Nonlinear quantum electrodynamics in strong electromagnetic elds. Felix Mackenroth. Graz, June 18 th, 2015

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1 Nonlinear quantum electrodynamics in strong electromagnetic elds Felix Mackenroth Graz, June 18 th, 2015

2 Introduction Field quantisation in strong external elds Fundamentals of strong- eld QED Refractive e ects Backup Motivation Strong electromagnetic elds are ubiquitous... Felix Mackenroth Nonlinear QED in strong electromagnetic elds 2 / 33

3 Introduction Field quantisation in strong external elds Fundamentals of strong- eld QED Refractive e ects Backup Motivation Strong electromagnetic elds are ubiquitous serve numerous technical purposes... Felix Mackenroth Nonlinear QED in strong electromagnetic elds 2 / 33

4 Introduction Field quantisation in strong external elds Fundamentals of strong- eld QED Refractive e ects Backup Motivation Strong electromagnetic elds are ubiquitous serve numerous technical purposes... Felix Mackenroth Nonlinear QED in strong electromagnetic elds 2 / 33

5 Introduction Field quantisation in strong external elds Fundamentals of strong- eld QED Refractive e ects Backup Motivation Strong electromagnetic elds are ubiquitous serve numerous technical purposes... Felix Mackenroth Nonlinear QED in strong electromagnetic elds 2 / 33

$Introduction Field quantisation in strong external elds Fundamentals of strong- eld QED Refractive e ects Backup Motivation Strong electromagnetic elds are$

6 Introduction Field quantisation in strong external elds Fundamentals of strong- eld QED Refractive e ects Backup Motivation Strong electromagnetic elds are ubiquitous serve numerous technical purposes and can even alter the vacuum Felix Mackenroth Nonlinear QED in strong electromagnetic elds 2 / 33

$Introduction Field quantisation in strong external elds Fundamentals of strong- eld QED Refractive e ects Backup Motivation Strong electromagnetic elds are$

7 Introduction Field quantisation in strong external elds Fundamentals of strong- eld QED Refractive e ects Backup Motivation Strong electromagnetic elds are ubiquitous serve numerous technical purposes and can even alter the vacuum Felix Mackenroth Nonlinear QED in strong electromagnetic elds 2 / 33

8 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 3 / 33 Outline 1 Introduction Motivation Scattering theory Quantum electrodynamics 2 Field quantisation in strong external elds Estimating the interaction strength The Furry picture of quantum dynamics Exact solutions 3 Fundamentals of strong-eld QED Monochromatic elds Quantum cascades Critical eld of QED 4 Refractive eects Vacuum birefringence

9 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 4 / 33 Introduction

10 Motivation Motivation II - The Klein paradox Strong electromagnetic elds aect the basic theory Scattering of a Dirac particle at a potential step (Klein (1929)) V (x) = V 0 Θ(x) E(x) = V 0 (x) "II = "I + V 0 Transmission of particles (innite electric eld) Felix Mackenroth Nonlinear QED in strong electromagnetic elds 5 / 33

11 Motivation Motivation II - The Klein paradox Strong electromagnetic elds aect the basic theory Scattering of a Dirac particle at a potential step (Klein (1929)) V (x) = V 0 Θ(x) E(x) = V 0 (x) "II = "I + V 0 Transmission of particles T (V 0! 1)! = 0 (innite electric eld) Felix Mackenroth Nonlinear QED in strong electromagnetic elds 5 / 33

12 Motivation Motivation II - The Klein paradox Strong electromagnetic elds aect the basic theory Scattering of a Dirac particle at a potential step (Klein (1929)) V (x) = V 0 Θ(x) E(x) = V 0 (x) "II = "I + V 0 Transmission of particles T (V 0! 1) = 2p " + p (innite electric eld) Felix Mackenroth Nonlinear QED in strong electromagnetic elds 5 / 33

Motivation Motivation II - The Klein paradox Strong electromagnetic elds aect the basic theory Scattering of a Dirac particle at a potential step (Klein (1929)) V (x) = V 0 Θ(x)

13 Motivation Motivation II - The Klein paradox Strong electromagnetic elds aect the basic theory Scattering of a Dirac particle at a potential step (Klein (1929)) V (x) = V 0 Θ(x) E(x) = V 0 (x) "II = "I + V 0 (innite electric eld) Transmission of particles T (V 0! 1) = 2p " + p 83% p = m; v 0:7c Felix Mackenroth Nonlinear QED in strong electromagnetic elds 5 / 33

14 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 5 / 33 Motivation Motivation II - The Klein paradox Strong electromagnetic elds aect the basic theory Scattering of a Dirac particle at a potential step (Klein (1929)) V (x) / x E(x) = const "II = "I + V 0 (nite electric eld) Calculation at a nite potential gradient (Sauter 1931) Transmission only for E & E cr = m2 e c 3 ~ jej

15 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 6 / 33 Scattering theory Scattering theory - I Need to obtain i state functions jdi jci Scattering probability P / hf j Ŝ jii time evolution (scattering operator) jbi jai f 2

16 Quantum electrodynamics Quantum electrodynamics Quantum elementary particles theory: Felix Mackenroth Nonlinear QED in strong electromagnetic elds 7 / 33

17 Quantum electrodynamics Quantum electrodynamics Quantum eld theory: elementary particles relativistic energy-mass conversion Felix Mackenroth Nonlinear QED in strong electromagnetic elds 7 / 33

18 Quantum electrodynamics Quantum electrodynamics Quantum eld theory: elementary particles relativistic energy-mass conversion connect spinor (Dirac) elds Ψ ;` and vector (photon) eld A Felix Mackenroth Nonlinear QED in strong electromagnetic elds 7 / 33

19 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 7 / 33 Quantum electrodynamics Quantum electrodynamics Quantum eld theory: elementary particles relativistic energy-mass conversion connect spinor (Dirac) elds Ψ ;` and vector (photon) eld A Notation and conventions: ~ = c = 1 4" 0 = 1 a b =: (ab) 6a =: a electrons: p = "; p ; mass: m e base unit: energy [ev]

20 Quantum electrodynamics Quantum electrodynamics - Field theory Quantum electrodynamics Lagrangian L = X` Ψ ;`(x)(i 6D D + iea (x) m)ψ ;`(x) A (x)@ A (x) 8 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 8 / 33

21 Quantum electrodynamics Quantum electrodynamics - Field theory Quantum electrodynamics Lagrangian L = X` Ψ ;`(x)(i 6D m)ψ ;`(x) A (x)@ A (x) 8 D + iea (x) Conjugate Ψ(x) (@tψ(x)) = iψy (x) A (@ta (x)) = (x) 4 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 8 / 33

22 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 8 / 33 Quantum electrodynamics Quantum electrodynamics - Field theory Quantum electrodynamics Lagrangian L = X` Ψ ;`(x)(i 6D D + iea (x) Conjugate momenta: Hamiltonian H(x) = Z m)ψ ;`(x) A (x)@ A (x) Ψ(x) (@tψ(x)) = iψy (x) A (@ta (x)) = (x) 4 dx Ψ(x)@tΨ(x) + A (x)@ ta (x) L Ψ(x); Ψ(x); A (x) = H Dirac (x) + H photon (x) +H int (x) {z } =:H 0(t)

23 Quantum electrodynamics Quantum electrodynamics - Quantisation Field quantisation by (anti-)commutation relations Ψp (x); Ψ q (y) + = i pq (x y) [A (x); A(y)] = ig (x y) Felix Mackenroth Nonlinear QED in strong electromagnetic elds 9 / 33

24 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 9 / 33 Quantum electrodynamics Quantum electrodynamics - Quantisation Field quantisation by (anti-)commutation relations Ψp (x); Ψ q (y) + = i pq (x y) [A (x); A(y)] = ig (x y) Mode decomposition Ψ (x) = A (x) = Z Z dp c p Ψ p (x) + dpψ y p (x) h dk a k A k (x) + ay k A k (x) i

25 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 9 / 33 Quantum electrodynamics Quantum electrodynamics - Quantisation Field quantisation by (anti-)commutation relations Ψp (x); Ψ q (y) + = i pq (x y) [A (x); A(y)] = ig (x y) Occupation number representation (non-interacting) Vacuum state jn q i = c y q n j0i Mode decomposition Ψ (x) = A (x) = Z Z c q j0i = 0 dp c p Ψ p (x) + dpψ y p (x) h dk a k A k (x) + ay k A k (x) i

26 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 10 / 33 Quantum electrodynamics Quantum electrodynamics - Interaction picture Recall Hamiltonian H(x) = H Dirac (x) + H photon (x) +H int (x) H int (x) := e Unitary transformation O int (t) = ˆT exp jψ(x); A(x); ti int = exp {z } =:H Z 0(t) i i Z t Z t dxj Dirac (x)a (x) t 0 d H 0 () t 0 d H 0 () O(t 0 ) ˆT exp jψ(x); A(x); t 0 i i Z t t 0 d H 0 ()

27 Quantum electrodynamics Quantum electrodynamics - Interaction picture Equations of motion Operators d dt O(t) = [O; H 0] Quantum states i d dt jψ(x); A(x); ti = H int(t) jψ(x); A(x); ti Felix Mackenroth Nonlinear QED in strong electromagnetic elds 11 / 33

28 Quantum electrodynamics Quantum electrodynamics - Interaction picture Equations of motion Operators Quantum states d dt O(t) = [O; H 0] (i m e ) Ψ(x) = 0 free solutions A = 0 i d dt jψ(x); A(x); ti = H int(t) jψ(x); A(x); ti Felix Mackenroth Nonlinear QED in strong electromagnetic elds 11 / 33

29 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 11 / 33 Quantum electrodynamics Quantum electrodynamics - Interaction picture Equations of motion Operators Quantum states d dt O(t) = [O; H 0] (i m e ) Ψ(x) = 0 free solutions A = 0 i d dt jψ(x); A(x); ti = H int(t) jψ(x); A(x); ti Integrate formally jψ(x); A(x); ti = ˆT exp i Z t d H int () jψ(x); A(x); t 0 i t 0

30 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 12 / 33 Quantum electrodynamics Scattering theory - II We now have at hand Initial states (non-interacting) pi ;j; k i ;l y = c j ay l j0i Time evolution jψ(x); A(x); ti = ˆT exp i Z t d H int () jψ(x); A(x); t 0 i t 0

31 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 12 / 33 Quantum electrodynamics Scattering theory - II We now have at hand Initial states (non-interacting) pi ;j; k i ;l y = c j ay l j0i Time evolution jψ(x); A(x); ti = ˆT exp i Z t t 0 d H int () LSZ reduction formula (properly normalised) Z S = p f ;j; k f ;l ie exp d 4 xj (x)a (x) p i ;j; k i ;l jψ(x); A(x); t 0 i

32 = Felix Mackenroth Nonlinear QED in strong electromagnetic elds 12 / 33 Quantum electrodynamics Scattering theory - II We now have at hand Initial states (non-interacting) pi ;j; k i ;l y = c j ay l j0i Time evolution jψ(x); A(x); ti = ˆT exp i Z t t 0 d H int () LSZ reduction formula (properly normalised) Z S = p f ;j; k f ;l ie exp ) Perturbative expansion S = * p f ;j; k f ;l X 1 + 1! d 4 xj (x)a (x) p i ;j; k i ;l Z ie jψ(x); A(x); t 0 i d 4 xj (x)a (x) + O e +1 p i ;j; k i ;l +

33 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 13 / 33 Quantisation in a background eld

34 Estimating the interaction strength Interaction strength Perturbative approach relies on e 1 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 14 / 33

35 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 14 / 33 Estimating the interaction strength Interaction strength Perturbative approach relies on Z d 4 x Estimates (unquantised) j (x)a (x)! e 1 j = Ψ(x) Ψ(x) = 1 (1; V v) one particle per unit volume A (x) = A (x) four potential r min C = 1 m e Compton wavelength t QED = C c = 1 m e QED coherence time

36 Estimating the interaction strength Interaction strength Perturbative approach relies on Z d 4 x Estimates (unquantised) j (x)a (x)! e 1 j = Ψ(x) Ψ(x) = 1 (1; V v) one particle per unit volume A (x) = A (x) four potential r min C = 1 m e Compton wavelength t QED = C c = 1 m e QED coherence time Strong electromagnetic elds ea constant eld dee h i 1 ; E 0: m e m d[cm] e Ze 2 Coulomb eld Ze2 1 ; E (H; r 0;H ) V cm rm e r 0 (Z)m e time varying eld ea m e = Ee!m e a 0 1 ; E nl =!me e 10 10! ev V cm V cm Felix Mackenroth Nonlinear QED in strong electromagnetic elds 14 / 33

37 Estimating the interaction strength Interaction strength Perturbative approach relies on Z d 4 x Estimates (unquantised) j (x)a (x)! e 1 j = Ψ(x) Ψ(x) = 1 (1; V v) one particle per unit volume A (x) = A (x) four potential r min C = 1 m e Compton wavelength t QED = C c = 1 m e QED coherence time Strong electromagnetic elds ea constant eld dee h i 1 ; E 0: m e m d[cm] e Ze 2 Coulomb eld Ze2 1 ; E (H; r 0;H ) V cm rm e r 0 (Z)m e time varying eld ea m e = ee!m e a 0 1 ; E nl =!me e 10 10! ev V cm V cm Felix Mackenroth Nonlinear QED in strong electromagnetic elds 14 / 33

38 Estimating the interaction strength Strong electromagnetic elds lightning E lightning 10 4 V cm Felix Mackenroth Nonlinear QED in strong electromagnetic elds 15 / 33

39 Estimating the interaction strength Strong electromagnetic elds lightning E lightning 10 4 V cm the sun E sun V cm Felix Mackenroth Nonlinear QED in strong electromagnetic elds 15 / 33

40 Estimating the interaction strength Strong electromagnetic elds lightning E lightning 10 4 V cm the sun E sun V cm lasers E max V cm Felix Mackenroth Nonlinear QED in strong electromagnetic elds 15 / 33

41 Estimating the interaction strength Strong electromagnetic elds lightning E lightning 10 4 V cm the sun E sun V cm lasers I max W cm 2 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 15 / 33

42 Estimating the interaction strength Strong electromagnetic elds lightning E lightning 10 4 V cm the sun E sun V cm lasers I max W cm 2 more E max V cm Felix Mackenroth Nonlinear QED in strong electromagnetic elds 15 / 33

0 1 lasers I max 10 22 W cm 2 a 0 10 2 E max. 10 16 V cm a 0?

43 Estimating the interaction strength Strong electromagnetic elds lightning E lightning 10 4 V cm a 0 1 the sun E sun V cm a 0 1 lasers I max W cm 2 a E max V cm a 0??? Felix Mackenroth Nonlinear QED in strong electromagnetic elds 15 / 33

44 The Furry picture of quantum dynamics Strong electromagnetic elds Limit of large photon numbers single photon quantum dynamics Felix Mackenroth Nonlinear QED in strong electromagnetic elds 16 / 33

45 The Furry picture of quantum dynamics Strong electromagnetic elds Limit of large photon numbers nonlinear eects Felix Mackenroth Nonlinear QED in strong electromagnetic elds 16 / 33

46 Introduction Field quantisation in strong external elds Fundamentals of strong- eld QED Refractive e ects Backup The Furry picture of quantum dynamics Strong electromagnetic elds Limit of large photon numbers classical elds Felix Mackenroth Nonlinear QED in strong electromagnetic elds 16 / 33

47 Introduction Field quantisation in strong external elds Fundamentals of strong- eld QED Refractive e ects Backup The Furry picture of quantum dynamics Strong electromagnetic elds Limit of large photon numbers (caveat: creation operators!) classical elds Treat strong elds as unquantised backgrounds Felix Mackenroth external currents Nonlinear QED in strong electromagnetic elds 16 / 33

48 The Furry picture of quantum dynamics Quantising with a strong background eld Split electromagnetic elds A (x) = A rad (x) + A ext(x) Felix Mackenroth Nonlinear QED in strong electromagnetic elds 17 / 33

49 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 17 / 33 The Furry picture of quantum dynamics Quantising with a strong background eld Split electromagnetic elds Hamiltonian Z H(x) = dx iψ y (x)@tψ(x) A (x) = A rad (x) + A ext(x) (x)@ta (x) L Ψ(x); Ψ(x); A (x) 4 = H Dirac (x) + H int,ext (x) + H photon,rad (x) +H int,rad (x) + const. {z } =:H Furry 0 (t)

50 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 17 / 33 The Furry picture of quantum dynamics Quantising with a strong background eld Split electromagnetic elds Hamiltonian Z H(x) = dx iψ y (x)@tψ(x) A (x) = A rad (x) + A ext(x) (x)@ta (x) L Ψ(x); Ψ(x); A (x) 4 = H Dirac (x) + H int,ext (x) + H photon,rad (x) +H int,rad (x) + const. {z } =:H Furry 0 (t) Furry picture (W.H. Furry, Phys. Rev. 81, 115 (1951)) U Furry (t; t 0 ) = exp i Z t t 0 dth Furry 0 (t) e.g. H Furry int (t) = U Furry y (t; t 0 ) Hint,rad (x)u Furry (t; t 0 ) = e Z dxψ Furry 6A Furry rad ΨFurry

51 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 18 / 33 The Furry picture of quantum dynamics Field operators in the Furry picture Equations of motion i d dt Field operators i d dt O(t) = ho(t); H Furry 0 (t) jψ(x); A(x); ti = HFurry int (t) jψ(x); A(x); ti e 6A ext m e ) Ψ Furry (x) = 0 i A rad (x) = 0 interacting theory with external elds exactly taken into account requires exact solution of wave equations ordinary S-Matrix theory applicable

52 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 19 / 33 The Furry picture of quantum dynamics Scattering theory - III Field operators Z Ψ Furry (A ext ; x) = Z A (x) = rad dp c p Ψ p (A ext ; x) + dpψ y p (A ext ; x) h dk a k A k (x) + ay k A k (x) i Non-interacting states Y Y Y + l k ; n p ; m e pe + pe p e + k = Y k a y k lk Y p e c y p e npe Y p e + d y mpe + p e j0i +

53 = Felix Mackenroth Nonlinear QED in strong electromagnetic elds 19 / 33 The Furry picture of quantum dynamics Scattering theory - III Field operators Z Ψ Furry (A ext ; x) = Z A (x) = rad dp c p Ψ p (A ext ; x) + dpψ y p (A ext ; x) h dk a k A k (x) + ay k A k (x) i Non-interacting states Y Y Y + l k ; n p ; m e pe + pe p e + k = Y k a y k Scattering matrix elements S = * p f ;j; k f ;l X 1 + 1! ie Z lk Y p e c y p e npe Y p e + d y mpe + p e j0i + d 4 xj Furry (x)a rad (x) p i ;j; k i ;l +

54 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 19 / 33 The Furry picture of quantum dynamics Scattering theory - III Field operators Z Ψ Furry (A ext ; x) = Z A (x) = rad dk Non-interacting states Y Y Y + l k ; n p ; m e pe + pe p e + k dp c p Ψ p (A ext ; x) + dpψ y p (A ext ; x) h = Y k a k A k (x) + ay k A k a y k lk Y p e c y p e (x) i npe Y p e + d y mpe + p e j0i +

55 Exact solutions Exact solutions of the wave equation Plane electromagnetic waves - The Volkov solution Dirac equation [i e 6A ext m e ] Ψ(x) = 0 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 20 / 33

56 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 20 / 33 Exact solutions Exact solutions of the wave equation Plane electromagnetic waves - The Volkov solution Dirac equation (quadratic) e 6A ext ) 2 m 2 i 2 ef Ψ(x) = 0

57 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 20 / 33 Exact solutions Exact solutions of the wave equation Plane electromagnetic waves - The Volkov solution Dirac equation (quadratic) e 6A ext ) 2 m 2 i 2 ef Ψ(x) = 0 Model eld as plane wave A ext(x ; t) = A ext(k 0; x = x ) Wave function - Ansatz Ψ p (x) = e ipx u F p (x p ) p 2"

58 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 20 / 33 Exact solutions Exact solutions of the wave equation Plane electromagnetic waves - The Volkov solution Dirac equation (quadratic) e 6A ext ) 2 m 2 i 2 ef Ψ(x) = 0 Model eld as plane wave A ext(x ; t) = A ext(k 0; x = x ) Wave function - Ansatz Ψ p (x) = e isv (x ;p) E p (x ) u p p 2" Solution F p (x ) = e i m2 R 2 p2 x i x 0 d e (pa()) (pk0 ) e 2 A 2 () 2(pk0 ) 1 + e 6k 0 6A(x ) 2(pk 0 )

59 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 20 / 33 Exact solutions Exact solutions of the wave equation Plane electromagnetic waves - The Volkov solution Dirac equation (quadratic) e 6A ext ) 2 m 2 i 2 ef Ψ(x) = 0 Model eld as plane wave A ext(x ; t) = A ext(k 0; x = x ) Wave function - Ansatz Ψ p (x) = e isv (x ;p) E p (x ) u p p 2" Exponential phase S V (x ; p) = m2 p 2 2 Z x + px + d x0 e (pa()) (pk 0 ) e 2 A 2 () 2(pk 0 )

60 Exact solutions Properties of the Volkov wave functions Classical phase - quasiclassical solution p class S V (x ; p) ea ; x class (t S V (x ; p): Volkov propagator - Green's function (4) (x y) = [i e 6A ext m e ] G(x ; y) Z d 4 p X G(x ; y) = Ψ (2) 4 p (x)ψ p (y)e ip(y x) Orthogonality and completeness (J. Phys. A - Math. Gen. 44, (2011)) Z 1 dxψ (2) 3 p 0(x) 0 Ψ p (x) = (3) (p 0 p) Z 1 dpψ (2) 3 p (x) 0 Ψ p (y) = (3) (x y) All requirements for building a quantum eld theory. Felix Mackenroth Nonlinear QED in strong electromagnetic elds 21 / 33

61 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 22 / 33 Strong-eld QED

62 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 23 / 33 Monochromatic elds Interaction with a monochromatic eld - Wave function Four potential Volkov phase A ext(x ) = ma 0 jej sin(x ) S V ;m.c. (x ) = cos(x ) + sin(2x ) + (qx) nonlinear dependence a 0 ; a 2 0 dressed momentum q = p + m2 a 2 0 4(pk 0 ) k 0 dressed mass m = p q 2 = m e r1 + a2 0 2 Fourier expansion Ψ p (x) = 1X n= 1 C 0;n + e 6k 0 6A 2(pk 0 ) C up 1;n p e i(q+nk0)x 2q 0

63 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 24 / 33 Monochromatic elds Nonlinear Compton scattering - Amplitude Scattering amplitude p f S = p i k 1 = ier 2 = 1X N= 1! 1 Z d 4 xψ pf 6 1 eik1x Ψ pi (x) M N (4) (q i + Nk 0 k 1 q f ) Harvey et al., Phys. Rev. A 79, (2009)

64 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 24 / 33 Monochromatic elds Nonlinear Compton scattering - Amplitude Scattering amplitude p f S = p i k 1 = ier 2 = 1X N= 1! 1 Z d 4 xψ pf 6 1 eik1x Ψ pi (x) M N (4) (q i + Nk 0 k 1 q f ) reminiscent of vacuum QED Harvey et al., Phys. Rev. A 79, (2009)

65 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 24 / 33 Monochromatic elds Nonlinear Compton scattering - Amplitude Scattering amplitude p f S = p i k 1 = ier 2 = 1X N= 1! 1 Z d 4 xψ pf 6 1 eik1x Ψ pi (x) M N (4) (q i + Nk 0 k 1 q f ) reminiscent of vacuum QED emitted frequencies!n = N "i! 0 (1 + ) (p i + (N + m2 a 2 0 4(pk 0) )k 0)n 1 Harvey et al., Phys. Rev. A 79, (2009)

66 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 24 / 33 Monochromatic elds Nonlinear Compton scattering - Amplitude Scattering amplitude p f S = p i k 1 = ier 2 = 1X N= 1! 1 Z d 4 xψ pf 6 1 eik1x Ψ pi (x) M N (4) (q i + Nk 0 k 1 q f ) reminiscent of vacuum QED emitted frequencies!n = N emission spectra "i! 0 (1 + ) (p i + (N + m2 a 2 0 4(pk 0) )k 0)n 1 Harvey et al., Phys. Rev. A 79, (2009) redshift

67 Monochromatic elds Nonlinear Compton scattering - Amplitude Scattering amplitude p f S = p i k 1 = ier 2 = 1X N= 1! 1 Z d 4 xψ pf 6 1 eik1x Ψ pi (x) M N (4) (q i + Nk 0 k 1 q f ) reminiscent of vacuum QED "i! 0 (1 + ) emitted frequencies!n = N (p i + (N + m2 a0 2 4(pk )k 0) 0)n 1 emission spectra P 1 higher order processes G(y ; x) / (q i k 1+Nk 0) 2 m 2 q i k 0 resonances:! 1 = N (q i + Nk 0 )n 1 Harvey et al., Phys. Rev. A 79, (2009) Felix Mackenroth Nonlinear QED in strong electromagnetic elds 24 / 33

68 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 25 / 33 Quantum cascades Pair production - Formalism Scattering amplitude S = k 1 = ier 2 = 1X N= 1! 1 Z p e + p e d 4 xψ p e e ik1x Ψ pe (x) em N (4) q pe + Nk 0 k 1 q pe +

69 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 25 / 33 Quantum cascades Pair production - Formalism Scattering amplitude S = k 1 = ier 2 = 1X N= 1! 1 Z p e + p e crossing symmetry d 4 xψ p e e ik1x Ψ pe (x) em N (4) q pe + Nk 0 k 1 q pe +

70 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 25 / 33 Quantum cascades Pair production - Formalism Scattering amplitude S = k 1 = ier 2 = 1X N= 1! 1 Z p e + p e crossing symmetry d 4 xψ p e e ik1x Ψ pe (x) em N (4) q pe + Nk 0 k 1 q pe + channel closing N! 0! 1 4m a2 0 2

71 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 25 / 33 Quantum cascades Pair production - Formalism Scattering amplitude S = k 1 = ier 2 = 1X N= 1! 1 Z p e + p e crossing symmetry d 4 xψ p e e ik1x Ψ pe (x) em N (4) q pe + Nk 0 k 1 q pe + channel closing N! 0! 1 4m a2 0 2 modelled experiment (problematic normalisation) Burke et al., Phys. Rev. Lett. 79, 1626 (1997)

72 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 25 / 33 Quantum cascades Pair production - Formalism Scattering amplitude S = k 1 = ier 2 = 1X N= 1! 1 Z p e + p e crossing symmetry d 4 xψ p e e ik1x Ψ pe (x) em N (4) q pe + Nk 0 k 1 q pe + channel closing N! 0! 1 4m a2 0 2 modelled experiment (problematic normalisation) photon + laser = matter Burke et al., Phys. Rev. Lett. 79, 1626 (1997)

73 Quantum cascades Pair production - Interpretation Matter from light Felix Mackenroth Nonlinear QED in strong electromagnetic elds 26 / 33

74 Quantum cascades Pair production - Interpretation Matter from light Felix Mackenroth Nonlinear QED in strong electromagnetic elds 26 / 33

75 Quantum cascades Pair production - Interpretation Matter from light Experiment Energy conservation (monochromatic eld) q i + Nk 0 = q f + q + + q Felix Mackenroth Nonlinear QED in strong electromagnetic elds 26 / 33

76 Quantum cascades Pair production - Interpretation Matter from light Experiment In the initial electron's rest frame N! 4m Felix Mackenroth Nonlinear QED in strong electromagnetic elds 26 / 33

Quantum cascades Pair production - Interpretation Matter from light Experiment E-144 @SLAC In the initial electron's rest frame N!

77 Quantum cascades Pair production - Interpretation Matter from light Experiment In the initial electron's rest frame N! 4m Photon energy seen by 50 GeV electron! 0:43 MeV N 6 nonlinearity necessary! Felix Mackenroth Nonlinear QED in strong electromagnetic elds 26 / 33

78 Quantum cascades Quantum cascades Two considerations in a strong laser eld charges radiate radiation decays into charges Felix Mackenroth Nonlinear QED in strong electromagnetic elds 27 / 33

79 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 27 / 33 Quantum cascades Quantum cascades Two considerations in a strong laser eld charges radiate radiation decays into charges Fundamental limitation on physical elds! Imax W cm 2

80 Critical eld of QED Critical eld of QED Break up of empty vacuum Felix Mackenroth Nonlinear QED in strong electromagnetic elds 28 / 33

81 Critical eld of QED Critical eld of QED Break up of empty vacuum Felix Mackenroth Nonlinear QED in strong electromagnetic elds 28 / 33

82 Particle can become real. Felix Mackenroth Nonlinear QED in strong electromagnetic elds 28 / 33 Critical eld of QED Critical eld of QED Break up of vacuum Pair production in constant elds (Schwinger (1951)) P exp E cr E ; E cr = m2 e Visualisation: Energy gain over a Compton wavelength " = ee cr 1 m e = m e

83 Particle can become real. Felix Mackenroth Nonlinear QED in strong electromagnetic elds 28 / 33 Critical eld of QED Critical eld of QED Break up of vacuum Pair production in constant elds (Schwinger (1951)) P exp E cr E ; E cr = m2 e V 1: cm Visualisation: Energy gain over a Compton wavelength " = ee cr 1 m e = m e

84 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 29 / 33 Refractive eects

85 Vacuum birefringence Euler-Heisenberg Lagrangian Integrate out Dirac eld dynamics Eective eld theory calculation gives Euler-Heisenberg Lagrangian L EH = 1 E 2 B 2 1 h + E 2 B (EB) 2i Ecr 2 Field equations (Euler-Lagrange equations) polarisation P = magnetisation M 1 4 E = E B = E 2 cr Nonlinear coupling of classical elds! h i 2 E 2 B 2 2 E + 7 (EB) 2 B h i 2 E 2 B 2 2 B 7 (EB) 2 E Felix Mackenroth Nonlinear QED in strong electromagnetic elds 30 / 33

86 Vacuum birefringence A matterless double slit Light can be scattered from light d 2 dt E(x 2 ; t) r2 E(x ; t) = j vac (x ; t) j vac (x ; t) = ; t) t 2 P(x ; t) r (rp(x ; t)) Bring two strong light elds (scatterers) close: B. King et al. Nature Photonics 4, 92 (2010) laseri strong 24 cm2 W 10 min ; I probe 16 W 10 cm 2 : signal n photons 3 10 after 4 hours Felix Mackenroth Nonlinear QED in strong electromagnetic elds 31 / 33

Vacuum birefringence A matterless double slit Light can be scattered from light d 2 dt E(x 2 ; t) r2 E(x ; t) = j vac (x ; t) j vac (x ; t) = r @tm(x ; t) + @ t 2 P(x ; t) r (rp(x ; t)) Bring two

87 Vacuum birefringence A matterless double slit Light can be scattered from light d 2 dt E(x 2 ; t) r2 E(x ; t) = j vac (x ; t) j vac (x ; t) = ; t) t 2 P(x ; t) r (rp(x ; t)) Bring two strong light elds (scatterers) close: double slit B. King et al. Nature Photonics 4, 92 (2010) laseri strong 24 cm2 W 10 min ; I probe 16 W 10 cm 2 : signal n photons 3 10 after 4 hours Felix Mackenroth Nonlinear QED in strong electromagnetic elds 31 / 33

88 Vacuum birefringence Open issues Current research innite spectra (-peaks, propagator poles) non-plane wave geometries higher order eects measurements Felix Mackenroth Nonlinear QED in strong electromagnetic elds 32 / 33

89 Vacuum birefringence Summary Take home strong electric elds alter fundamental entities perturbative expansion of eld theories breaks down nonperturbative backgrounds: exact solution lasers oer an ideal testing ground Felix Mackenroth Nonlinear QED in strong electromagnetic elds 33 / 33

90 Vacuum birefringence Summary Take home strong electric elds alter fundamental entities perturbative expansion of eld theories breaks down nonperturbative backgrounds: exact solution lasers oer an ideal testing ground Thank you Felix Mackenroth Nonlinear QED in strong electromagnetic elds 33 / 33

91 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 34 / 33 Backup Backup

92 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 35 / 33 Backup Motivation II - The Klein paradox Strong electromagnetic elds aect the basic theory Scattering of a Dirac particle at a potential step (Klein (1929)) 2 = 1 V 0 (2" V 0 ) p 2 (1 ) 2 T (V 0 ) = 1 (1 + ) = 4 2 (1 + ) 2 jt (V 0! 1)j 2 = 2p (" + p) Calculation at a nite potential gradient (Sauter 1931) Transmission only for E & E cr = m2 c 3 ~ jej 2 " + m e " V 0 + m e

93 Backup Quantising with a strong background eld Hamiltonian Z H(x) = dx iψ y (x)@tψ(x) = Z (x)@ta (x) L Ψ(x); Ψ(x); A (x) 4 dxi Ψ (x) t Ψ(x) 1 t A µ (x) rad ta ;rad (x) 4 Ψ ;`(x)(i A rad m)ψ ;`(x) Ψ ;`(x)(i A ext m)ψ ;`(x) + 1 ν A ;rad (x) µ A ν rad(x) Ψ ;`(x)(i m)ψ ;`(x) 8 1 ext(x)@ta ;ext rad (x)@ ta ;ext (x) 4 4 ext(x)@ta ;rad (x) + A ;rad (x)@ A ext(x) 4 + A ;ext (x)@ A rad(x) + A ;ext (x)@ A ext(x) 8 8 = H Dirac (x) + H int,ext (x) +H int,rad (x) + H photon,rad (x) {z } =:H 0(t) 8 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 36 / 33

94 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 37 / 33 Backup Laboratory elds Nonlinear optical eld threshold a 0 > 0 1 ev Atomic elds 92+ E U V cm dicult to control E nl h! ev i V cm

95 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 37 / 33 Backup Laboratory elds Nonlinear optical eld threshold a 0 > 0 1 ev Atomic elds E U V cm dicult to control Laser elds Z E nl h! ev i V cm critical intensity I nl = 1 h dx E 2 (x ; t) + B 2 (x ; t) 10 18! W 8 ev cm 2 peak intensities I peak W ) a cm energy " laser 10 J, duration 10 fs, focus w cm i 2

96 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 37 / 33 Backup Laboratory elds Nonlinear optical eld threshold a 0 > 0 1 ev Atomic elds E U V cm dicult to control Laser elds future development E nl h! ev i V cm nonlinear physics!

97 Felix Mackenroth Nonlinear QED in strong electromagnetic elds 38 / 33 Backup Exact solutions of the wave equation Plane electromagnetic waves - The Volkov solution Wave function - Ansatz Ψ p (x) = e ipx F p (x ) u p p 2" Dirac equation = ( )=2 2i(pk 0 )F 0 (x ) + (p 2 m 2 ) 2(pA) + e 2 A 2 ie 6k (@ 6A(x 6 )) F (x ) = 0 Solution F p (x ) = exp " i Z d x0 e p A () (pk 0 ) with 0 = e 2 6k 0 6A(x ) 6k 0 6A(x ) 4(pk 0 ) 2 / k 2 # e 2 A 2 () + e 6k 0 6A(x ) 2(pk 0 ) 2(pk 0 )

98 Backup Euler-Heisenberg Lagrangian Integrate out Dirac eld dynamics Path integral formalism (assuming proper normalisation) Assume Lagrangian to distinguish between dynamic (light) and hidden (heavy) elds L [l ; h] = L l + L h + J [l] h hl ; h; t f j l ; h; t i i = n-point correlation functions = c n (x 1 ; ; x n ) = Z Z Dl Z R Dle i t f t i Dhe i R t f t i d 4 xl[ l ; h ] d 4 xl l e 2 3 Y 4 n J(x j ) W [J] 5 j=1 Lowest order give classical eld evolution c h = W [J] J iw [J] J=0 = h0 j hj 0i h0 j 0i Felix Mackenroth Nonlinear QED in strong electromagnetic elds 39 / 33

99 Backup Expand around classical solution Z d 4 x (L h + J [l] h) = Energy functional W [J] = Z With QED Lagrangian Z d 4 x (L c h + J [l] c h) L QED = Functional derivative d 4 x (L c h + J [l] c h)+ Z i 2 T r log A + Ψ (6D 8 d 4 y h(y) (L h + J [l] h) 2 L h(y 1 )h(y 2 ) m) Ψ 2 L QED Ψ(x 1 )Ψ(x 2 ) = [6D m] (4) (x 1 x 2 ) QED energy functional (normalised by the vacuum propagator as exponential phase) W QED [J] = Z d 4 x L c + Ψ;Ψ J [ A] c Ψ;Ψ i [i 6D m] 2 T r log : [i 6@ m] Integrated result Felix Mackenroth Nonlinear QED in strong electromagnetic elds 40 / 33 :

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