Electron-Positron Pair Production in Strong Electric Fields

Transcription

1 Electron-Positron Pair Production in Strong Electric Fields Christian Kohlfürst PhD Advisor: Reinhard Alkofer University of Graz Institute of Physics Dissertantenseminar Graz, November 21, 2012

2 Outline 1 Motivation Phenomenology Quantum Electrodynamics Concepts 2 Theoretical Method Background Information Derivation 3 Calculations Aspects of Pair Production Optimal Control Theory 4 Summary 5 Outlook

3 Outline 1 Motivation Phenomenology Quantum Electrodynamics Concepts 2 Theoretical Method Background Information Derivation 3 Calculations Aspects of Pair Production Optimal Control Theory 4 Summary 5 Outlook

4 Phenomenology Why Pair Production? Cite:

5 Phenomenology Pair Production in Physics Cite: SLAC Experiment 144 a a D. L. Burke et al., Phys. Rev. Lett. 79(9), 1997

6 Phenomenology Pair Production in Physics Cite: bnl.gov SLAC Experiment 144 a Heavy Ion Collisions(Glasma Flux Tube) b a D. L. Burke et al., Phys. Rev. Lett. 79(9), 1997 b N. Tanji, Ann. Phys. 325(9), 2010

7 Phenomenology Pair Production in Physics Cite: Upcoming high-intensity Laser experiments a a ELI, Grand Challenges Meeting, 2009

8 Phenomenology Pair Production in Physics Cite: Upcoming high-intensity Laser experiments Neutronstars, Pulsars a a J. Y. Kim, arxiv: , 2012

9 Phenomenology Pair Production in Physics Cite: www-naweb.iaea.org Upcoming high-intensity Laser experiments Neutronstars, Pulsars Atomic Ionization a a J. R. Oppenheimer, Phys. Rev. 31, 1928

10 Quantum Electrodynamics QED Vacuum Cite: Quantum fluctuations Dynamical object Various Experiments: Casimir effect, Lamb shift

11 Quantum Electrodynamics Casimir Effect Cite: commons.wikipedia.org Attractive force between uncharged plates in vacuum Zero-point energy

12 Quantum Electrodynamics Lamb Shift Corrections to hydrogen spectrum Interaction between electrons and vacuum

13 Concepts LASER Physics Cite: commons.wikipedia.org Light Amplification by Stimulated Emission of Radiation Various possibilities for gain medium

14 Concepts LASER Beams Field Strength Spatial Coordinate Gaussian shape Pulsed/Continous wave operation Interaction region approximately spatial homogeneuos

15 Concepts Free Electron Laser Cite: commons.wikipedia.org Electron Beam as lasing medium Widely tuneable frequency X-Ray

16 Concepts Pair Production Energy Particle Band 0. Band Gap Antiparticle Band Treatment of vacuum as medium Electrons(blue band) and positrons(red band)

17 Concepts Applying an External Field Energy Multiphoton ( ) 4mτ Photon absorption P eeτ 2m Perturbative process N. Narozhnyi: Sov. J. Nucl. Phys. 11(596), 1970

18 Concepts Applying an External Field Energy Schwinger Electron tunneling P exp( πm 2 /ee) Non-perturbative process F. Sauter: Z. Phys. 69(742), 1931 J. S. Schwinger: Phys. Rev. 82(664), 1951

19 Concepts Characteristic Scales Pair Production E cr = (m 2 c 3 )/(e h) P M ( eeτ 2m ) 4mτ P S exp( πm 2 /ee) γ = (mτε) 1 E cr V /m m = keV m s ELI(Goal) vs. Optical Laser E peak 0.01E cr τ min s E /2mc E peak 10 4 E cr τ min s E /2mc

20 Outline 1 Motivation Phenomenology Quantum Electrodynamics Concepts 2 Theoretical Method Background Information Derivation 3 Calculations Aspects of Pair Production Optimal Control Theory 4 Summary 5 Outlook

21 Background Information Considerations Goal Describe e and e + in strong electric field Requirement Describe dynamical pair creation Show time evolution of non-equilibrium process Quantum fluctuations Particle statistics

22 Background Information Boltzmann Equation Advantage Semi-classical method Describes behaviour of particle distribution (fluid) Capable of describing non-eqilibrium processes One partial differential equation ( ) 1 t F(t,x,p) = m p x + F p F(t,x,p) + F Collision (1) Disadvantage Problems in describing plasma Not applicable to Coulomb interaction

23 Background Information Vlasov Equation Improvement Expands Boltzmann equation by Maxwell equations Capable of describing distribution of charged particles Provides self-consistent way to treat EM-fields Simplifies in non-relativistic limit ( 1 t F i (t,x,p) = p m i x + e ) i E pi F i (t,x,p) (2) i m i E = 4πρ (3)

24 Derivation Problem Statement Quantum electrodynamic Only QED-Lagrangian and Dirac equation is known L = ψ(iγ µ D µ m)ψ 1 4 F µνf µν (4) iγ µ µ ψ mψ = eγ µ A µ ψ (5) Covariant derivative D µ = µ + iea µ Field strength tensor F µν = µ A ν ν A µ Open question How to obtain statistical quantity F (q, t) from particle description?

25 Derivation Quantisation Vector Potential Spatial-independent vector potential in one direction A µ = (0,A(t)e 3 ) (6) Very strong fields regarded as classical Mean field approximation(background field) S. Schmidt et al.: Int. J. Mod. Phys. E 7(709), 1998 F. Hebenstreit: Dissertation, 2011

26 Derivation Quantisation Dirac Field Fully quantized Ψ χ + a(q) + χ b (q) (7) Equation of motion ( ) t 2 + m 2 + (q 3 ea(t)) 2 + iee(t) χ ± = 0 (8) Creation/Annihilation Operators Operators a(q), b (q) hold information about particle statistics Fermions anti-commutation relations

27 Derivation Operator Transformation Hamiltonian Non-vanishing off-diagonal elements Diagonalization by Bogoliubov transformation Switching to quasi-particle picture A(q,t) = α (q,t)a(q) β (q,t)b ( q) B ( q,t) = β (q,t)a(q) + α (q,t)b ( q) (9a) (9b) Bogoliubov coefficients have to fulfill α (q,t) 2 + β (q,t) 2 = 1

28 Derivation Particle Distribution One-particle distribution function F (q,t) = A (q,t)a(q,t) (10) Fulfills equation of motion t F (q,t) = S (q,t) (11) Gives distribution in momentum space Time-dependent quantity Interpretation as electron/positron distribution for t ± only

29 Derivation Quantum Vlasov Equation t t F (q,t) = W (q,t) dt W ( q,t )( 1 F ( q,t )) cos ( 2θ ( q,t,t )) t Vac θ ( q,t,t ) t = ω ( q,t ) dt, t W (q,t) = ee (t)ε (q,t) ω 2, ε 2 (q,t) (q,t) = m2 + q 2, ω 2 (q,t) = ε 2 (q,t) + (q3 ea(t))2 (12) Non-Markovian integral equation Fermions and Pauli statistics Rapidly oscillating term

30 Derivation Differential Equation Auxiliary functions t G = dt W ( q,t )( 1 F ( q,t )) cos ( 2θ ( q,t,t )) t Vac H = t t Vac dt W ( q,t )( 1 F ( q,t )) sin ( 2θ ( q,t,t )) (13a) (13b) Coupled differential equation Ḟ 0 W 0 F 0 Ġ = W 0 2ω G + W (14) Ḣ 0 2ω 0 H 0 J. C. R. Bloch et al.: Phys. Rev. D 60(116011), 1999

31 Derivation Recapitulation of Quantum Kinetic Positive Aspects Works for arbitrary time-dependent fields Insight into time evolution of system Gives momentum space distribution of particles Particle density easily calculable Negative Aspects Collisions between particles neglected Quantum nature of electric field omitted No spatial inhomogenity No magnetic field

32 Derivation Magnetic Fields Pure Magnetic Fields Particle creation not possible Collinear EM-Fields Space-dependent vector potential A µ (x,t) = (0,0,xB,A(t)) Additional constraints compared to pure electric case Quantised particle distribution ε 2 m2 + eb (2n ( 1) s ) N. Tanji: Ann. Phys. 8(324), 2009

33 Outline 1 Motivation Phenomenology Quantum Electrodynamics Concepts 2 Theoretical Method Background Information Derivation 3 Calculations Aspects of Pair Production Optimal Control Theory 4 Summary 5 Outlook

34 Aspects of Pair Production Single Pulse 0.08 E t Electric field: E (t) = εe cr sech 2 ((t t 0 )/τ) cos (ωt + φ)

35 Aspects of Pair Production Single Pulse 0.08 E t Electric field: E (t) = εe cr sech 2 ((t t 0 )/τ) Three parameters: ε, τ, t 0

36 Aspects of Pair Production Single Pulse F q,t Electric field: E (t) = εe cr sech 2 ((t t 0 )/τ) Three parameters: ε, τ, t 0 Particle momentum p = (q ea)

37 Aspects of Pair Production Single Pulse Particle Number Electric field: E (t) = εe cr sech 2 ((t t 0 )/τ) Three parameters: ε, τ, t 0 Particle momentum p = (q ea) Particle number N [dq]f (q)

38 Aspects of Pair Production Dynamically Assisted Schwinger Effect Τ1 501 m Enhancement of Number Density Τ m Τ m Τ m Combination of long&strong pulse (ε 1 = 0.1,τ 1 ) with short&weak pulse (ε 2 = 0.01,τ 2 ) Dynamical Assistance observable at γ = (mε 1 τ 2 ) 1 R. Schützhold et al.: Phys. Rev. Lett. 101(130404), 2008

39 Aspects of Pair Production Interferences in Multiphoton Regime t m t m Particle Distribution Particle Distribution ε = 0.02,τ = 3[1/m] Blue curve: Combination of short pulses ε = ±0.02,τ = 3[1/m] Red curve: 100 particle distribution rate of single pulse Equally signed fields Interference width Alternatingly signed fields Interference height E. Akkerman et al.: Phys. Rev. Lett. 108(030401), 2012

40 Aspects of Pair Production Interferences in Multiphoton Regime t m t m Particle Distribution Particle Distribution ε = 0.02,τ = 3[1/m] Blue curve: Combination of short pulses ε = ±0.02,τ = 3[1/m] Red curve: 100 particle distribution rate of single pulse Equally signed fields Interference width Alternatingly signed fields Interference height E. Akkerman et al.: Phys. Rev. Lett. 108(030401), 2012

41 Aspects of Pair Production Interferences in Multiphoton Regime t m t m Particle Distribution Particle Distribution ε = 0.02,τ = 3[1/m] Blue curve: Combination of short pulses ε = ±0.02,τ = 3[1/m] Red curve: 100 particle distribution rate of single pulse Equally signed fields Interference width Alternatingly signed fields Interference height E. Akkerman et al.: Phys. Rev. Lett. 108(030401), 2012

42 Aspects of Pair Production Combination of Effects Particle Number Τ2 31 m Red curve: Just short pulses Blue curve: Short pulses with additional long pulse (ε max = 0.1,τ 1 100[1/m]) Black lines: 1,10 and 100 single pulse result, asymptotic blue curve Interferences observable

43 Optimal Control Theory Optimal Control Theory Description Form of constrained variational calculus Maximization of particle number by optimization of pulse shapes Technicalities Cost functional holds as basis for calculation Initial field configuration as input Two different ways of formulation

44 Optimal Control Theory Optimization Process Single electric field: E (t) = εe cr sech 2 ((t t 0 )/τ) Optimization parameters ε, τ, t 0 Model full configuration by sum of single fields

45 Optimal Control Theory Optimization Process Single electric field: E (t) = εe cr sech 2 ((t t 0 )/τ) Optimization parameters ε, τ, t 0 Model full configuration by sum of single fields

46 Optimal Control Theory Optimization Process Single electric field: E (t) = εe cr sech 2 ((t t 0 )/τ) Optimization parameters ε, τ, t 0 Model full configuration by sum of single fields

47 Optimal Control Theory Optimization Process Single electric field: E (t) = εe cr sech 2 ((t t 0 )/τ) Optimization parameters ε, τ, t 0 Model full configuration by sum of single fields

48 Optimal Control Theory Enhancement in Particle Number ε 2 τ 2 [1/m] particle number e e e e e-04 Optimization with different initial conditions Procedure finds optimal pulse shape Huge gain in particle number

49 Summary Particle pair production is connected to various fields of research Schwinger effect exponentially suppressed e e + pair production is highly dynamical process Pulse shaping by optimal control theory is possible

50 Outlook Improve in optimization Perform calculations with realistic setups Find formalism to include magnetic fields

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

52 Appendix Optimal Control Theory Cost functional J (F (q,t),g (q,t),h (q,t),a) = γ F (q,t )dq + p[a] (15)

53 Appendix Optimal Control Theory Cost functional J (F (q,t),g (q,t),h (q,t),a) = γ F (q,t )dq + p[a] (15) Constraints λ F (q,t) = Ḟ (q,t) W (q,t)g (q,t) = 0 (16) λ G (q,t) = Ġ (q,t) + W (q,t)f (q,t) + 2ω (q,t)h (q,t) W (q,t) = 0 (17) λ H (q,t) = Ḣ (q,t) 2ω (q,t)g (q,t) = 0 (18)

54 Appendix Optimal Control Theory Lagrange function L = J (F,G,H,A) + λ F, µ F + λ G, µ G + λ H, µ H (19)

55 Appendix Optimal Control Theory Lagrange function L = J (F,G,H,A) + λ F, µ F + λ G, µ G + λ H, µ H (19) Adjoint differential equation µ F (q,t) = W (q,t) µ G (q,t) (20) µ G (q,t) = W (q,t) µ F (q,t) 2ω (q,t) µ H (q,t) (21) µ H (q,t) = 2ω (q,t) µ G (q,t) (22)

56 Appendix Optimal Control Theory Gradient J = δl δa = δp δa 2 + dq ( µ H G + µ G H) q A ω (23) dq ε ω 2 t (µ G F µ F G µ G ) (24)

57 Appendix Optimal Control Theory Gradient J = δl δa = δp δa 2 + dq ( µ H G + µ G H) q A ω (23) dq ε ω 2 t (µ G F µ F G µ G ) (24) Wolfe conditions ( ) J A k (t) + α k J J (A k (t)) + c 1 α k J (A), J (A) (25) ( ) J A k (t) + α k J, J (A k ) c 2 J (A k ), J (A k ) (26)

58 Appendix Internal Electric Field Internal electric field E int (t) due to electron-positron pairs: E(t) = E ext (t) + E int (t) Ė(t) = j ext (t) j int (t)

59 Appendix Internal Electric Field Internal electric field E int (t) due to electron-positron pairs: E(t) = E ext (t) + E int (t) Ė(t) = j ext (t) j int (t) Internal electromagnetic current j int (t) in the e3 direction: j int (t) = 4e d 3 k (2π) 3 p3(t) ω (q,t) F (q,t) + 4 E (t) d 3 k ω (q,t)ḟ (q,t) 3 (2π)

60 Appendix Internal Electric Field Internal electric field E int (t) due to electron-positron pairs: E(t) = E ext (t) + E int (t) Ė(t) = j ext (t) j int (t) Internal electromagnetic current j int (t) in the e3 direction: j int (t) = 4e d 3 k (2π) 3 p3(t) ω (q,t) F (q,t) + 4 E (t) d 3 k ω (q,t)ḟ (q,t) 3 (2π) Conduction current stays regular for all momenta!

61 Appendix Internal Electric Field Internal electric field E int (t) due to electron-positron pairs: E(t) = E ext (t) + E int (t) Ė(t) = j ext (t) j int (t) Internal electromagnetic current j int (t) in the e3 direction: j int (t) = 4e d 3 k (2π) 3 p3(t) ω (q,t) F (q,t) + 4 E (t) d 3 k ω (q,t)ḟ (q,t) 3 (2π) Conduction current stays regular for all momenta! Polarization current exhibits a logarithmic UV-divergence

62 Appendix Internal Electric Field Internal electric field E int (t) due to electron-positron pairs: E(t) = E ext (t) + E int (t) Ė(t) = j ext (t) j int (t) Internal electromagnetic current j int (t) in the e3 direction: j int (t) = 4e d 3 k (2π) 3 p3(t) ω (q,t) F (q,t) + 4 E (t) d 3 k ω (q,t)ḟ (q,t) 3 (2π) Conduction current stays regular for all momenta! Polarization current exhibits a logarithmic UV-divergence Charge renormalization

63 Appendix Internal Electric Field Internal electric field E int (t) due to electron-positron pairs: E(t) = E ext (t) + E int (t) Ė(t) = j ext (t) j int (t) Internal electromagnetic current j int (t) in the e3 direction: j int (t) = 4e d 3 k (2π) 3 p3(t) ω (q,t) F (q,t) + 4 E (t) d 3 k ω (q,t)ḟ (q,t) 3 (2π) Conduction current stays regular for all momenta! Polarization current exhibits a logarithmic UV-divergence Charge renormalization Properly renormalized internal electromagnetic current: 4e d 3 k (2π) 3 p3(t) ω (q,t) [ ] F (q,t) + ω2 (q,t) ee (t)p3(t) Ḟ (q,t) eė (t)ε2 8ω 4 (q,t)p3(t)

64 Appendix Scattering picture [ t 2 + ω 2 (q,t)]g (q,t) = 0 1-dimensional scattering problem [ ] H ψ(x) = h2 2m x 2 + V (x) ψ(x) = Eψ(x)

65 Appendix Scattering picture [ t 2 + ω 2 (q,t)]g (q,t) = 0 1-dimensional scattering problem [ ] H ψ(x) = h2 2m x 2 + V (x) ψ(x) = Eψ(x) Formal similarity Scattering potential: V (t) ω 2 (q,t) Reflection coefficient Produced pairs