Electron-Positron Pair Production in Strong Electric Fields -
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Motivation Theoretical Method Calculations Summary 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 Outlook Motivation Theoretical Method Calculations 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 Summary Outlook Motivation Theoretical Method Calculations 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 Summary Outlook Motivation Theoretical Method Phenomenology Why Pair Production? Cite: www.slac.stanford.edu/exp/e144 Calculations Summary Outlook Motivation Theoretical Method Phenomenology Pair Production in Physics Cite: www.slac.stanford.edu/exp/e144 SLAC Experiment 144a a D. L. Burke et al., Phys. Rev. Lett. 79(9), 1997 Calculations Summary Outlook Motivation Theoretical Method Calculations Phenomenology Pair Production in Physics Cite: bnl.gov SLAC Experiment 144a 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 Summary Outlook Motivation Theoretical Method Calculations Summary Phenomenology Pair Production in Physics Cite: www.extreme-light-infrastructure.eu Upcoming high-intensity Laser experiments a ELI, Grand Challenges Meeting, 2009 a Outlook Motivation Theoretical Method Calculations Phenomenology Pair Production in Physics Cite: www.nrao.edu Upcoming high-intensity Laser experiments Neutronstars, Pulsarsa a J. Y. Kim, arXiv:1208.1319 , 2012 Summary Outlook Motivation Theoretical Method Calculations Phenomenology Pair Production in Physics Cite: www-naweb.iaea.org Upcoming high-intensity Laser experiments Neutronstars, Pulsars Atomic Ionizationa a J. R. Oppenheimer, Phys. Rev. 31, 1928 Summary Outlook Motivation Theoretical Method Calculations Summary Quantum Electrodynamics QED Vacuum Cite: www.extreme-light-infrastructure.eu Quantum fluctuations Dynamical object Various Experiments: Casimir effect, Lamb shift Outlook Motivation Theoretical Method Calculations Summary Quantum Electrodynamics Casimir Effect Cite: commons.wikipedia.org Attractive force between uncharged plates in vacuum Zero-point energy Outlook Motivation Theoretical Method Calculations Quantum Electrodynamics Lamb Shift Corrections to hydrogen spectrum Interaction between electrons and vacuum Summary Outlook Motivation Theoretical Method Calculations Summary Concepts LASER Physics Cite: commons.wikipedia.org Light Amplification by Stimulated Emission of Radiation Various possibilities for gain medium Outlook Motivation Theoretical Method Calculations Summary Concepts LASER Beams Field Strength Spatial Coordinate Gaussian shape Pulsed/Continous wave operation Interaction region approximately spatial homogeneuos Outlook Motivation Theoretical Method Calculations Concepts Free Electron Laser Cite: commons.wikipedia.org Electron Beam as lasing medium Widely tuneable frequency → X-Ray Summary Outlook Motivation Theoretical Method Calculations Summary Concepts Pair Production Dirac Sea Picture Energy Particle Band Band Gap 0. Antiparticle Band Treatment of vacuum as medium Electrons(blue band) and positrons(red band) Outlook Motivation Theoretical Method Calculations Concepts Applying an External Field Energy Multiphoton Photon absorption P ≈ Perturbative process N. Narozhnyi: Sov. J. Nucl. Phys. 11(596), 1970 eEτ 2m 4mτ Summary Outlook Motivation Theoretical Method Calculations Concepts Applying an External Field Energy Schwinger Electron tunneling P ≈ exp(−πm2 /eE) Non-perturbative process F. Sauter: Z. Phys. 69(742), 1931 J. S. Schwinger: Phys. Rev. 82(664), 1951 Summary Outlook Motivation Theoretical Method Calculations Summary Concepts Characteristic Scales Pair Production Ecr = (m2 c 3 )/(eh̄) 4mτ PM ≈ eEτ 2m Ecr ∼ 1.3 · 1018 V /m PS ≈ exp(−πm2 /eE) m−1 ∼ 1.288 · 10−21 s γ = (mτε) −1 m = 510.998keV ELI(Goal) vs. Optical Laser Epeak ∼ 0.01Ecr Epeak ∼ 10−4 Ecr τmin ∼ 10−19 s τmin ∼ 10−15 s E /2mc 2 ∼ 0.005 E /2mc 2 ∼ 10−6 Outlook Motivation Theoretical Method Calculations 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 Summary Outlook Motivation Theoretical Method Calculations Summary 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 Outlook Motivation Theoretical Method Calculations Summary Outlook 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) = − p · ∇x + F · ∇p F (t, x, p) + FCollision (1) m Disadvantage Problems in describing plasma Not applicable to Coulomb interaction Motivation Theoretical Method Calculations Summary Outlook 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 e ∂t Fi (t, x, p) = − pi · ∇x + i E · ∇pi Fi (t, x, p) mi mi ∇ · E = 4πρ (2) (3) Motivation Theoretical Method Calculations Summary Outlook Derivation Problem Statement Quantum electrodynamic Only QED-Lagrangian and Dirac equation is known 1 L = ψ̄(iγ µ Dµ − m)ψ − Fµν F µν 4 µ iγ ∂µ ψ − mψ = eγµ Aµ ψ Covariant derivative Dµ = ∂µ + ieAµ Field strength tensor Fµν = ∂µ Aν − ∂ν Aµ Open question How to obtain statistical quantity F (q, t) from particle description? (4) (5) Motivation Theoretical Method Calculations Summary Outlook Derivation Quantisation Vector Potential Spatial-independent vector potential in one direction Aµ = (0, A(t)e3 ) 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 (6) Motivation Theoretical Method Calculations Summary Outlook Derivation Quantisation Dirac Field Fully quantized Ψ ∼ χ + a(q) + χ − b† (q) Equation of motion ∂t2 + m2 + (q3 − eA(t))2 + ieE(t) χ ± = 0 Creation/Annihilation Operators Operators a(q), b† (q) hold information about particle statistics Fermions → anti-commutation relations (7) (8) Motivation Theoretical Method Calculations Summary Outlook 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 Motivation Theoretical Method Calculations Summary Outlook Derivation Particle Distribution One-particle distribution function F (q, t) = hA† (q, t) A (q, t)i (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 Motivation Theoretical Method Calculations Summary Outlook Derivation Quantum Vlasov Equation ∂t F (q, t) = W (q, t) Z t dt 0 W q, t 0 1 − F q, t 0 cos 2θ q, t, t 0 tVac (12) W (q, t) = eE (t) ε⊥ (q, t) , ω 2 (q, t) Zt ω q, t 00 dt 00 , θ q, t, t 0 = t0 2 ω 2 (q, t) = ε⊥ (q, t) + (q3 − eA (t))2 Non-Markovian integral equation Fermions and Pauli statistics Rapidly oscillating term 2 ε⊥ (q, t) = m2 + q⊥2 , Motivation Theoretical Method Calculations Summary Outlook Derivation Differential Equation Auxiliary functions G= H= Z t tVac Z t dt 0 W q, t 0 dt 0 W q, t 0 1 − F q, t 0 cos 2θ q, t, t 0 (13a) sin 2θ q, t, t 0 (13b) W 0 F 0 0 −2ω G + W 2ω 0 H 0 (14) 1 − F q, t 0 tVac Coupled differential equation Ḟ 0 −W = Ġ 0 Ḣ J. C. R. Bloch et al.: Phys. Rev. D 60(116011), 1999 Motivation Theoretical Method Calculations Summary 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 Outlook Motivation Theoretical Method Calculations Summary Derivation Magnetic Fields Pure Magnetic Fields Particle creation not possible Collinear EM-Fields Space-dependent vector potential Aµ (x, t) = (0, 0, xB, A(t)) Additional constraints compared to pure electric case Quantised particle distribution ε⊥2 → m2 + eB (2n + 1 + (−1)s ) N. Tanji: Ann. Phys. 8(324), 2009 Outlook Motivation Theoretical Method Calculations 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 Summary Outlook Motivation Theoretical Method Calculations Summary Aspects of Pair Production Single Pulse 0.10 0.08 EH t L 0.06 0.04 0.02 0.00 - 40 - 20 0 t @1 mD 20 40 Electric field: E (t) = εEcr sech2 ((t − t0 ) /τ) × cos (ωt + φ ) Outlook Motivation Theoretical Method Calculations Summary Aspects of Pair Production Single Pulse 0.10 0.08 EH t L 0.06 0.04 0.02 0.00 - 40 - 20 0 t @1 mD 20 40 Electric field: E (t) = εEcr sech2 ((t − t0 ) /τ) Three parameters: ε, τ, t0 Outlook Motivation Theoretical Method Calculations Summary Aspects of Pair Production Single Pulse 1. ´ 10 - 11 FH q ,t®¥L 8. ´ 10 - 12 6. ´ 10 - 12 4. ´ 10 - 12 2. ´ 10 - 12 0 -2 -1 0 1 q@ mD Electric field: E (t) = εEcr sech2 ((t − t0 ) /τ) Three parameters: ε, τ, t0 Particle momentum p = (q − eA) 2 Outlook Motivation Theoretical Method Calculations Summary Aspects of Pair Production Single Pulse Particle Number 0.001 10 - 5 10 - 7 10 - 9 10 -11 10 -13 0.001 0.01 0.1 1 10 100 1000 Pulse Length Τ@1 mD Electric field: E (t) = εEcr sech2 ((t − t0 ) /τ) Three parameters: ε, τ, t0 Particle momentum p = (q − eA) R Particle number N ∼ [dq]F (q) Outlook Motivation Theoretical Method Calculations Summary Aspects of Pair Production Enhancement of Number Density Dynamically Assisted Schwinger Effect 100 70 50 æ 30 æ 20 æàò 15 æ ò à æ æ æ æ ò ò ò à à æ à à æ æ ò 0.5 Τ1= 50@1 mD ò Τ1=100@1 mD à Τ1=150@1 mD æ Τ1= 250@1 mD à æ à æ ò à æ à æ æ 0.0 æ æ ò ò 10 æ ò à æ ò æ æ æææææ æ æ òòòòòæ æ ò òæ ò ààààààòæ æ àò à àò æææ àæ ò à ææ æææò à æ æ æ à æ æ 1.0 1.5 2.0 Combined Keldysh Parameter 2.5 Combination of long&strong pulse (ε1 = 0.1, τ1 ) with short&weak pulse (ε2 = 0.01, τ2 ) Dynamical Assistance observable at γ = (mε1 τ2 )−1 R. Schützhold et al.: Phys. Rev. Lett. 101(130404), 2008 Outlook Motivation Theoretical Method Calculations Summary Outlook Aspects of Pair Production Interferences in Multiphoton Regime Dt=21.79@1mD Particle Distribution Particle Distribution Dt=21.79@1mD -2 -1 0 1 Parallel Momentum@mD 2 -2 -1 0 1 Parallel Momentum@mD 10 × ε = 0.02, τ = 3[1/m] 10 × ε = ±0.02, τ = 3[1/m] Blue curve: Combination of short pulses Red curve: 100 × particle distribution rate of single pulse Equally signed fields Alternatingly signed fields Interference → width Interference → height E. Akkerman et al.: Phys. Rev. Lett. 108(030401), 2012 2 Motivation Theoretical Method Calculations Summary Outlook Aspects of Pair Production Interferences in Multiphoton Regime Dt=22.66@1mD Particle Distribution Particle Distribution Dt=22.66@1mD -2 -1 0 1 Parallel Momentum@mD 2 -2 -1 0 1 Parallel Momentum@mD 10 × ε = 0.02, τ = 3[1/m] 10 × ε = ±0.02, τ = 3[1/m] Blue curve: Combination of short pulses Red curve: 100 × particle distribution rate of single pulse Equally signed fields Alternatingly signed fields Interference → width Interference → height E. Akkerman et al.: Phys. Rev. Lett. 108(030401), 2012 2 Motivation Theoretical Method Calculations Summary Outlook Aspects of Pair Production Interferences in Multiphoton Regime Dt=23.53@1mD Particle Distribution Particle Distribution Dt=23.53@1mD -2 -1 0 1 Parallel Momentum@mD 2 -2 -1 0 1 Parallel Momentum@mD 10 × ε = 0.02, τ = 3[1/m] 10 × ε = ±0.02, τ = 3[1/m] Blue curve: Combination of short pulses Red curve: 100 × particle distribution rate of single pulse Equally signed fields Alternatingly signed fields Interference → width Interference → height E. Akkerman et al.: Phys. Rev. Lett. 108(030401), 2012 2 Motivation Theoretical Method Calculations Summary Aspects of Pair Production Combination of Effects Particle Number æ æ 1 ´ 10 - 7 5 ´ 10 - 8 ¶2=+ 0.005 Τ2= 3@1 mD a ÷æ ÷ ÷ ÷ ÷æ ææææ ÷÷ æ÷ ææ ÷÷ ÷÷ ææ æ ÷÷ ÷ ææææææææ æ ææ ÷ ÷ æ æ æ ÷ ææææææ ÷ ææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ æ æ ÷ ææ ÷ æ æ æ æ æ æ æææææææ ÷ ÷ æ æ æ ÷ æ ÷ ÷ æ ÷ ÷ ÷ æ æ ææ æ÷ ÷ ÷ æ÷ ÷ ææææ÷ ÷ ÷ ææ ÷ ÷÷ ÷ ÷æ 1 ´ 10 - 8 5 ´ 10 - 9 æ æ æ ÷ ÷÷æ ÷ æ ææ æ æ ÷æææ æææ ÷÷÷ æ æ æ æ ÷ ÷ ææææ 1 ´ 10 - 9 5 ´ 10 -10 ÷÷÷ ÷÷ ÷÷ ÷ ÷÷÷ ÷ ÷ ÷ 0 2 ÷ ÷ ÷ ÷ ÷÷ ÷ ÷÷ ÷ ÷÷÷ ÷÷ ÷ ÷÷ ÷ ÷ ÷÷ ÷ 4 ÷ ÷ ÷÷ ÷ ÷ ÷÷ ÷÷ ÷÷÷ ÷÷ ÷÷ ÷÷ ÷÷ ÷ ÷÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷÷ ÷÷÷÷÷÷ ÷ 6 8 10 Time Interval@1 mD Red curve: Just short pulses Blue curve: Short pulses with additional long pulse (εmax = 0.1, τ1 ≈ 100[1/m]) Black lines: 1, 10 and 100 × single pulse result, asymptotic blue curve Interferences observable 12 ÷ ÷ ÷÷ ÷ ÷÷ ÷÷÷ ÷÷÷÷÷÷÷ ÷ ÷ ÷ 14 Outlook Motivation Theoretical Method Calculations Summary 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 Outlook Motivation Theoretical Method Calculations Summary Optimal Control Theory Optimization Process EH t L -4 -2 2 4 t @1 mD Single electric field: E (t) = εEcr sech2 ((t − t0 ) /τ) Optimization parameters ε, τ, t0 Model full configuration by sum of single fields Outlook Motivation Theoretical Method Calculations Summary Optimal Control Theory Optimization Process EH t L -4 -2 2 4 t @1 mD Single electric field: E (t) = εEcr sech2 ((t − t0 ) /τ) Optimization parameters ε, τ, t0 Model full configuration by sum of single fields Outlook Motivation Theoretical Method Calculations Summary Optimal Control Theory Optimization Process EH t L -4 -2 2 4 t @1 mD Single electric field: E (t) = εEcr sech2 ((t − t0 ) /τ) Optimization parameters ε, τ, t0 Model full configuration by sum of single fields Outlook Motivation Theoretical Method Calculations Summary Optimal Control Theory Optimization Process EH t L -4 -2 2 4 t @1 mD Single electric field: E (t) = εEcr sech2 ((t − t0 ) /τ) Optimization parameters ε, τ, t0 Model full configuration by sum of single fields Outlook Motivation Theoretical Method Calculations Summary Optimal Control Theory Enhancement in Particle Number particle number ε2 τ2 [1/m] 0.0001 0.0001 0.005 0.02 0.1 2 2 4 2.93e-10 2.98e-11 7.46e-08 5.42e-11 0.0205 0.546 1.18e-04 Optimization with different initial conditions Procedure finds optimal pulse shape Huge gain in particle number Outlook Motivation Theoretical Method Calculations Summary 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 Outlook Motivation Theoretical Method Calculations Outlook Improve in optimization Perform calculations with realistic setups Find formalism to include magnetic fields Summary Outlook Motivation Theoretical Method Calculations Summary Outlook Thank you! supported by FWF Doctoral Program on Hadrons in Vacuum, Nuclei and Stars (FWF DK W1203-N16) Appendix Optimal Control Theory Cost functional J (F (q, t) , G (q, t) , H (q, t) , A) = −γ Z F (q, T )dq + p[A] (15) Appendix Optimal Control Theory Cost functional J (F (q, t) , G (q, t) , H (q, t) , A) = −γ Z F (q, T )dq + p[A] (15) Constraints λF (q, t) = Ḟ (q, t) − W (q, t) G (q, t) = 0 (16) λG (q, t) = Ġ (q, t) + W (q, t) F (q, t) + 2ω (q, t) H (q, t) − W (q, t) = 0 (17) λH (q, t) = Ḣ (q, t) − 2ω (q, t) G (q, t) = 0 (18) Appendix Optimal Control Theory Lagrange function L = J (F , G, H, A) + hλF , µF i + hλG , µG i + hλH , µH i (19) Appendix Optimal Control Theory Lagrange function L = J (F , G, H, A) + hλF , µF i + hλG , µG i + hλH , µH i (19) Adjoint differential equation µ̇F (q, t) = W (q, t) µG (q, t) (20) µ̇G (q, t) = −W (q, t) µF (q, t) − 2ω (q, t) µH (q, t) (21) µ̇H (q, t) = 2ω (q, t) µG (q, t) (22) Appendix Optimal Control Theory Gradient ∇J̃ = δL δp = −2 δA δA Z + Z dq (−µH G + µG H) dq q −A ω ε⊥ ∂t (µG F − µF G − µG ) ω2 (23) (24) Appendix Optimal Control Theory Gradient ∇J̃ = δL δp = −2 δA δA Z + Z dq (−µH G + µG H) dq q −A ω ε⊥ ∂t (µG F − µF G − µG ) ω2 Wolfe conditions J Ak (t) + αk ∇J̃ ≤ J (Ak (t)) + c1 αk h∇J̃ (A) , ∇J̃ (A)i h∇J̃ Ak (t) + αk ∇J̃ , ∇J̃ (Ak )i ≤ c2 h∇J̃ (Ak ) , ∇J̃ (Ak )i (23) (24) (25) (26) Appendix Internal Electric Field Internal electric field Eint (t) due to electron-positron pairs: E (t) = Eext (t) + Eint (t) → Ė (t) = −jext (t) − jint (t) Appendix Internal Electric Field Internal electric field Eint (t) due to electron-positron pairs: E (t) = Eext (t) + Eint (t) → Ė (t) = −jext (t) − jint (t) Internal electromagnetic current jint (t) in the e3 direction: jint (t) = 4e Z d 3k p3 (t) 4 F (q, t) + E (t) (2π) ω (q, t) 3 Z d 3k (2π)3 ω (q, t) Ḟ (q, t) Appendix Internal Electric Field Internal electric field Eint (t) due to electron-positron pairs: E (t) = Eext (t) + Eint (t) → Ė (t) = −jext (t) − jint (t) Internal electromagnetic current jint (t) in the e3 direction: jint (t) = 4e Z d 3k p3 (t) 4 F (q, t) + E (t) (2π) ω (q, t) 3 Z d 3k (2π)3 ω (q, t) Ḟ (q, t) Conduction current stays regular for all momenta! Appendix Internal Electric Field Internal electric field Eint (t) due to electron-positron pairs: E (t) = Eext (t) + Eint (t) → Ė (t) = −jext (t) − jint (t) Internal electromagnetic current jint (t) in the e3 direction: jint (t) = 4e Z d 3k p3 (t) 4 F (q, t) + E (t) (2π) ω (q, t) 3 Z d 3k (2π)3 ω (q, t) Ḟ (q, t) Conduction current stays regular for all momenta! Polarization current exhibits a logarithmic UV-divergence Appendix Internal Electric Field Internal electric field Eint (t) due to electron-positron pairs: E (t) = Eext (t) + Eint (t) → Ė (t) = −jext (t) − jint (t) Internal electromagnetic current jint (t) in the e3 direction: jint (t) = 4e Z d 3k p3 (t) 4 F (q, t) + E (t) (2π) ω (q, t) 3 Z d 3k (2π)3 ω (q, t) Ḟ (q, t) Conduction current stays regular for all momenta! Polarization current exhibits a logarithmic UV-divergence Charge renormalization Appendix Internal Electric Field Internal electric field Eint (t) due to electron-positron pairs: E (t) = Eext (t) + Eint (t) → Ė (t) = −jext (t) − jint (t) Internal electromagnetic current jint (t) in the e3 direction: jint (t) = 4e Z d 3k p3 (t) 4 F (q, t) + E (t) (2π) ω (q, t) 3 Z d 3k (2π)3 ω (q, t) Ḟ (q, t) Conduction current stays regular for all momenta! Polarization current exhibits a logarithmic UV-divergence Charge renormalization Properly renormalized internal electromagnetic current: 4e Z " # 2 eĖ (t) ε⊥ p3 (t) ω 2 (q, t) F (q, t) + Ḟ (q, t) − eE (t) p3 (t) 8ω 4 (q, t) p3 (t) (2π)3 ω (q, t) d 3k Appendix Scattering picture [∂t2 + ω 2 (q, t)]g (q, t) = 0 1-dimensional scattering problem " # h̄2 2 H ψ(x) = − ∂ + V (x) ψ(x) = Eψ(x) 2m x Appendix Scattering picture [∂t2 + ω 2 (q, t)]g (q, t) = 0 1-dimensional scattering problem " # h̄2 2 H ψ(x) = − ∂ + V (x) ψ(x) = Eψ(x) 2m x Formal similarity → Scattering potential: V (t) ∼ −ω 2 (q, t) Reflection coefficient ↔ Produced pairs
