Pulse shape dependence in the dynamically assisted Sauter-Schwinger effect

Transcription

1 Pulse shape dependence in the dynamically assisted Sauter-Schwinger effect Joachim Sicking, Nikodem Szpak, Ralf Schützhold Fakultät für Physik, Universität Duisburg-Essen 20 March 2014, DPG Tagung Berlin THE IDEA Small perturbations can enhance the pair creation probability significantly! 1

2 Sauter-Schwinger effect Sauter-Schwinger effect: strong constant electric field E creation of e e + pairs { } { P e + e exp π m2 = exp π E } S, qe E Same result for slow (ω 0) homogeneous pulses, e.g. E(t) = E 1 cosh 2 (ωt) Characteristic field strength E S V/m very large this fundamental QED prediction not confirmed experimentally yet! Nikodem Szpak, Univ. Duisburg-Essen Pulse shape dependence in the dynamically assisted Sauter-Schwinger effect 2

3 Dynamically assisted Sauter-Schwinger effect Addition of small & fast perturbation E(t) = E 1 cosh 2 (ω 1 t) + E 2 cosh 2 (ω 2 t) with E 2 E 1 and ω 2 ω 1, significantly enhances the pair creation probability { } P e + e exp π m2 χ qe 1 with χ 2 π π 1 2γ c and γ c = (mω 2 )/(qe 1 ) combined Keldysh parameter ( ) 2 ( ) π π + arcsin 2γ c 2γ c γ c π/2 χ 2/γ c < 1 enhances the pair production exponentially! Nikodem Szpak, Univ. Duisburg-Essen Pulse shape dependence in the dynamically assisted Sauter-Schwinger effect 3

4 Dynamically assisted Sauter-Schwinger effect (cd.) To understand better χ we consider E(t) = E 1 cosh 2 (ω 1 t) + E 2 f(ω 2 t) with a) f(t) = sin(t) b) f(t) = exp( t 2 ) E t E t t Main result: P e + e depends sensitively on the form of the function f(t)! mainly via growth rate in imaginary time τ = it (related to parameters of instanton appearing also as turning points in the WKB-method) Nikodem Szpak, Univ. Duisburg-Essen Pulse shape dependence in the dynamically assisted Sauter-Schwinger effect 4

5 Dirac equation with time-dependent electric field In external-field-qed: scattering operator in Fock space (describing completely particle creation and annihilation) Bogoliubov coefficients α k and β k classical Dirac equation as reflection R k and transmission T k amplitudes The probability for electron-positron pair creation: P e + e = β k 2 d N k Dirac equation in 1+1 dimensions Fourier transformed in space i t ψ k = ([k + qa(t)]σ x + mσ z ) ψ k = H k ψ k with the electric field in the temporal gauge E = E(t)e x = A Expanding the wave-function into instantaneous eigenvectors of H k with phases and eigenvalues ϕ k (t) = ψ k (t) = α(t)e +iϕ k(t) u + k (t) + β(t)e iϕ k(t) u k (t) t t 0 dt Ω k (t ), Ω k (t) = m 2 + [k + qa(t)] 2 Nikodem Szpak, Univ. Duisburg-Essen Pulse shape dependence in the dynamically assisted Sauter-Schwinger effect 5

6 Probability of pair creation leads to Riccati equation for R k = β k /α k ( ) Ṙ k (t) = Ξ k (t) e 2iϕk(t) + Rk(t)e 2 2iϕ k(t), Ξ k (t) = Ω k (t) 2Ω k (t) For small R k (t) 1 we skip Rk 2(t) and integrate with R k( ) = 0 to R k := R k (+ ) + Ξ k (t) e 2iϕ k(t) dt Deform integration contour to complex plane: Ξ k (t) has poles at t i C +, φ k (t) is analytic Cauchy theorem sum over the residua plus the shifted contour (to Im(t) = λ > 0) R k = t i C i e 2iϕ k(t i ) + Ce λω 0 The result is dominated by the lowest laying pole t of Ξ(t) satisfying R k e 2 Im ϕ k(t ) e 2χ qa(t ) = k ± im Nikodem Szpak, Univ. Duisburg-Essen Pulse shape dependence in the dynamically assisted Sauter-Schwinger effect 6

7 Poles For the original Sauter pulse E(t) = E 1 cosh 2 (ω 1 t) the pole is at t m ( i + k ) qe 1 m For the assisted case E(t) = E 1 cosh 2 (ω 1 t) + E 2 cosh 2 (ω 2 t) (with E 2 E 1, ω 2 ω 1 ) another pole exists at t iπ 2ω 2 which strongly contributes to R its position does not depend on E 2! Here, we look at other perturbations with poles depending on E 2 /E 1 : Goal: maximize R E(t) = E(t) = E 1 cosh(ω 1 t) + E 2 sin(ω 2 t) E 1 cosh(ω 1 t) + E 2e (ω 2 2t) (Sinus profile) (Gauß profile) Nikodem Szpak, Univ. Duisburg-Essen Pulse shape dependence in the dynamically assisted Sauter-Schwinger effect 7

8 The Sinus profile The positions of the poles satisfy τ ɛ cos(τ ) = iγ c with τ = ω 2 t, ɛ = E 2 /E 1 1 and γ c = (mω 2 )/(qe 1 ) i Im Τ Beyond the main pole τ iγ c there appears a series ( τ (n) i log ɛ + π n + 1 ) Τ Poles τ and τ (n) as functions of ɛ = 10 s, s = 15 (red)... 1 (blue) γ c = 15 (constant) For γ c < γ c,critical log ɛ : the exponent dominated by the main pole χ π/4 For γ c > γ c,critical : we have Im(τ ) < Im(τ (n) ) poles τ (n) push τ down χ decreases below π/4 dynamical assistance of the small perturbation! Nikodem Szpak, Univ. Duisburg-Essen Pulse shape dependence in the dynamically assisted Sauter-Schwinger effect 8

9 The Sinus profile (cd.) 30 4 i Im Τ Χ Τ Poles τ and τ (n) as functions of ɛ = 10 s, s = 15 (red)... 1 (blue) χ as function of γ c for ɛ = 10 s, s = Γ c Nikodem Szpak, Univ. Duisburg-Essen Pulse shape dependence in the dynamically assisted Sauter-Schwinger effect 9

10 The Gauß profile 8 In this case, the poles satisfy π τ + 2 ɛ erf(τ ) = iγ c Here, beyond the main pole τ iγ c we have a series τ (n) i πn log ɛ + 2 log ɛ i Im Τ Re Τ Poles τ and τ (n) as functions of ɛ = 10 s, s = 15 (red)... 1 (blue) γ c = 5 (constant) For γ c < γ c,critical log ɛ : the exponent dominated by the main pole χ π/4 Nikodem Szpak, Univ. Duisburg-Essen Pulse shape dependence in the dynamically assisted Sauter-Schwinger effect 10

11 For γ c > γ c,critical we find log ɛ χ 2γ c for γ c γ c,critical 1 ( log ɛ γ c ) arcsin ( ) log ɛ 2γ c log ɛ γ c 4 Χ χ (γ c ) for ɛ = 10 s, s = 3, 6, 9, 12, 15 (dotted) compared to analytical approximation (solid) Γ c Nikodem Szpak, Univ. Duisburg-Essen Pulse shape dependence in the dynamically assisted Sauter-Schwinger effect 11

12 Discussion and outlook We observe universal dependence for γ c γ c,critical χ C(ɛ)/γ c C(ɛ) depends on the weaker-and-faster pulse, related to the growth rate of E(iτ) o) f(t) = cosh 2 (t) f(iτ) (τ τ 0 ) 2 a) f(t) = sin(t) f(iτ) exp(τ) b) f(t) = exp( t 2 ) f(iτ) exp(τ 2 ) Calculation of τ requires inversion τ = ia 1 (iγ c ) for A(t) = t 0 E(t ) dt For large γ c only the rate of growth of f(iτ) is relevant (choice of A(t)!) and therefore, asymptotically ɛf(iτ ) iγ c leads to o) τ τ 0 = π/2... a) τ log(γ c /ɛ) +... b) τ log(γ c /ɛ) +... Nikodem Szpak, Univ. Duisburg-Essen Pulse shape dependence in the dynamically assisted Sauter-Schwinger effect 12