Jun 23, 2009 ... Vacuum fluctuations, virtual particles, vacuum energy. ... Electron Line. Photon
Line. Elementary Vertex. NLO (1-Loop) Diagrams ↔ QED Corrections ..... Monte
Carlo Simulations. • . ..... V. Popov, JETP Letters 74 (2001).
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Schwinger Pair Production in Strong Electric Fields Florian Hebenstreit Advisers: R. Alkofer (KFU Graz), H. Gies (FSU Jena)
Seminar des Graduiertenkollegs FSU Jena 23.06.2009
Summary
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
Historical overview: Does the nothingness exist? Does the nothingness exist? What is the meaning of the vacuum? • Democritus, Leucippus: ∼ 450BC
Vacuum necessary for the motion of particles
• Aristotle: ∼ 350BC
Nature abhors a vacuum → horror vacui
• Torricelli, Pascal, von Guericke: ∼ 1650AD
Experiments (diluted gases) → vacuum exists
• Quantum Field Theory: ∼ 1950AD
Vacuum fluctuations, virtual particles, vacuum energy...
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Outline
QED Vacuum Schwinger Effect for E (x, t) = E (t) Results Schwinger Effect for E (x, t) Summary
Schwinger Effect for E(x , t)
Summary
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Outline
QED Vacuum Schwinger Effect for E (x, t) = E (t) Results Schwinger Effect for E (x, t) Summary
Schwinger Effect for E(x , t)
Summary
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Schwinger Effect for E(x , t)
Results
Summary
The Vacuum of QuantumElectroDynamics Vacuum of QFT: State in which no real particles are present BUT Vacuum fluctuations: Virtual particles do exist
Quantum Mechanics
→
↓
Uncertainty Principle: ∆E · ∆t & ~
Quantum Field Theory
←
l
→
Vacuum Fluctuations: Virtual Particles
Virtual particles exist for ∆t ≈
~ mc 2
Special Relativity ↓
←
Mass-Energy Equivalence: E = mc2
≈ 10−21 s
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
Feynman Diagrams for QED
¯ ∂/ − eA / + m)ψ • QED Lagrangian: L = − 14 F µν Fµν + ψ(i • Amplitudes → Cross sections, Decay rates...
• Feynman Diagrams: Graphical representation
Electron Line
Photon Line
Elementary Vertex
NLO (1-Loop) Diagrams ↔ QED Corrections
Electron Self Energy
Vacuum Polarization
Vertex Function
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
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 Vacuum
Schwinger Effect for E(x , t) = E(t)
Schwinger Effect for E(x , t)
Results
Summary
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
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
QED effects in atomic physics Lamb Shift Lift of degenerate energy levels in the H-atom Non-relativistic QM
Relativistic QM
QED Corrections
2p (n = 2, l = 1) 2s (n = 2, l = 0)
1s (n = 1, l = 0) Most important: Emission and re-absorption of virtual photons
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
QED effects in atomic physics Lamb Shift Lift of degenerate energy levels in the H-atom Non-relativistic QM
Relativistic QM
QED Corrections
2p (n = 2, l = 1)
H 2p3/2 (j = 3/2, l = 1) 2s (n = 2, l = 0) A H A A 2p (j = 1/2, l = 1) A 1/2
2s1/2 (j = 1/2, l = 0) Most important: Emission and re-absorption of virtual photons
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
QED effects in atomic physics Lamb Shift Lift of degenerate energy levels in the H-atom Non-relativistic QM
Relativistic QM
QED Corrections
2p (n = 2, l = 1) 2s (n = 2, l = 0) A
2s
(j = 1/2, l = 0)
1/2 A A 2p (j = 1/2, l = 1) A 1/2 2s1/2 (j = 1/2, l = 0)@ 2p1/2 (j = 1/2, l = 1) @
Most important: Emission and re-absorption of virtual photons
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
QED effects in pure vacuum Casimir effect • Outside: Fluctuations with all frequencies/wavelengths • Conducting plates → Boundary condition: Ek = 0 • Inside: Possible frequencies/wavelengths restricted vac ǫvac outside > ǫinside
~cπ 2 FC =− A 240d 4 A. Lambrecht, Physik in unserer Zeit 36 (2005)
Attractive force between the plates
Summary
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
QED effects in perturbed vacuum Vacuum Birefringence • Strong background field polarizes the vacuum • Vacuum polarization → refractive index n • Different n for different polarization states
α n± = 1 + 45π
E Ecr
2
(11±3)
T. Heinzl and A. Ilderton, arXiv:0809.3348 (2008)
Linear polarized probe beam → eliptically polarized
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
QED effects in perturbed vacuum Schwinger Effect: The Analogy to Atomic Physics • Sub-barrier Tunnelling: Ionization of H-atom by E-field
J. Oppenheimer, Phys. Rev. 31 (1928) 4
• Ground state: Electron bound with Eb = me = −13.6 eV 2~2 • Perturbation: Constant electric field → ΦE ∼ −E z VHrL @eVD r @a.uD
-13.6
2 m2 e5 P ∼ exp − 3 E ~4
4 ∼ exp − 3
√
3/2
2mEb eE ~
!
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
QED effects in perturbed vacuum Schwinger Effect: The Analogy to Atomic Physics • Sub-barrier Tunnelling: Ionization of H-atom by E-field
J. Oppenheimer, Phys. Rev. 31 (1928) 4
• Ground state: Electron bound with Eb = me = −13.6 eV 2~2 • Perturbation: Constant electric field → ΦE ∼ −E z VHrL @eVD r @a.uD
-13.6
2 m2 e5 P ∼ exp − 3 E ~4
4 ∼ exp − 3
√
3/2
2mEb eE ~
!
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
QED effects in perturbed vacuum Schwinger Effect: The Analogy to Atomic Physics • Sub-barrier Tunnelling: Ionization of H-atom by E-field
J. Oppenheimer, Phys. Rev. 31 (1928) 4
• Ground state: Electron bound with Eb = me = −13.6 eV 2~2 • Perturbation: Constant electric field → ΦE ∼ −E z VHrL @eVD r @a.uD
-13.6
2 m2 e5 P ∼ exp − 3 E ~4
4 ∼ exp − 3
√
3/2
2mEb eE ~
!
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Schwinger Effect for E(x , t)
Results
Summary
QED effects in perturbed vacuum Schwinger Effect • Sub-barrier Tunnelling: Production of e+ e− pairs by E-field
F. Sauter, Z. Phys. 69, 742 (1931) W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1935)
• Vacuum state: Dirac sea picture → Eb = 2mc 2 • Perturbation: Constant electric field → ΦE ∼ −E z +mc2 -mc2
pz
m2 c 3 P ∼ exp −π eE ~
z
π ∼ exp − 4
√
3/2
2mEb eE ~
!
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Schwinger Effect for E(x , t)
Results
Summary
QED effects in perturbed vacuum Schwinger Effect • Sub-barrier Tunnelling: Production of e+ e− pairs by E-field
F. Sauter, Z. Phys. 69, 742 (1931) W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1935)
• Vacuum state: Dirac sea picture → Eb = 2mc 2 • Perturbation: Constant electric field → ΦE ∼ −E z +mc2 -mc2
pz
m2 c 3 P ∼ exp −π eE ~
z
π ∼ exp − 4
√
3/2
2mEb eE ~
!
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Schwinger Effect for E(x , t)
Results
Summary
QED effects in perturbed vacuum Schwinger Effect • Sub-barrier Tunnelling: Production of e+ e− pairs by E-field
F. Sauter, Z. Phys. 69, 742 (1931) W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1935)
• Vacuum state: Dirac sea picture → Eb = 2mc 2 • Perturbation: Constant electric field → ΦE ∼ −E z +mc2 -mc2
pz
m2 c 3 P ∼ exp −π eE ~
z
π ∼ exp − 4
√
3/2
2mEb eE ~
!
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
QED effects in perturbed vacuum Yet another Analogy: Oscillatory Inhomogeneity L. Keldysh, Sov. Phys. JETP. 20 (1965)
• 1st (laser) time scale ω: E (t) = E cos(ω t) √ • 2nd (tunnelling) time scale ωT : ωT ∼
v L
• Keldysh adiabaticity parameter γ: γ =
∼
ω ωT
Eb /2m Eb /eE
=
ω
√
= √ eE
2mEb
2mEb eE
Non-perturbative regime: ω ≪ ωT ↔ γ ≪ 1 Low frequency/Strong fields ↔ ’Instantaneous’; Tunneling √ 3/2 2mE P ∼ exp − 43 eE~b
Perturbative regime: ω ≫ ωT ↔ γ ≫ 1 High frequency/Weak fields ↔ ’No time to tunnel’; Multi-Photon 2Eb /~ω P ∼ 2ω√eE2mE b
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
QED effects in perturbed vacuum 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 ∼ vL ∼ mc 2c/eE = • Keldysh adiabaticity parameter γ: γ =
ω ωT
=
eE mc
ωmc eE
Non-perturbative regime: ω ≪ ωT ↔ γ ≪ 1 Low frequency/Strong fields ↔ ’Instantaneous’; Tunneling 2 c3 P ∼ exp −π meE~
Perturbative regime: ω ≫ ωT ↔ γ ≫ 1 High frequency/Weak fields ↔ ’No time to tunnel’; Multi-Photon 2 eE 4mc /~ω P ∼ 2ωmc
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
QED effects in perturbed vacuum
Effect:
Atomic Physics
Schwinger Effect
Ionization of Atoms
Vacuum Pair Production
Keldysh γ:
γ=
γ ≪ 1:
P ∼ exp
γ ≫ 1:
P∼
√ ω 2mEb eE
− 34
√
√eE 2ω 2mEb
γ= 3/2
2mEb eE~
2Eb /~ω
ωmc eE
2 c3 P ∼ exp −π meE~ P∼
2 eE 4mc /~ω 2ωmc
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Outline
QED Vacuum Schwinger Effect for E (x, t) = E (t) Results Schwinger Effect for E (x, t) Summary
Schwinger Effect for E(x , t)
Summary
Schwinger Effect for E(x , t) = E(t)
QED Vacuum
Schwinger Effect for E(x , t)
Results
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
m2 e ≈ 10−21 s
• Critical field strength: Ecr = • Compton time: tc =
1 m
≈
1018 V/m
Summary
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
Effective Action Approach (Imaginary Time) • Vacuum persistence amplitude: h0|0iA = eiSeff • Pair production probability: P ≃ 1 − e−2Im[Seff ] ≈ 2Im[Seff ] • General expression: Seff = log det[iγ µ (∂µ − ieAµ ) + m]
How to calculate Im[Seff ]?! • Exact result: Constant field E (t) = E :
∞ e2E 2 X 1 nπm2 Im[Seff ] = V T exp − eE 8π 2 n2 n=1
J. Schwinger, Phys. Rev. 82 (1951)
• Exact result: Sauter-type field E (t) = E sech2 (t/τ ):
1 Im[Seff ] = −V 8π 2
Z
d 3 k ln
i h 1 − e−πω+ (k) 1 − e−πω− (k)
N. Narozhnyi and A. Nikishov, Sov. J. Nucl. Phys. 11 (1970)
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Schwinger Effect for E(x , t)
Results
Summary
Effective Action Approach (Imaginary Time) • Vacuum persistence amplitude: h0|0iA = eiSeff • Pair production probability: P ≃ 1 − e−2Im[Seff ] ≈ 2Im[Seff ] • General expression: Seff = log det[iγ µ (∂µ − ieAµ ) + m]
How to calculate Im[Seff ]?! • Exact result: Constant field E (t) = E :
m4 Im[Seff ] = V T 8π 2
E Ecr
2 X ∞ n=1
1 Ecr exp −nπ E n2 J. Schwinger, Phys. Rev. 82 (1951)
• Exact result: Sauter-type field E (t) = E sech2 (t/τ ):
Im[Seff ] = −V
1 8π 2
Z
d 3 k ln
i h 1 − e−πω+ (k) 1 − e−πω− (k)
N. Narozhnyi and A. Nikishov, Sov. J. Nucl. Phys. 11 (1970)
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Schwinger Effect for E(x , t)
Results
Summary
Quantum Kinetic Theory (Real Time) • Single particle distribution function: f (k, t)
d • Boltzmann-type equation: 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 − 41 F µν Fµν • Quantization: Classical vector potential Aµ = (0, A(t)e3 )
• Canonical Quantization: Aµ classical ↔ φ(x, t) quantized
φ(x, t) =
Z
d 3k † [gp (t)ak + gp∗ (t)b−k ]eik·x (2π)3
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
Quantum Kinetic Theory (Real Time) • Equation of motion: In general not exactly solvable
[∂t2 + m2 + k2⊥ + (k3 − eA(t))2 ]gp (t) = 0 • Hamiltonian operator: Off-diagonal • Bogoliubov transformation: Quasi-particle representation † ˜ † (t) ˜p (t)a ˜k (t) + g ˜p∗ (t)b gp (t)ak + gp∗ (t)b−k =g −k
CAUTION: Particle interpretation ONLY for t → ±∞ ! ˜†k (t)a ˜k (t)i • Distribution function: f (k, t) = ha • Equation of motion:
eE (t)pk (t) 2ωp2 (t)
Z
t
dt ′
d dt f (k, t)
= S(k, t) in sQED
eE (t ′ )pk (t ′ ) ωp2 (t ′ )
Z t dt ′′ ωp (t ′′ ) [1 + 2f (k, t ′ )] cos 2 t′
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
Quantum Kinetic Theory (Real Time) • Equation of motion: In general not exactly solvable
[∂t2 + ǫ2⊥ + (k3 − eA(t))2 ]gp (t) = 0 • Hamiltonian operator: Off-diagonal • Bogoliubov transformation: Quasi-particle representation † ˜ † (t) ˜p (t)a ˜k (t) + g ˜p∗ (t)b gp (t)ak + gp∗ (t)b−k =g −k
CAUTION: Particle interpretation ONLY for t → ±∞ ! ˜†k (t)a ˜k (t)i • Distribution function: f (k, t) = ha • Equation of motion:
eE (t)pk (t) 2ωp2 (t)
Z
t
dt ′
d dt f (k, t)
= S(k, t) in sQED
eE (t ′ )pk (t ′ ) ωp2 (t ′ )
Z t dt ′′ ωp (t ′′ ) [1 + 2f (k, t ′ )] cos 2 t′
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
Quantum Kinetic Theory (Real Time) • Equation of motion: In general not exactly solvable
[∂t2 + ǫ2⊥ + pk2 (t)]gp (t) = 0
• Hamiltonian operator: Off-diagonal • Bogoliubov transformation: Quasi-particle representation † ˜ † (t) ˜p (t)a ˜k (t) + g ˜p∗ (t)b gp (t)ak + gp∗ (t)b−k =g −k
CAUTION: Particle interpretation ONLY for t → ±∞ ! ˜†k (t)a ˜k (t)i • Distribution function: f (k, t) = ha • Equation of motion:
eE (t)pk (t) 2ωp2 (t)
Z
t
dt ′
d dt f (k, t)
= S(k, t) in sQED
eE (t ′ )pk (t ′ ) ωp2 (t ′ )
Z t dt ′′ ωp (t ′′ ) [1 + 2f (k, t ′ )] cos 2 t′
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
Quantum Kinetic Theory (Real Time) • Equation of motion: In general not exactly solvable
[∂t2 + ωp2 (t)]gp (t) = 0 • Hamiltonian operator: Off-diagonal • Bogoliubov transformation: Quasi-particle representation † ˜ † (t) ˜p (t)a ˜k (t) + g ˜p∗ (t)b gp (t)ak + gp∗ (t)b−k =g −k
CAUTION: Particle interpretation ONLY for t → ±∞ ! ˜†k (t)a ˜k (t)i • Distribution function: f (k, t) = ha • Equation of motion:
eE (t)pk (t) 2ωp2 (t)
Z
t
dt ′
d dt f (k, t)
= S(k, t) in sQED
eE (t ′ )pk (t ′ ) ωp2 (t ′ )
Z t dt ′′ ωp (t ′′ ) [1 + 2f (k, t ′ )] cos 2 t′
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
Quantum Kinetic Theory (Real Time) • Equation of motion: In general not exactly solvable
[∂t2 + ωp2 (t)]gp (t) = 0 • Hamiltonian operator: Off-diagonal • Bogoliubov transformation: Quasi-particle representation † ˜ † (t) ˜p (t)a ˜k (t) + g ˜p∗ (t)b gp (t)ak + gp∗ (t)b−k =g −k
CAUTION: Particle interpretation ONLY for t → ±∞ ! ˜†k (t)a ˜k (t)i • Distribution function: f (k, t) = ha • Equation of motion:
eE (t)pk (t) 2ωp2 (t)
Z
t
dt ′
d dt f (k, t)
= S(k, t) in sQED
eE (t ′ )pk (t ′ ) ωp2 (t ′ )
Z t dt ′′ ωp (t ′′ ) [1 + 2f (k, t ′ )] cos 2 t′
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
Quantum Kinetic Theory (Real Time) • Equation of motion: In general not exactly solvable
[∂t2 + ωp2 (t)]gp (t) = 0 • Hamiltonian operator: Off-diagonal • Bogoliubov transformation: Quasi-particle representation † ˜ † (t) ˜p (t)a ˜k (t) + g ˜p∗ (t)b gp (t)ak + gp∗ (t)b−k =g −k
CAUTION: Particle interpretation ONLY for t → ±∞ ! ˜†k (t)a ˜k (t)i • Distribution function: f (k, t) = ha • Equation of motion:
eE (t)ǫ⊥ 2ωp2 (t)
Z
t
dt ′
d dt f (k, t)
= S(k, t) in QED
Z t eE (t ′ )ǫ⊥ ′′ ′′ ′ [1−2f dt ω (t ) (k, t )] cos 2 p ωp2 (t ′ ) t′
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
Quantum Kinetic Theory (Real Time) Quantum kinetic equation (QED) ↔ Integro-differential equation Z t Z ′ eE(t)ǫ⊥ t d ′′ ′′ ′ ′ eE(t )ǫ⊥ dt ωp (t ) dt [1 − 2f(k, t )] cos 2 f(k, t) = dt 2ωp2 (t) −∞ ωp2 (t′ ) t′ • Non-Markovian equation: Statistical factor & Cosine term • Reformulation: First order differential equation system • Backreaction mechanism: E (t) = Eext (t) + Eint (t)
E˙ int (t) = −4e
Z
d3 k (2π)3
2 ˙ pk (t) ωp (t) d eE(t)ǫ ⊥ f(k, t)− f(k, t) + ωp (t) eE(t) dt 8ωp5 (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
R
[dk]f (k, ∞)
!
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Outline
QED Vacuum Schwinger Effect for E (x, t) = E (t) Results Schwinger Effect for E (x, t) Summary
Schwinger Effect for E(x , t)
Summary
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Schwinger Effect for E(x , t)
Results
Electric Field: Pulse-Shaped Time dependent field: E (t) = E sech2 (t/τ ) EHtLE 1.0
0.8
0.6
0.4
0.2
tΤ -4
-2
2
4
Exactly solvable in different approaches!
Summary
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
Electric Field: Pulse-Shaped REMINDER: Instantaneous approximation should be valid for γ≪1 πm2 e2E 2 exp − ninst [e e ] ≃ V T eE 4π 2 + −
REMINDER: Keldysh adiabaticity parameter γ γ=
m eE τ
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?
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
Electric Field: Pulse-Shaped REMINDER: Instantaneous approximation should be valid for γ≪1 ∞
e2 E (t ′ )2 πm2 exp − ninst [e e ] ≃ V dt eE (t ′ ) 4π 2 −∞ + −
Z
′
REMINDER: Keldysh adiabaticity parameter γ γ=
m eE τ
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?
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
Electric Field: Pulse-Shaped REMINDER: Instantaneous approximation should be valid for γ≪1 + −
ninst [e e ] ≃ V
Z
∞
πm2 e2 E (t ′ )2 exp − dt eE (t ′ ) 4π 2 −∞ ′
REMINDER: Keldysh adiabaticity parameter γ γ=
Ecr tc E τ
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?
Schwinger Effect for E(x , t) = E(t)
QED Vacuum
Schwinger Effect for E(x , t)
Results
Summary
Electric Field: Pulse-Shaped FH, R. Alkofer and H. Gies, Phys. Rev. D 78 (2008)
E = 0.1 Ecr
5 ´ 10
τ = 10 tc 10
-5
8
à
à æ
à æ
à æ
à æ
0.9
1
à æ
-5
5 ´ 10
-6
æ
inst.
à
q.k.t.
10
à
1 ´ 10
-6
5 ´ 10
-7
à
à
à æ
à æ
à æ
à æ
à æ
æ à
number density @nm-3 D
number density @nm-3 D
à æ
1 ´ 10
à æ à æ
100
æ
inst.
à
q.k.t.
à æ
0.1
æ æ
5
10
-4
10
-7
à
æ
1 ´ 10
-7
æ
10
20
30
40
50
60 Τ @tc D
70
80
90
100
æ
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
EEcr
• Instantaneous approximation: Huge deviation for γ = 1 • γ & 0.2: ’Overlap region’ → Multi-photon effects set in
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Schwinger Effect for E(x , t)
Results
Summary
Electric Field: Oscillation with Gaussian Envelope Time dependent field: E (t) = E cos(ω t) exp(−t 2 /2τ 2 ) EHtLE 1.0 Ω×Τ = 0 Ω×Τ = 5 Ω×Τ = 10
0.5
tΤ -4
2
-2
4
-0.5
-1.0
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
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
Electric Field: Oscillation with Gaussian Envelope 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.u.D 0
0 x @a.u.D
Crossed laser beams: 2 counter-propagating wave packages
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
Electric Field: Oscillation with Gaussian Envelope 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 1.0
0.5
t @a.uD
-0.5
Interaction region x = 0: Oscillation with Gaussian Envelope
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Summary
Electric Field: Oscillation with Gaussian Envelope V. Popov, JETP Letters 74 (2001)
• WKB / Scattering Theory: Semiclassical treatment
• Gaussian approximation: Production probability for γ ≪ 1
i Ecr 1 2 1 h 2 2 2 γ˜ pk + p⊥ P(p) ∼ exp −π 1 − γ˜ ·exp − E 8 eE with γ˜ =
s
(ωτ )2 + 1 γ (ωτ )2
Accuracy of the Gaussian approximation?
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Schwinger Effect for E(x , t)
Results
Summary
Electric Field: Oscillation with Gaussian Envelope 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
5. ´ 10
-14
4. ´ 10
-14
3. ´ 10
-14
2. ´ 10
-14
1. ´ 10
-14
p° @keVD -400
-200
0
200
400
600
• Width: f (k, ∞) NOT Gaussian → Steeper decay! • Structure: Oscillatory behaviour with ∆pkmax = ω
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Schwinger Effect for E(x , t)
Results
Summary
Electric Field: Oscillation with Gaussian Envelope 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
5. ´ 10
-14
4. ´ 10
-14
3. ´ 10
-14
2. ´ 10
-14
1. ´ 10
-14
p° @keVD -400
-200
0
200
400
600
• Width: f (k, ∞) NOT Gaussian → Steeper decay! • Structure: Oscillatory behaviour with ∆pkmax = ω
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Schwinger Effect for E(x , t)
Results
Electric Field: Additional Phase Shift Time dependent field: E (t) = E cos(ω t + φ) exp(−t 2 /2τ 2 ) EHtLE 1.0 Gaussian Φ=0 Φ = -Π4 0.5
Φ = -Π2
tΤ -4
2
-2
4
-0.5
-1.0
• 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?
Summary
Schwinger Effect for E(x , t) = E(t)
QED Vacuum
Schwinger Effect for E(x , t)
Results
Summary
Electric Field: Additional Phase Shift E = 0.1Ecr , τ = 100 tc , ω = 25 keV −→ ωτ = 5 and γ = 0.5 φ = −π/4
φ = −0 -14
φ = −π/2 5. ´ 10
-14
4. ´ 10
-14
3. ´ 10
-14
-14
-14
2. ´ 10
-14
-14
1. ´ 10
-14
-14
5. ´ 10
5. ´ 10
-14
-14
4. ´ 10
4. ´ 10
-14
3. ´ 10
3. ´ 10
-14
2. ´ 10
2. ´ 10
-14
1. ´ 10
1. ´ 10
p° @keVD -400
-200
0
200
400
600
p° @keVD
p° @keVD -400
-200
0
102
200
400
600
Huge qualitative difference!
-400
-200
0
137
200
400
600
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Electric Field: Additional Phase Shift Explanation: Scattering picture • REMINDER: [∂t2 + ωp2 (t)]gp (t) = 0 • 1-dimensional scattering problem
~2 2 Hψ(x) = − ∂ + V (x) ψ(x) = E ψ(x) 2m x
• Formal similarity → ’Scattering potential’: V (t) ∼ −ωp2 (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’: ωp2 (t) = ωp2 (−t)
Resonances ↔ perfect transmission ↔ No pairs produced!
Summary
Schwinger Effect for E(x , t) = E(t)
QED Vacuum
Schwinger Effect for E(x , t)
Results
Summary
Electric Field: Additional Phase Shift
sQED : QED :
5. ´ 10
REMINDER: Source term for pair production R ′ ′ eE(t)pk (t) R t ′ eE(t )pk (t ) [1+2f(k, t′ )] cos 2 t dt′′ ω (t′′ ) dt ′ p 2 2 −∞ t 2ωp (t) ωp (t′ ) R R ′ eE(t)ǫ⊥ t ′ eE(t )ǫ⊥ [1−2f(k, t′ )] cos 2 t dt′′ ω (t′′ ) dt ′ p t 2ω 2 (t) −∞ ω 2 (t′ ) p
p
-14
5. ´ 10
-14
4. ´ 10
-14
QED
QED
sQED
sQED
4. ´ 10
-14
3. ´ 10
-14
3. ´ 10
-14
2. ´ 10
-14
2. ´ 10
-14
1. ´ 10
-14
1. ´ 10
-14
p° @keVD -400
-200
0
200
400
600
p° @keVD -100
0
100
Effect of particle statistics becomes obvious!
200
300
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Outline
QED Vacuum Schwinger Effect for E (x, t) = E (t) Results Schwinger Effect for E (x, t) Summary
Schwinger Effect for E(x , t)
Summary
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
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)
+ 0 • Cαβ = h0| ψα (x~1 , t), ψ¯β (x~2 , t) |0i = δ3 (x~1 − x~2 )γαβ
− • Cαβ = h0| ψα (x~1 , t), ψ¯β (x~2 , t) |0i
• Wigner transform: Fourier transform w.r.t. ~s = x~1 − x~2 − • DHW function Wαβ (~ x , ~p, t): Wigner transform of Cαβ
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
Generalization of Quantum Kinetic Theory Equation of motion for DHW function • Hartree approximation: Mean electric field ~ =0 • Vanishing magnetic field: B
i i o h n 1 h Dt W αβ = − ∇ γ 0~γ , W − i mγ 0 , W − i γ 0~γ~p, W 2 αβ αβ αβ with
Dt = ∂t + e
Z
1/2
~ ~x + iλ∂p , t)∂p d λE(
−1/2
• Basis set for DHW function: {1, γ5 , γ µ , γ5 γ µ , σ µν } • 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!
Summary
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Outline
QED Vacuum Schwinger Effect for E (x, t) = E (t) Results Schwinger Effect for E (x, t) Summary
Schwinger Effect for E(x , t)
Summary
QED Vacuum
Schwinger Effect for E(x , t) = E(t)
Results
Schwinger Effect for E(x , t)
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(−t 2 /2τ 2 ): • Crossed laser beams: Realistic model at interaction region • Momentum space: Oscillatory structure; non-Gaussian • E (t) = E cos(ωt + φ)exp(−t 2 /2τ 2 ): • 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