Dynamically assisted Sauter-Schwinger effect

Transcription

1 Dynamically assisted Sauter-Schwinger effect Ralf Schützhold Fachbereich Physik Universität Duisburg-ssen Dynamically assisted Sauter-Schwinger effect p.1/16

2 Dirac Sea Schrödinger equation (non-relativistic) i t ψ = 2 2m 2 ψ +Vψ = p2 2m +V Dirac equation (relativistic) γ µ (i µ +qa µ )Ψ = mcψ = V ± c 2 p 2 2 c 4 positive and negative energy levels! filled up in vacuum (Pauli principle) holes: positrons (prediction!) Dynamically assisted Sauter-Schwinger effect p.2/16

3 Sauter-Schwinger ffect Constant electric field potential V(x) = qx tilt of level spectrum tunnelling from Dirac sea! Creation of e + e pairs out of the vacuum { P e+ e exp L V } exp { 2mc2 c q O(mc 2 } ) c Dynamically assisted Sauter-Schwinger effect p.3/16

4 Critical Field Strength xact calculation yields exponent P e+ e exp { π c3 m 2 } { = exp π } S q Non-perturbative QD vacuum effect ( later) m 2 S = c3 q V/m Corresponds to intensity I S = O(10 29 W/cm 2 ) Planned ultra-strong lasers I = O(10 26 W/cm 2 ) xponential suppression exp{ π S /} = O(10 61 ) Four-volume factor(λ laser /λ Compton ) Further enhancement? Dynamically assisted Sauter-Schwinger effect p.4/16

5 Assisted Tunnelling Borrow idea from quantum optics: assisted tunnelling Shorter way to tunnel reduction of exponent Tunnelling: non-perturbativeexp{ O(1/[q])} vs multi-photon: perturbativeo([q] 2n ) R. S., H. Gies, G. Dunne, Phys. Rev. Lett. 101, (2008) Dynamically assisted Sauter-Schwinger effect p.5/16

6 Non-constant Fields? Problem: const, e.g., = (t), = (x) or = (t,r) Very few analytic solutions (Dirac operator), e.g., (t) = cosh 2 (Ωt) e z F. Sauter, Z. Phys. 69, 742 (1931); ibid. 73, 547 (1931). Keldysh parameterγ = 1/a (note: S,Ω m ) γ = mω { q P exp{ πs /} : γ 1 e + e (q/[mω]) 4m/Ω : γ 1 Non-perturbative vs multi-photon (perturbative) SLAC experiment: D.L. Burke et al, Phys. Rev. Lett. 79, 1626 (1997). Different time scales? Dynamically assisted Sauter-Schwinger effect p.6/16

7 Toy Model Strong & slow plus weak & fast pulse with S ε and Ω m ( = c = 1) (t) = cosh 2 (Ωt) e ε z + cosh 2 (t) e z xact solution for each pulse separately: P e+ e (,Ω) exp{ π S /} 1 P e+ e (ε,) (qε/[m]) 4m/ 1 Non-perturbative vs multi-photon (e.g., SLAC) Combined impact? Dynamically assisted Sauter-Schwinger effect p.7/16

8 Worldline Instanton Method uclidean path integral with closed worldlines dχ 1 [ m2ẋ2 ] Γ[A µ ] = χ e χ Dx exp dτ 4χ +iqa ẋ 0 x(0)=x(1) Saddle-point method with m as large parameter P e+ e I(Γ[A µ ]) e A inst Worldline instantonsx µ (τ) as tunnelling events mẍ µ = iqf µν ẋ ν Problem: analytic continuation of F µν and solution... (t) = cosh 2 (Ωt) e z + 0 ε cosh 2 (t) e z Dynamically assisted Sauter-Schwinger effect p.8/16

9 nhanced Tunnelling Instanton action A inst tunnelling P e+ e e A inst Combined Keldysh parameter γ = m q xponential enhancement forγ > π/2 inst Γ (t) = cosh 2 (Ωt) e z + ε cosh 2 (t) e z R. S., H. Gies, G. Dunne, Phys. Rev. Lett. 101, (2008) Dynamically assisted Sauter-Schwinger effect p.9/16

10 Scattering Analogy Scalar wave equation for = (t) ( ) d 2 dt +[k 2 x qa(t)] 2 +k 2 2 φ k = 0 Schrödinger scattering problem ( 1 d 2 ) 2M dx +V(x) Ψ(x) = Ψ(x) 2 Analogy t x and φ k (t) Ψ(x) as well as 2M[ V(x)] [k x qa(t)] 2 +k 2 2 > 0 scattering above the barrier complex turning points in WKB φ out k (t) = α k exp{ i k t}+β k exp{+i k t} Reflection particle creation P e+ e k = β k 2 Dynamically assisted Sauter-Schwinger effect p.10/16

11 Momentum Dependence Strong & slow: = S /4 Ω = 10 4 ev normal turning pointt R Weak & fast: ε = /10 = ev additional anomalous WKB turning point t P e+ e k = c exp{ 2ϕ(t )}+c exp{ 2ϕ(t )} 2 C. Fey and R. S., Phys. Rev. D 85, (2012). Numerical results (quantum kinetic approach) p M. Orthaber, F. Hebenstreit, R. Alkofer, Phys. Lett. B 698, 80 (2011). Dynamically assisted Sauter-Schwinger effect p.11/16

12 Realistic Model Plane-wave X-ray + laser focus Imaginary part of polarisation tensor α QD P e+ e I(Π µν ) q exp π(π 2) For X-ray frequency 2m { m2 G. V. Dunne, H. Gies, R. Schützhold, Phys. Rev. D 80, (2009). } (π 2) q still purely non-perturbative (cf. conservation laws) huge enhancement (exponent...) exp{ (π 2)m 2 /(q)} exp{ πm 2 /(q)} Observable? Dynamically assisted Sauter-Schwinger effect p.12/16

13 Alternatively: Wave Functions Interaction picture: laser V, X-ray A(t, r) Ĥ 0 = ˆΨ ( iα β +V ) ˆΨ Ĥ 1 = qˆψ α A(t,r)ˆΨ lectron-positron pair creation amplitude A IJ = q d 4 xu I (r)α A(t,r)v J(r)e i It+i J t Overlap integral of Whittaker functions A 2 IJ ( I + J = 0) exp{ πm 2 /(q)} A 2 IJ( I + J = 2m) exp{ (π 2)m 2 /(q)} Confirmation of exponential enhancement Dynamically assisted Sauter-Schwinger effect p.13/16

14 Observability? 1e+18 1e+15 1e+12 1e+09 1e e-06 # of pairs, /m = 1 # of pairs, /m = 2 catalysis ratio, /m = 1 catalysis ratio, /m = 2 1e e/m X-ray photons per pulse with 1 MeV (/m = 2) laser: I W/cm 2 over L = 1µm onee + e pair after pulses (one day with 1 Hz) Dynamically assisted Sauter-Schwinger effect p.14/16

15 Summary 1e+18 1e+15 1e+12 1e+09 1e # of pairs, /m = 1 # of pairs, /m = 2 catalysis ratio, /m = 1 catalysis ratio, /m = 2 1e-06 1e e/m 2 Dynamically assisted Sauter-Schwinger effect strong & slow+weak & fast Sauter pulse laser focus+plane wave X-ray general field profiles = (t,x)??? inst Γ R p Dynamically assisted Sauter-Schwinger effect p.15/16

16 So What? Perturbation theory Feynman diagrams out Ŝ in = a 0 +a 1 q +a 2 q For example q k q + k q q k k q q q q +... k k q q q +... Sauter-Schwinger: non-perturbative QD effect P e+ e exp { π c3 m 2 } { = exp π } S q no Taylor expansion in q or Cf. quantum chromo-dynamics (QCD) ffect of interactions between e + and e? Dynamically assisted Sauter-Schwinger effect p.16/16