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Quantum Reflection as a New Signature of the Quantum Vacuum Nonlinearity Nico Seegert TPI Uni Jena & Helmholtz-Institut Jena February 4th, 2014 Advances in Strong-Field Electrodynamics @ Etvs University, Budapest, Feb.4th 2014 Contents

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SLIDE 1

Quantum Reflection as a New Signature of the Quantum Vacuum Nonlinearity

Nico Seegert

TPI Uni Jena & Helmholtz-Institut Jena

February 4th, 2014

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 2

Contents

1

Properties of the Quantum Vacuum

2

Photon Propagation in Inhomogeneous Fields

3

Quantum Reflection

4

Time-independent, one-dimensional Inhomogeneity

5

Outlook: Time-dependent Inhomogeneities

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 3

Properties of the Quantum Vacuum

Light Propagation in Vacuum

Classical vacuum is empty

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 4

Properties of the Quantum Vacuum

Light Propagation in Vacuum

Classical vacuum is empty Light propagation in vacuum governed by (linear) Maxwell equations ∂µF µν = 0 → superposition principle

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 5

Properties of the Quantum Vacuum

Light Propagation in Vacuum

QED vacuum: Zero-point energy fluctuations

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 6

Properties of the Quantum Vacuum

Light Propagation in Vacuum

Real electromagnetic fields couple to e−e+-loops = ⇒ Nonlinear interactions Fµν

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 7

Properties of the Quantum Vacuum

Heisenberg-Euler Effective Action

First quantitative description: Heisenberg-Euler Lagrangian (1936)

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 8

Properties of the Quantum Vacuum

Heisenberg-Euler Effective Action

First quantitative description: Heisenberg-Euler Lagrangian (1936) 1-loop effective action

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 9

Properties of the Quantum Vacuum

Heisenberg-Euler Effective Action

First quantitative description: Heisenberg-Euler Lagrangian (1936) 1-loop effective action Valid for “locally” constant EM-fields Fµν Scales: λ = 1

m ∼

=

  • 3.9 × 10−13 m

1.3 × 10−21 s

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 10

Properties of the Quantum Vacuum

Heisenberg-Euler Effective Action

First quantitative description: Heisenberg-Euler Lagrangian (1936) 1-loop effective action Valid for “locally” constant EM-fields Fµν Scales: λ = 1

m ∼

=

  • 3.9 × 10−13 m

1.3 × 10−21 s Critical field strengths: Ecr Bcr

  • = m2

e ∼ = 1.3 × 1018V/m 4.4 × 109T

  • Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014
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SLIDE 11

Properties of the Quantum Vacuum

Heisenberg-Euler Effective Action

First quantitative description: Heisenberg-Euler Lagrangian (1936) 1-loop effective action Valid for “locally” constant EM-fields Fµν Scales: λ = 1

m ∼

=

  • 3.9 × 10−13 m

1.3 × 10−21 s Critical field strengths: Ecr Bcr

  • = m2

e ∼ = 1.3 × 1018V/m 4.4 × 109T

  • Splitting of Fµν = F µν + f µν, with F µν ≫ f µν and F µν = const.

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 12

Properties of the Quantum Vacuum

Optical Signatures of the Quantum Vacuum

Nonlinear interactions between probe photons and electromagnetic background field: ∝ (f µν)2 Magnetic birefringence

(Toll’52) Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 13

Properties of the Quantum Vacuum

Optical Signatures of the Quantum Vacuum

Nonlinear interactions between probe photons and electromagnetic background field: ∝ (f µν)2 Magnetic birefringence

(Toll’52)

∝ (f µν)3 Photon splitting (Adler’71)

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 14

Properties of the Quantum Vacuum

Optical Signatures of the Quantum Vacuum

Nonlinear interactions between probe photons and electromagnetic background field: ∝ (f µν)2 Magnetic birefringence

(Toll’52)

∝ (f µν)3 Photon splitting (Adler’71) ∝ (f µν)4 Light-by-light scattering

(Karplus’51) Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 15

Properties of the Quantum Vacuum

Optical Signatures of the Quantum Vacuum

Nonlinear interactions between probe photons and electromagnetic background field: ∝ (f µν)2 Magnetic birefringence

(Toll’52)

∝ (f µν)3 Photon splitting (Adler’71) ∝ (f µν)4 Light-by-light scattering

(Karplus’51)

Pair production

(Sauter’31,HE’35,Schwinger’51) Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 16

Properties of the Quantum Vacuum

Optical Signatures of the Quantum Vacuum

This talk: Introduction of a new optical signature related to photon propagation Quantum vacuum modified by background field = effective potential for probe photons

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 17

Properties of the Quantum Vacuum

Optical Signatures of the Quantum Vacuum

This talk: Introduction of a new optical signature related to photon propagation Quantum vacuum modified by background field = effective potential for probe photons “Quantum Vacuum Reflection” Reflection of probe photons at a strong electromagnetic background field

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 18

Properties of the Quantum Vacuum

Optical Signatures of the Quantum Vacuum

This talk: Introduction of a new optical signature related to photon propagation Quantum vacuum modified by background field = effective potential for probe photons “Quantum Vacuum Reflection” Reflection of probe photons at a strong electromagnetic background field Requires manifestly inhomogeneous background field → energy/momentum transfer k′, ω′ k, ω

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 19

Photon Propagation in Inhomogeneous Fields

Outline

1

Properties of the Quantum Vacuum

2

Photon Propagation in Inhomogeneous Fields

3

Quantum Reflection

4

Time-independent, one-dimensional Inhomogeneity

5

Outlook: Time-dependent Inhomogeneities

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 20

Photon Propagation in Inhomogeneous Fields

Effective Action and Equations of Motion

Generalized effective action for photon propagation in a slowly varying electromagnetic background field:

(cf. Dittrich,Gies’00)

Γ[a] = −1 4

  • d4x FµνFµν
  • a+A

− 1 2

  • d4x d4y aµ(x)Πµν(x, y|A)aν(y)

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 21

Photon Propagation in Inhomogeneous Fields

Effective Action and Equations of Motion

Generalized effective action for photon propagation in a slowly varying electromagnetic background field:

(cf. Dittrich,Gies’00)

Γ[a] = −1 4

  • d4x FµνFµν
  • a+A

− 1 2

  • d4x d4y aµ(x)Πµν(x, y|A)aν(y)

with the “photon polarization tensor” in an electromagnetic background field at

  • ne loop order

Πµν(x, y|A) = ν µ

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 22

Photon Propagation in Inhomogeneous Fields

Effective Action and Equations of Motion

Generalized effective action for photon propagation in a slowly varying electromagnetic background field:

(cf. Dittrich,Gies’00)

Γ[a] = −1 4

  • d4x FµνFµν
  • a+A

− 1 2

  • d4x d4y aµ(x)Πµν(x, y|A)aν(y)

with the “photon polarization tensor” in an electromagnetic background field at

  • ne loop order

Πµν(x, y|A) = ν µ Equations of Motion (k2gµν − kµkν)aν(k) = −

  • d4k′

(2π)4 ˜

Πµν(k, −k′|A)aν(k′)

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 23

Photon Propagation in Inhomogeneous Fields

The Photon Polarization Tensor

Explicit expressions for the photon polarization tensor in momentum space have been obtained at

(Batalin,Shabad’71)

  • ne loop order,

for arbitrary constant electromagnetic fields involving external couplings to all orders.

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 24

Photon Propagation in Inhomogeneous Fields

The Photon Polarization Tensor

Explicit expressions for the photon polarization tensor in momentum space have been obtained at

(Batalin,Shabad’71)

  • ne loop order,

for arbitrary constant electromagnetic fields involving external couplings to all orders. Here: Purely magnetic case Πµν(x, x′|B) = Πµν(x − x′|B) ← → Πµν(k|B)

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 25

Photon Propagation in Inhomogeneous Fields

The Photon Polarization Tensor

Explicit expressions for the photon polarization tensor in momentum space have been obtained at

(Batalin,Shabad’71)

  • ne loop order,

for arbitrary constant electromagnetic fields involving external couplings to all orders. Here: Purely magnetic case Πµν(x, x′|B) = Πµν(x − x′|B) ← → Πµν(k|B) Inclusion of the inhomogeneity by means of Fourier transformations Πµν(k′) (2π)4 δ(4)(k + k′)

F.T.

− − − → Πµν(x − x′)

B→B(x)

− − − − − − → Πµν(x, x′)

F.T.−1

− − − − → Πµν(k, k′)

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 26

Photon Propagation in Inhomogeneous Fields

The Photon Polarization Tensor II

Inclusion of the inhomogeneity: Perturbative regime eB/m2 ≪ 1 Πµν(k′|B) =

∞

  • n=0

Πµν

(2n)(k′) (eB)2n

= + + + O

  • (eB)6

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 27

Photon Propagation in Inhomogeneous Fields

The Photon Polarization Tensor II

Inclusion of the inhomogeneity: Perturbative regime eB/m2 ≪ 1 Πµν(k′|B) =

∞

  • n=0

Πµν

(2n)(k′) (eB)2n

= + + + O

  • (eB)6

Procedure: B → B(x) Πµν(k, k′|B) =

∞

  • n=0

Πµν

(2n)(k′)

  • d4x e−i(k+k′)x(eB(x))2n

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 28

Photon Propagation in Inhomogeneous Fields

The Photon Polarization Tensor II

Inclusion of the inhomogeneity: Perturbative regime eB/m2 ≪ 1 Πµν(k′|B) =

∞

  • n=0

Πµν

(2n)(k′) (eB)2n

= + + + O

  • (eB)6

Procedure: B → B(x) Πµν(k, k′|B) =

∞

  • n=0

Πµν

(2n)(k′)

  • d4x e−i(k+k′)x(eB(x))2n

Equations of Motion (k2gµν − kµkν)aν(k) = −

  • d4k′

(2π)4 ˜

Πµν(k, −k′|B)aν(k′)

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 29

Photon Propagation in Inhomogeneous Fields

The Photon Polarization Tensor II

Inclusion of the inhomogeneity: Perturbative regime eB/m2 ≪ 1 Πµν(k′|B) =

∞

  • n=0

Πµν

(2n)(k′) (eB)2n

= + + + O

  • (eB)6

Symmetrization: ˜ Πµν(k, k′|B) = 1 2

∞

  • n=0
  • Πµν

(2n)(k′) + Πµν (2n)(k)

d4x e−i(k+k′)x(eB(x))2n Equations of Motion (k2gµν − kµkν)aν(k) = −

  • d4k′

(2π)4 ˜

Πµν(k, −k′|B)aν(k′)

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 30

Quantum Reflection

Outline

1

Properties of the Quantum Vacuum

2

Photon Propagation in Inhomogeneous Fields

3

Quantum Reflection

4

Time-independent, one-dimensional Inhomogeneity

5

Outlook: Time-dependent Inhomogeneities

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 31

Quantum Reflection

Atomic Quantum Reflection (AQR)

Zhao et al., Science Magazin, 18.Feb 2011

Repulsion of atoms from an attractive potential - “Above-barrier-scattering”

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 32

Quantum Reflection

Atomic Quantum Reflection (AQR)

Zhao et al., Science Magazin, 18.Feb 2011

Repulsion of atoms from an attractive potential - “Above-barrier-scattering” Very sensitive means to study long-range surface forces: Van der Waals-force, Casimir-force

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 33

Quantum Reflection

Atomic Quantum Reflection (AQR)

Zhao et al., Science Magazin, 18.Feb 2011

Repulsion of atoms from an attractive potential - “Above-barrier-scattering” Very sensitive means to study long-range surface forces: Van der Waals-force, Casimir-force Shining of probe atoms on the surface at grazing incidence: Classical reflection + Quantum reflection

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 34

Quantum Reflection

Optical Quantum Reflection

Zhao et al., Science Magazin, 18.Feb 2011

Pump

P r

  • b

e

Detector

Transmission Reflection

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 35

Time-independent, one-dimensional Inhomogeneity

Outline

1

Properties of the Quantum Vacuum

2

Photon Propagation in Inhomogeneous Fields

3

Quantum Reflection

4

Time-independent, one-dimensional Inhomogeneity

5

Outlook: Time-dependent Inhomogeneities

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 36

Time-independent, one-dimensional Inhomogeneity

Optical Quantum Reflection: Static, One-dimensional Case

Goal: Analytical insights into Optical Quantum Reflection

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 37

Time-independent, one-dimensional Inhomogeneity

Optical Quantum Reflection: Static, One-dimensional Case

Goal: Analytical insights into Optical Quantum Reflection Simplifications: Purely magnetic background field B Static case One-dimensional inhomogeneity in the x-direction

β x y ky k′

x

Transmission Reflection Field Inhomogeneity B(x) k′ k ky kx k′ Gies,Karbstein,Seegert, NJP, Aug. 2013 Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 38

Time-independent, one-dimensional Inhomogeneity

Optical Quantum Reflection: Static, One-dimensional Case

Goal: Analytical insights into Optical Quantum Reflection Simplifications: Purely magnetic background field B Static case One-dimensional inhomogeneity in the x-direction

β x y ky k′

x

Transmission Reflection Field Inhomogeneity B(x) k′ k ky kx k′ Gies,Karbstein,Seegert, NJP, Aug. 2013

Furthermore: Perturbative treatment in eB/m2 = B/Bcr ≪ 1 Bcr = m2 e ≈ 4 × 109T High-intensity lasers (Petawatt-regime): B = O

  • 106T − 107T
  • Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014
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SLIDE 39

Time-independent, one-dimensional Inhomogeneity

Basic Setup

EoM: Complicated tensor structure B(x) = B(x)eB sets global reference direction Scalar EoM for specific modes p =, ⊥:

β x y ky k′

x

Transmission Reflection Field Inhomogeneity B(x) k′ k ky kx k′

  • a

a⊥

  • = Polarization modes
  • parallel

perpendicular

  • to plane spanned by k′ and B

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 40

Time-independent, one-dimensional Inhomogeneity

Basic Setup

EoM: Complicated tensor structure B(x) = B(x)eB sets global reference direction Scalar EoM for specific modes p =, ⊥:

β x y ky k′

x

Transmission Reflection Field Inhomogeneity B(x) k′ k ky kx k′

  • a

a⊥

  • = Polarization modes
  • parallel

perpendicular

  • to plane spanned by k′ and B
  • k2

x − ˜

ω2 ap(ω, kx, ky) = − dk′

x

2π ˜ Πp(kx, −k′

x, ky|B)ap(ω, k′ x, ky)

with “reduced” frequency ˜ ω2 = ω2 − k2

y = ω2 cos2 β

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 41

Time-independent, one-dimensional Inhomogeneity

Basic Setup

EoM: Complicated tensor structure B(x) = B(x)eB sets global reference direction Scalar EoM for specific modes p =, ⊥:

β x y ky k′

x

Transmission Reflection Field Inhomogeneity B(x) k′ k ky kx k′

  • a

a⊥

  • = Polarization modes
  • parallel

perpendicular

  • to plane spanned by k′ and B
  • k2

x − ˜

ω2 ap,ind(ω, kx, ky) = − dk′

x

2π ˜ Πp(kx, −k′

x, ky|B)ap,in(ω, k′ x, ky)

“induced” ր “incoming” ր

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 42

Time-independent, one-dimensional Inhomogeneity

Reflection Coefficient

Incoming photons: plane wave of fixed, arbitrary frequency ω and angle β

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 43

Time-independent, one-dimensional Inhomogeneity

Reflection Coefficient

Incoming photons: plane wave of fixed, arbitrary frequency ω and angle β Reflection coefficient: Ratio of amplitudes at asymptotic distances Rp =

  • ˜

Πp(−˜ ω, −˜ ω, ky|B) 2˜ ω

  • 2

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

slide-44
SLIDE 44

Time-independent, one-dimensional Inhomogeneity

Reflection Coefficient

Incoming photons: plane wave of fixed, arbitrary frequency ω and angle β Reflection coefficient: Ratio of amplitudes at asymptotic distances Rp =

  • ˜

Πp(−˜ ω, −˜ ω, ky|B) 2˜ ω

  • 2

Resulting Reflection coefficient Rp =

  • cp

π ˜ ω

  • dx ei2˜

ωx

eB(x) m2 2

  • 2

+ O

  • ( eB

m2 )6

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

slide-45
SLIDE 45

Time-independent, one-dimensional Inhomogeneity

Reflection Coefficient

Incoming photons: plane wave of fixed, arbitrary frequency ω and angle β Reflection coefficient: Ratio of amplitudes at asymptotic distances Rp =

  • ˜

Πp(−˜ ω, −˜ ω, ky|B) 2˜ ω

  • 2

Resulting Reflection coefficient Rp =

  • cp

π ˜ ω

  • dx ei2˜

ωx

eB(x) m2 2

  • 2

+ O

  • ( eB

m2 )6

cp =

α 180 sin2 θ+sin2 θ′ cos2 β

7 4

  • θ

B = BeB k θ′(β) =

  • k′, B
  • θ(β) =
  • k, B
  • Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014
slide-46
SLIDE 46

Time-independent, one-dimensional Inhomogeneity

Quantum Mechanical Analogy

EoM can be cast into Schrödinger-type equation

  • − d2

dx2 + V (x)

  • ap(x, ky, ω) = ˜

ω2 ap(x, ky, ω) V (x) = −2 cp

π ˜

ω2

eB(x) m2

2 x eikxx re−ikxx teikxx

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

slide-47
SLIDE 47

Time-independent, one-dimensional Inhomogeneity

Quantum Mechanical Analogy

EoM can be cast into Schrödinger-type equation

  • − d2

dx2 + V (x)

  • ap(x, ky, ω) = ˜

ω2 ap(x, ky, ω) V (x) = −2 cp

π ˜

ω2

eB(x) m2

2 x eikxx re−ikxx teikxx "Above-barrier scattering"

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

slide-48
SLIDE 48

Time-independent, one-dimensional Inhomogeneity

Gaussian beam profile

Reflection coefficient Rp =

  • cp

π ˜ ω

  • dx ei2˜

ωx

eB(x) m2 2

  • 2

+ O

  • ( eB

m2 )6

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

slide-49
SLIDE 49

Time-independent, one-dimensional Inhomogeneity

Gaussian beam profile

Reflection coefficient Rp =

  • cp

π ˜ ω

  • dx ei2˜

ωx

eB(x) m2 2

  • 2

+ O

  • ( eB

m2 )6

Example 1: Gaussian beam profile characterized by width 2w0 B(x) = B e

−

  • x

w0

2

− → Rp =

  • cp

√ 2π eB m2 2 ˜ ωw0e− 1

2 (˜

ωw0)2

  • 2

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

slide-50
SLIDE 50

Time-independent, one-dimensional Inhomogeneity

Gaussian beam profile

Reflection coefficient Rp =

  • cp

π ˜ ω

  • dx ei2˜

ωx

eB(x) m2 2

  • 2

+ O

  • ( eB

m2 )6

Example 1: Gaussian beam profile characterized by width 2w0 B(x) = B e

−

  • x

w0

2

− → Rp =

  • cp

√ 2π eB m2 2 ˜ ωw0e− 1

2 (˜

ωw0)2

  • 2

Exponential suppression analogous to AQR: Rp ∝ e−w0ω cos β → β provides handle to overcome suppression

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

slide-51
SLIDE 51

Time-independent, one-dimensional Inhomogeneity

Gaussian beam profile

Numerical values for high-intensity laser facilities to be available in Jena: Design Parameters POLARIS: Pulse energy E = 150J, Duration τ = 150fs, Wavelength λ = 1035nm JETI200: Pulse energy E = 4J, Duration τ = 20fs, Wavelength λ = 800nm

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

slide-52
SLIDE 52

Time-independent, one-dimensional Inhomogeneity

Gaussian beam profile

Numerical values for high-intensity laser facilities to be available in Jena: Design Parameters POLARIS: Pulse energy E = 150J, Duration τ = 150fs, Wavelength λ = 1035nm JETI200: Pulse energy E = 4J, Duration τ = 20fs, Wavelength λ = 800nm Pump: POLARIS , Probe: JETI200

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

slide-53
SLIDE 53

Time-independent, one-dimensional Inhomogeneity

Gaussian beam profile

Numerical values for high-intensity laser facilities to be available in Jena: Design Parameters POLARIS: Pulse energy E = 150J, Duration τ = 150fs, Wavelength λ = 1035nm JETI200: Pulse energy E = 4J, Duration τ = 20fs, Wavelength λ = 800nm Pump: POLARIS , Probe: JETI200 Here: β = 83° R R⊥

  • ≈

2.5 0.8

  • · 10−19

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

slide-54
SLIDE 54

Time-independent, one-dimensional Inhomogeneity

Gaussian beam profile

Numerical values for high-intensity laser facilities to be available in Jena: Design Parameters POLARIS: Pulse energy E = 150J, Duration τ = 150fs, Wavelength λ = 1035nm JETI200: Pulse energy E = 4J, Duration τ = 20fs, Wavelength λ = 800nm Pump: POLARIS , Probe: JETI200 Here: β = 83° R R⊥

  • ≈

2.5 0.8

  • · 10−19

Number of reflected photons per shot Np ≈ Rp Nin ≈ Rp · 1.6 × 1019 Ph./shot

  • N

N⊥

  • ≈
  • 4.1

1.3

  • Photons per shot

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

slide-55
SLIDE 55

Time-independent, one-dimensional Inhomogeneity

Crossed Fields Configuration

Example 2: 2w0 λm e1 e1,⊥ e2 e2,⊥ B1(x, y) B2(x, y) δ ain(x) y x Superposition of two identical beams under angle δ = ⇒ Modulated inhomogeneity along x-axis with frequency ωm cos δ = cos δ 2π/λm

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

slide-56
SLIDE 56

Time-independent, one-dimensional Inhomogeneity

Crossed Fields Configuration II

Resulting magnetic background field: B(x, y = 0) = B e

−

  • x

w0 sin δ

2

cos(ωm cos δ x)

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 57

Time-independent, one-dimensional Inhomogeneity

Crossed Fields Configuration II

Resulting magnetic background field: B(x, y = 0) = B e

−

  • x

w0 sin δ

2

cos(ωm cos δ x) Reflection coefficient: Rp ≈

  • cp

√ 32π eB m2 2 ωw0 sin δ e− 1

2( w0 sin δ) 2(ω−ωm cos δ)2

  • 2

Overcome exponential suppression by matching ω = ωm cos δ (for ω ≤ ωm)

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 58

Time-independent, one-dimensional Inhomogeneity

Crossed Fields Configuration II

Resulting magnetic background field: B(x, y = 0) = B e

−

  • x

w0 sin δ

2

cos(ωm cos δ x) Reflection coefficient: Rp ≈

  • cp

√ 32π eB m2 2 ωw0 sin δ e− 1

2( w0 sin δ) 2(ω−ωm cos δ)2

  • 2

Overcome exponential suppression by matching ω = ωm cos δ (for ω ≤ ωm) Background beam: POLARIS (SHG) Probe beam: JETI200 → δmax ≈ 48.5° R ≈ 0.5 · 10−20 N ≈ 0.1 Photons per shot

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 59

Outlook: Time-dependent Inhomogeneities

Outline

1

Properties of the Quantum Vacuum

2

Photon Propagation in Inhomogeneous Fields

3

Quantum Reflection

4

Time-independent, one-dimensional Inhomogeneity

5

Outlook: Time-dependent Inhomogeneities

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 60

Outlook: Time-dependent Inhomogeneities

Optical Quantum Reflection: Time-dependent, 1-dim-Case

Real lasers fields are time-dependent

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 61

Outlook: Time-dependent Inhomogeneities

Optical Quantum Reflection: Time-dependent, 1-dim-Case

Real lasers fields are time-dependent Similar treatment for time-dependent, one-dimensional inhomogeneities B(x, t)

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 62

Outlook: Time-dependent Inhomogeneities

Optical Quantum Reflection: Time-dependent, 1-dim-Case

Real lasers fields are time-dependent Similar treatment for time-dependent, one-dimensional inhomogeneities B(x, t) Scalar equation of motion only for ⊥ component

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 63

Outlook: Time-dependent Inhomogeneities

Optical Quantum Reflection: Time-dependent, 1-dim-Case

Real lasers fields are time-dependent Similar treatment for time-dependent, one-dimensional inhomogeneities B(x, t) Scalar equation of motion only for ⊥ component First Investigation: B(x, t) = B(x) cos(ωmt)

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 64

Outlook: Time-dependent Inhomogeneities

Optical Quantum Reflection: Time-dependent, 1-dim-Case

Real lasers fields are time-dependent Similar treatment for time-dependent, one-dimensional inhomogeneities B(x, t) Scalar equation of motion only for ⊥ component First Investigation: B(x, t) = B(x) cos(ωmt) = ⇒ Frequency-mixing Sum-frequency (“+”): ω → ω + 2ωm Difference-frequency (“−”): ω → ω − 2ωm No energy exchange: ω → ω

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 65

Outlook: Time-dependent Inhomogeneities

Optical Quantum Reflection: Time-dependent, 1-dim-Case

ω − 2ωm ω − 2ωm ω + 2ωm ω + 2ωm ω ω ω kx −kx kx −k− k− −k+ k+ ky ky ky ky ky ky ky

B(x, t) = B(x) cos(ωmt)

y x

Reflection Transmission Field Inhomogeneity

Incoming beam β Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 66

Outlook: Time-dependent Inhomogeneities

Optical Quantum Reflection: Time-dependent, 1-dim-Case

Reflection coefficient (R⊥)± =

  • α

90π k2

y

k±

  • dx eix(kx+k±)

eB(x) m2 2

  • 2

with k± :=

  • (ω ± 2ωm)2 − k2

y

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 67

Outlook: Time-dependent Inhomogeneities

Optical Quantum Reflection: Time-dependent, 1-dim-Case

Reflection coefficient (R⊥)± =

  • α

90π k2

y

k±

  • dx eix(kx+k±)

eB(x) m2 2

  • 2

with k± :=

  • (ω ± 2ωm)2 − k2

y

Effects of the same order of magnitude as in static case

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 68

Outlook: Time-dependent Inhomogeneities

Optical Quantum Reflection: Time-dependent, 1-dim-Case

Reflection coefficient (R⊥)± =

  • α

90π k2

y

k±

  • dx eix(kx+k±)

eB(x) m2 2

  • 2

with k± :=

  • (ω ± 2ωm)2 − k2

y

Effects of the same order of magnitude as in static case Similar procedure works for polarization for two-dimensional, static inhomogeneities B(x, y)

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 69

Outlook: Time-dependent Inhomogeneities

Optical Quantum Reflection: Time-dependent, 1-dim-Case

Reflection coefficient (R⊥)± =

  • α

90π k2

y

k±

  • dx eix(kx+k±)

eB(x) m2 2

  • 2

with k± :=

  • (ω ± 2ωm)2 − k2

y

Effects of the same order of magnitude as in static case Similar procedure works for polarization for two-dimensional, static inhomogeneities B(x, y) These three scenarios are the only ones with configurations without mode mixing

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 70

Outlook: Time-dependent Inhomogeneities

Summary

Photons in strong inhomogeneous magnetic background fields experience “quantum reflection”

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 71

Outlook: Time-dependent Inhomogeneities

Summary

Photons in strong inhomogeneous magnetic background fields experience “quantum reflection” First estimates: Promising candidate to probe the quantum vacuum nonlinearity with current laser technology due to clear signal/background separation

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 72

Outlook: Time-dependent Inhomogeneities

Summary

Photons in strong inhomogeneous magnetic background fields experience “quantum reflection” First estimates: Promising candidate to probe the quantum vacuum nonlinearity with current laser technology due to clear signal/background separation Effect ∝ field strength ratio (eB/m2)4 + exponential suppression in ω, which can partly be overcome by suitable choice of different beam profiles and incidence angles β

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 73

Outlook: Time-dependent Inhomogeneities

Summary

Photons in strong inhomogeneous magnetic background fields experience “quantum reflection” First estimates: Promising candidate to probe the quantum vacuum nonlinearity with current laser technology due to clear signal/background separation Effect ∝ field strength ratio (eB/m2)4 + exponential suppression in ω, which can partly be overcome by suitable choice of different beam profiles and incidence angles β Ultimate goal: Study photon propagation for general inhomogeneous magnetic background fields B(x, t) ⇒ polarization-mode mixing between ⊥ and components

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 74

Outlook: Time-dependent Inhomogeneities

Summary

Photons in strong inhomogeneous magnetic background fields experience “quantum reflection” First estimates: Promising candidate to probe the quantum vacuum nonlinearity with current laser technology due to clear signal/background separation Effect ∝ field strength ratio (eB/m2)4 + exponential suppression in ω, which can partly be overcome by suitable choice of different beam profiles and incidence angles β Ultimate goal: Study photon propagation for general inhomogeneous magnetic background fields B(x, t) ⇒ polarization-mode mixing between ⊥ and components

Köszönöm a figyelmet!

see also H.Gies,F.Karbstein,N.Seegert; NJP 15 (2013) 083002 Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 75

Appendix

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 76

The Photon Polarization Tensor

The photon polarization tensor for a constant, magnetic field B is given by Πµν(k|B) = P µν

0 Π0(k|B) + P µν Π(k|B) + P µν ⊥ Π⊥(k|B)

with    Π0 Π Π⊥    = α 2π ∞ ds s 1

−1

dν 2  e−is ˜

φ0

   k2 N0 N0k2

⊥ + N1k2

  • N2k2

⊥ + N0k2

  − e−im2sk2(1 − ν2)   where N0 = cos νz − ν sin νz cot z, N1 = (1 − ν2) cos z, N2 = 2cos νz − cos z sin2 z and ˜ φ0 = m2 + 1 − ν2 4 k2

+ 1

2 cos νz − cos z z sin z k2

⊥.

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 77

The Photon Polarization Tensor II

Weak field expansion eB/m2 ≪ 1 Πµν(k|B) = Πµν

(0)(k) + Πµν (2)(k) (eB)2 + O

  • (eB)4

Zero-th order: Πp,(0)(k) =

  • k22 α

4π 1 dν ν2 3 − 1 ν2 φ0 Second order: Πp,(2)(k) = − α 12π 1 dν (1 − ν2)2 φ2      1 −

2 1−ν2

1    k2

+

     1 1

5−ν2 2(1−ν2)

   − k2(1 − ν2) 4φ0   k2

⊥

  with φ0 = m2 − iǫ + 1−ν2

4

k2 and p = (0, , ⊥)

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 78

Decomposition

Constant magnetic field sets global reference direction: Decomposition of the photon momentum vector kµ θ B = BeB k⊥ = k − k k = (keB)eB k kµ

= (ω, k)

kµ

⊥ = (0, k⊥)

Projectors span the transversal eigenspace: P µν

  • = gµν
  • −

kµ

kν

  • k2
  • ,

P µν

⊥ = gµν ⊥ − kµ ⊥kν ⊥

k2

⊥

, P µν = gµν − kµkν k2 − P µν

  • − P µν

⊥

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 79

Photon Polarization Modes

Parallel Case: (∇B) · eB = 0 → P µν

(k) = P µν (k′)

B = Byey + Bzez and for By = 0 : θ = θ′ = θ(β) Perpendicular Case: k′

⊥ · (∇B) = 0

→ P µν

⊥ (k) = P µν ⊥ (k′)

For fixed angle β, magnetic field must fulfill

  • B2

y + B2 z

  • = tan β BxBy

→ β → 0 : By, Bz = 0

β 2 4 x 0.5 1.0 1.5 z y 1 2 3 k′ B

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 80

Quantum Mechanical Reflection Coefficient

QM reflection and transmission in one dimension for arbitrary smooth potential: tn+1eikn+1x rn+1e−ikn+1x tneiknx rne−iknx Vn+1 Vn xn+1 xn+2 ǫ Limit: kn → k(x) Reflection: R =

  • ∞

−∞ dx e2ikx k′ 2k

1 + ∞

−∞ dx

k′

2k + ixk′

  • 2

Transmission: T =

  • 1

1 + ∞

−∞ dx

k′

2k + ixk′

  • 2

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 81

Laser Parameters: POLARIS and JETI200

Setup (a) POLARIS: Background beam, JETI200: Probe beam B = 2.86 · 108eV2, w0 = 5.25eV−1, ω = 1.55eV, Nin = 1, 61 · 1019 Photons per shot, fint = 1, Setup (b) POLARIS: Probe beam, JETI200: Background beam B = 1.65 · 108eV2, w0 = 4.06eV−1, ω = 1.20eV, Nin = 7.8 · 1020 Photons per shot, fint = 0.13 .

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014

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SLIDE 82

Optical Quantum Reflection: Static, two-dimensional Case

y x k′ k β φ B(x, y) "Scattered" "Incoming" |k| = |k′| Differential cross section dσ dφ(β, φ, ω) = 1 2πω

  • 7

45 αω2 π

  • dx dy ei(k′−k)x

eB m2 (x, y) 2

  • 2

Advances in Strong-Field Electrodynamics @ Eötvös University, Budapest, Feb.4th 2014