum


Results


Summary


Schwinger Effect: Various Methods

Various methods:

• WKB / Scattering Theory

• Effective Action Approach

• Quantum Kinetic Theory

• Monte Carlo Simulations

• ...

Only one-dimensional inhomogeneities: E(x) or E(t)!

• Technical remark: ~ = c = 1

• Critical field strength: Ecr = m2

e ≈ 1018 V/m

• Compton time: tc = 1m ≈ 10−21 s


Effective Action Approach (Imaginary Time)

• Vacuum persistence amplitude: 〈0|0〉A = eiSeff

• Pair production probability: P ≃ 1− e−2Im[Seff] ≈ 2Im[Seff]

• General expression: Seff = log det[iγµ(∂µ − ieAµ) + m]

How to calculate Im[Seff]?!

• Exact result: Constant field E(t) = E :

Im[Seff ] = V Te2E2

8π2

∞∑

n=1

1n2 exp

(

−nπm2

eE

)

J. Schwinger, Phys. Rev. 82 (1951)

• Exact result: Sauter-type field E(t) = E sech2(t/τ):

Im[Seff] = −V1

8π2

∫

d3k ln[(

1− e−πω+(k))(

1− e−πω−(k))]

N. Narozhnyi and A. Nikishov, Sov. J. Nucl. Phys. 11 (1970)


Effective Action Approach (Imaginary Time)

• Vacuum persistence amplitude: 〈0|0〉A = eiSeff

• Pair production probability: P ≃ 1− e−2Im[Seff] ≈ 2Im[Seff]

• General expression: Seff = log det[iγµ(∂µ − ieAµ) + m]

How to calculate Im[Seff]?!

• Exact result: Constant field E(t) = E :

Im[Seff ] = V Tm4

8π2

(

EEcr

)2 ∞∑

n=1

1n2 exp

(

−nπEcr

E

)

J. Schwinger, Phys. Rev. 82 (1951)

• Exact result: Sauter-type field E(t) = E sech2(t/τ):

Im[Seff] = −V1

8π2

∫

d3k ln[(

1− e−πω+(k))(

1− e−πω−(k))]

N. Narozhnyi and A. Nikishov, Sov. J. Nucl. Phys. 11 (1970)


Quantum Kinetic Theory (Real Time)

• Single particle distribution function: f (k, t)• Boltzmann-type equation: d

dt f (k, t) = C(k, t) + S(k, t)• C(k, t): Collission term→ Negligible for low densities!• S(k, t): Source term for pair production

f (k, t) and S(k, t) from first principles?!

S. Schmidt et al., Int. J. Mod. Phys. E 7 (1998)

• Simplification: sQED instead of QED• Lagrangian: |(∂ + ieA)φ(x, t)|2 −m2 |φ(x, t)|2 − 1

4FµνFµν

• Quantization: Classical vector potential Aµ = (0,A(t)e3)

• Canonical Quantization: Aµ classical↔ φ(x, t) quantized

φ(x, t) =

∫

d3k(2π)3 [gp(t)ak + g∗

p(t)b†−k]e

ik·x



• Equation of motion: In general not exactly solvable

[∂2t + m2 + k2

⊥ + (k3 − eA(t))2]gp(t) = 0

• Hamiltonian operator: Off-diagonal• Bogoliubov transformation: Quasi-particle representation

gp(t)ak + g∗p(t)b†

−k = gp(t)ak(t) + g∗p(t)b†

−k(t)

CAUTION: Particle interpretation ONLY for t→ ±∞ !

• Distribution function: f (k, t) = 〈a†k(t)ak(t)〉

• Equation of motion: ddt f (k, t) = S(k, t) in sQED

eE(t)p‖(t)

2ω2p(t)

∫ t

dt ′eE(t ′)p‖(t ′)

ω2p(t ′)

[1 + 2f (k, t ′)] cos(

2∫ t

t ′dt ′′ωp(t ′′)

)




[∂2t + ǫ2⊥ + (k3 − eA(t))2]gp(t) = 0




−k(t)




eE(t)p‖(t)

2ω2p(t)

∫ t

dt ′eE(t ′)p‖(t ′)

ω2p(t ′)

[1 + 2f (k, t ′)] cos(

2∫ t

t ′dt ′′ωp(t ′′)

)




[∂2t + ǫ2⊥ + p2

‖(t)]gp(t) = 0




−k(t)




eE(t)p‖(t)

2ω2p(t)

∫ t

dt ′eE(t ′)p‖(t ′)

ω2p(t ′)

[1 + 2f (k, t ′)] cos(

2∫ t

t ′dt ′′ωp(t ′′)

)




[∂2t + ω2

p(t)]gp(t) = 0




−k(t)




eE(t)p‖(t)

2ω2p(t)

∫ t

dt ′eE(t ′)p‖(t ′)

ω2p(t ′)

[1 + 2f (k, t ′)] cos(

2∫ t

t ′dt ′′ωp(t ′′)

)




[∂2t + ω2

p(t)]gp(t) = 0




−k(t)



• Equation of motion: ddt f (k, t) = S(k, t) in QED

eE(t)ǫ⊥2ω2

p(t)

∫ t

dt ′eE(t ′)ǫ⊥ω2

p(t ′)[1−2f (k, t ′)] cos

(

2∫ t

t ′dt ′′ωp(t ′′)

)



Quantum kinetic equation (QED)↔ Integro-differential equation

ddt

f(k, t) =eE(t)ǫ⊥2ω2

p(t)

∫ t

−∞dt′

eE(t′)ǫ⊥ω2

p(t′)[1− 2f(k, t′)] cos

(

2∫ t

t′dt′′ωp(t′′)

)

• Non-Markovian equation: Statistical factor & Cosine term• Reformulation: First order differential equation system• Backreaction mechanism: E(t) = Eext(t) + Eint(t)

Eint(t) = −4e∫

d3k(2π)3

(

p‖(t)

ωp(t)f(k, t) +

ωp(t)eE(t)

ddt

f(k, t)−eE(t)ǫ2⊥

8ω5p(t)

)

• Advantage (1): Valid for any time-dependency E(t)• Advantage (2): Momentum space distribution f (k, t)• Advantage (3): Density nqk [e+e−] = 2

∫

[dk ]f (k,∞)


Outline

QED Vacuum


Results


Summary


Electric Field: Pulse-Shaped

Time dependent field: E(t) = E sech2(t/τ )

-4 -2 2 4tΤ

0.2

0.4

0.6

0.8

1.0EHtLE

Exactly solvable in different approaches!



REMINDER: Instantaneous approximation should be valid forγ ≪ 1

ninst[e+e−] ≃ V Te2E2

4π2 exp(

−πm2

eE

)

REMINDER: Keldysh adiabaticity parameter γ

γ =m

eEτ

For field strengths of the order of E ≃ Ecr:

• γ ≪ 1: Long pulse lengths• γ ≫ 1: Short pulse lengths

What happens in a region for which γ ≈ 1?




ninst[e+e−] ≃ V∫ ∞

−∞dt ′

e2E(t ′)2

4π2 exp(

− πm2

eE(t ′)

)


γ =m

eEτ







ninst[e+e−] ≃ V∫ ∞

−∞dt ′

e2E(t ′)2

4π2 exp(

− πm2

eE(t ′)

)


γ =Ecr

Etcτ






FH, R. Alkofer and H. Gies, Phys. Rev. D 78 (2008)

E = 0.1 Ecr

æ

æ

æ

ææ

ææ

ææ

æ

à

à

àà à à à

à à à

10 20 30 40 50 60 70 80 90 100

1´10-7

5´10-7

1´10-6

5´10-6

1´10-5

5´10-5

Τ @tcD

nu

mb

erd

ensi

ty@n

m-

3D

à q.k.t.

æ inst.

τ = 10 tc

æ

æ

æ

æ

æ

ææ

ææ

æ

à

à

à

à

à

àà

àà

à

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110-7

10-4

0.1

100

105

108

EEcr

nu

mb

erd

ensi

ty@n

m-

3D

à q.k.t.

æ inst.

• Instantaneous approximation: Huge deviation for γ = 1• γ & 0.2: ’Overlap region’→ Multi-photon effects set in


Electric Field: Oscillation with Gaussian Envelope

Time dependent field: E(t) = E cos(ω t) exp(−t2/2τ2)

-4 -2 2 4tΤ

-1.0

-0.5

0.5

1.0EHtLE

Ω×Τ = 10

Ω×Τ = 5

Ω×Τ = 0

Single pulse: 1 scale τ ↔ Envelope pulse: 2 scales ω, τ

• Time scale τ : Total pulse length• Time scale ω: Laser frequency• Combined scale ωτ : Number of oscillations in the pulse



Field strengths of the order of Ecr via crossed laser beams

• XFEL (DESY): E ≃ 0.1Ecr reachable→ focusing?!

• Optical Laser (ELI): Probably ’only’ E ≃ 0.01Ecr reachable

00x @a.u.D

00

t @a.u.D

Crossed laser beams: 2 counter-propagating wave packages



Field strengths of the order of Ecr via crossed laser beams

• XFEL (DESY): E ≃ 0.1Ecr reachable→ focusing?!

• Optical Laser (ELI): Probably ’only’ E ≃ 0.01Ecr reachable

t @a.uD

-0.5

0.5

1.0

Interaction region x = 0: Oscillation with Gaussian Envelope



V. Popov, JETP Letters 74 (2001)

• WKB / Scattering Theory: Semiclassical treatment

• Gaussian approximation: Production probability for γ ≪ 1

P(p) ∼ exp(

−πEcr

E

[

1− 18γ2])

·exp(

− 1eE

[

γ2p2‖ + p2

⊥]

)

with

γ =

√

(ωτ)2 + 1(ωτ)2 γ

Accuracy of the Gaussian approximation?



FH, R. Alkofer, G. Dunne and H. Gies, Phys. Rev. Lett. 102 (2009)

E = 0.1Ecr , τ = 100 tc , ω = 25 keV −→ ωτ = 5 and γ = 0.5

0-200-400 200 400 600p° @keVD

1.´10-14

2.´10-14

3.´10-14

4.´10-14

5.´10-14

• Width: f (k,∞) NOT Gaussian → Steeper decay!• Structure: Oscillatory behaviour with ∆pmax

‖ = ω


Electric Field: Additional Phase Shift

Time dependent field: E(t) = E cos(ω t + φ) exp(−t2/2τ2)

-4 -2 2 4tΤ

-1.0

-0.5

0.5

1.0

EHtLE

Φ = -Π2

Φ = -Π4

Φ = 0

Gaussian

• For φ = 0: Time symmetric field E(t) = E(−t)• For φ 6= 0,±π/2: Mixed time symmetry• For φ = ±π/2: Time antisymmetric field E(t) = −E(−t)

Effect of the phase shift φ?



E = 0.1Ecr , τ = 100 tc , ω = 25 keV −→ ωτ = 5 and γ = 0.5

φ = −0

0-200-400 200 400 600p° @keVD

1.´10-14

2.´10-14

3.´10-14

4.´10-14

5.´10-14

φ = −π/4

1020-200-400 200 400 600p° @keVD

1.´10-14

2.´10-14

3.´10-14

4.´10-14

5.´10-14

φ = −π/2

1370-200-400 200 400 600p° @keVD

1.´10-14

2.´10-14

3.´10-14

4.´10-14

5.´10-14

Huge qualitative difference!



Explanation: Scattering picture

• REMINDER: [∂2t + ω2

p(t)]gp(t) = 0• 1-dimensional scattering problem

Hψ(x) =

[

− ~2

2m∂2

x + V (x)

]

ψ(x) = Eψ(x)

• Formal similarity→ ’Scattering potential’: V (t) ∼ −ω2p(t)

• Reflection coefficient↔ Produced pairs

Schwinger effect↔ Over-barrier-scattering!

• Asymmetric electric field: E(t) = −E(−t)• Symmetric vector potential: A(t) = A(−t)• Symmetric ’scattering potential’: ω2

p(t) = ω2p(−t)

Resonances↔ perfect transmission↔ No pairs produced!



REMINDER: Source term for pair production

sQED :eE(t)p‖(t)

2ω2p(t)

∫ t−∞ dt′

eE(t′)p‖(t′)

ω2p(t′)

[1+2f(k, t′)] cos(

2∫ t

t′ dt′′ωp(t′′))

QED : eE(t)ǫ⊥2ω2

p(t)

∫ t−∞ dt′ eE(t′)ǫ⊥

ω2p(t′)

[1−2f(k, t′)] cos(

2∫ t

t′ dt′′ωp(t′′))

0-200-400 200 400 600p° @keVD

1.´10-14

2.´10-14

3.´10-14

4.´10-14

5.´10-14

sQED

QED

0 100 200 300-100p° @keVD

1.´10-14

2.´10-14

3.´10-14

4.´10-14

5.´10-14

sQED

QED

Effect of particle statistics becomes obvious!


Outline

QED Vacuum


Results


Summary


Generalization of Quantum Kinetic Theory

Phase-Space formulation of Schwinger effect: ~x , ~p, t

• Quantum Kinetic Theory so far: E(x , t) = E(t)

• ~k conjugate variable of ~x → No direct generalization!

Approach: Dirac-Heisenberg-Wigner (DHW) function

I. Bialynicki-Birula, P. Gornicki and J. Rafelski, Phys. Rev. D 44 (1991)

• C+αβ = 〈0|

ψα( ~x1, t), ψβ(~x2, t)

|0〉 = δ3( ~x1 − ~x2)γ0αβ

• C−αβ = 〈0|[

ψα( ~x1, t), ψβ( ~x2, t)]

|0〉• Wigner transform: Fourier transform w.r.t. ~s = ~x1 − ~x2

• DHW functionWαβ(~x , ~p, t): Wigner transform of C−αβ


Generalization of Quantum Kinetic Theory

Equation of motion for DHW function

• Hartree approximation: Mean electric field• Vanishing magnetic field: ~B = 0

DtWαβ = −12∇[

γ0~γ,W]

αβ− i[

mγ0,W]

αβ− i

γ0~γ~p,W

αβ

with

Dt = ∂t + e∫ 1/2

−1/2dλ~E(~x + iλ∂p, t)∂p

• Basis set for DHW function: 1, γ5, γµ, γ5γ

µ, σµν• PDE for 16 generalized phase space functions ci(~x , ~p, t)

Dt~c(~x , ~p, t) =M(m, ~p,∇)~c(~x , ~p, t)

For E(x , t) = E(t)→ Equivalent to Quantum Kinetic Theory!


Outline

QED Vacuum


Results


Summary


Summary

QED vacuum is not empty↔ New physics with new lasers?!

• Schwinger effect: Spectacular effect in perturbed vacuum

• E(t) = E sech2(t/τ):• Instantaneous approach: Breakdown at short time scales

• E(t) = E cos(ωt)exp(−t2/2τ2):• Crossed laser beams: Realistic model at interaction region• Momentum space: Oscillatory structure; non-Gaussian

• E(t) = E cos(ωt + φ)exp(−t2/2τ2):• Phase shift: Strong dependence on φ• Particle statistics: sQED ↓↑ ←→ QED ↑↓

• Formalism for general E(x , t):• For E(x , t) = E(t): Identical to quantum kinetic approach

schwinger pair production in strong electric fields€¦ · · 2009-08-04qed vacuum schwinger...

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