Dynamically assisted Sauter-Schwinger effect in inhomogeneous electric fields

Via the worldline instanton method, we study electron-positron pair creation by a strong electric field of the profile $E/\cosh^2(kx)$ superimposed by a weaker pulse $E'/\cosh^2(\omega t)$. If the temporal Keldysh parameter $\gamma_\omega=m\omega/(qE)$ exceeds a threshold value $\gamma_\omega^{\rm crit}$ which depends on the spatial Keldysh parameter $\gamma_k=mk/(qE)$, we find a drastic enhancement of the pair creation probability -- reporting on what we believe to be the first analytic non-perturbative result for the interplay between temporal and spatial field dependences $E(t,x)$ in the Sauter-Schwinger effect.


I. INTRODUCTION
Despite the tremendous progress of quantum field theory as a fundamental description of nature, our understanding of its non-perturbative properties is still disappointingly incomplete. In quantum electrodynamics (QED), for example, a striking non-perturbative phenomenon is the Sauter-Schwinger effect predicting the creation of electron-positron pairs out of the vacuum by a strong electric field [1][2][3][4]. In case of a constant electric field E, the pair creation probability behaves as ( = c = 1) where E crit = m 2 /q ≈ 1.3 × 10 18 V/m denotes the Schwinger critical field. Unfortunately, the dependence of this non-perturbative phenomenon on the field profile E(t, r) away from the constant field approximation is still mostly terra incognita. There are several results for fields which depend on one coordinate only, such as space x or time t [5] or one of the the light-cone coordinates x ± = t ± x [6]. In these cases, the underlying (Dirac or Klein-Fock-Gordon) equation simplifies to an ordinary differential equation. However, to the best of our knowledge, there are no analytic results for fields E(t, x) which genuinely depend on space x and time t. So far, this case has only been treated numerically via the Wigner formalism [7] or a direct integration of the Dirac equation [8]. This lack of understanding is not only unsatisfactory from a theoretical point of view, a deeper insight into the impact of space-time dependent fields is also highly desirable in view of experimental efforts aiming at a verification of this non-perturbative pair-creation effect [9]. In the following, we venture a first step into this direction and employ the worldline instanton technique [5,[10][11][12][13] in order to study the superposition of a spatial and a temporal field pulse as an example for a genuinely space-time dependent field. * ralf.schuetzhold@uni-due.de

II. WORLDLINE INSTANTON METHOD
Let us start with a brief review of the worldline instanton method. Since the electron spin does not affect the exponent of the pair creation probability, we consider the vacuum persistence amplitude of scalar QED with the covariant derivative D µ = ∂ µ + iqA µ . After analytic continuation to Euclidean space, this functional path integral can be translated into the worldline representation where Dφ Dφ * is replaced by the sum over all closed loops x µ (s) in Euclidean space. Then, via the saddle point method (with the electron mass m playing the role of the large expansion parameter), the pair creation probability can be estimated as with the worldline instanton action Hereẋ µ = dx µ /ds denotes the derivative of a closed x µ (s = 0) = x µ (s = 1) worldline loop x µ (s) as a solution of the instanton equations

III. SUM OF SAUTER PULSES
Now let us apply the worldline instanton method to a space-time dependent electric field consisting of a strong spatial Sauter [2] pulse ∝ E and a weaker temporal Sauter pulse ∝ E where both field arXiv:1407.3584v1 [hep-th] 14 Jul 2014 Furthermore, in order to be in the non-perturbative regime, we assume slowly varying pulses ω, k m. For convenience, we introduce the spatial and temporal Keldysh parameters via The Euclidean vector potential reads As a result, the instanton equations (5) assume the form and are analogous to the planar motion of a charged particle in a magnetic field B(r) = B(x, y)e z . Due to E /E 1, the second term is negligible unless cos 2 (ωx 0 ) becomes very small near the poles of E(x 0 , x 1 ) at ωx 0 = ±π/2. Away from these poles, we may omit the second term and the above equations can be integrateḋ As mentioned after Eq. (5), the constant a is given bẏ x νẋ ν = a 2 = const. The other integration constant b determines the velocityẋ 0 just before (or just after) crossing the x 0 -axis, see Fig 1 Near the poles ωx 0 ≈ ±π/2, on the other hand, the second term becomes important. Similar to the reflection of a charged particle at the region of a very strong magnetic field, the instanton trajectory is basically reflected by the "wall" at ωx 0 ≈ ±π/2 if it reaches out far enough. Since this reflection occurs during a very short proper time ∆s, we may neglect the regular terms in Eq. (9) and keep only the divergent contributions. Then, the equation for x 1 can be integrated approximately tȯ and thus the equation for x 0 becomes As a result, the perpendicular velocityẋ 0 is reversed by that reflection while the parallel velocityẋ 1 has the same valueẋ in 1 before and after the reflection.

IV. TUNNELLING PROBABILITY
Again due to E E , the instanton action reads In order to calculate the above integral, we split the closed loop into four quarters: from x 1 = 0 to the spatial turning point x * 1 , from x * 1 to x 1 = 0, from x 1 = 0 to −x * 1 , and finally back to x 1 = 0, see Fig... Since each quarter yields the same contribution, we get where x * 1 denotes the spatial turning point given by i.e., the zero of the square root in the integral in Eq. (14) where dx 1 /dx 0 = 0. The constant a is determined bẏ x νẋ ν = a 2 and x µ (s = 0) = x µ (s = 1) which gives The remaining integration constant b depends on the frequency ω. If ω is too small and thus the poles at ωx 0 = ±π/2 are too far away, the instanton trajectory is not reflected at all and thus we have b = 0. In case of reflection, the integration constant b is non-zero and determined by the implicit condition Together with the above equations for x * 1 , a, and b, Eq. (14) is the main result of this paper.
The threshold condition b = 0 translates into If the frequency is too low γ ω < γ crit ω , the instanton trajectory is basically not affected by the poles at ωx 0 = ±π/2 leading to b = 0 and thus the weak temporal pulse ∝ E has negligible impact. In this case b = 0, we get x * 1 = artanh(γ k )/k and all the integrals can be carried our analytically, yielding the same results as for a static Sauter pulse, which have already been obtained in [5]. If the frequency exceeds this treshold value γ ω > γ crit ω , on the other hand, the instanton trajectory is reflected at the poles (i.e., b > 0) and thus the instanton action (14) is reduced by the weak temporal pulse ∝ E , leading to a significant enhancement of the pair creation probability. In the homogeneous limit γ k ↓ 0, the threshold value (18) approaches γ crit ω = π/2 consistent with the results of [14]. For γ k ↑ 1, the threshold γ crit ω scales as γ crit ω ≈ 1 − γ 2 k , i.e., very small frequencies ω can have a significant impact in this case.
Unfortunately, due to the implicit nature of the condition for b, we cannot provide a closed analytical expression for S. However, near but above threshold, we can Taylor expand the involved quantities and obtain the following approximate formula for the instanton action The zeroth order S 0 = [m 2 /(qE)]2π/(1+ 1 − γ 2 k ) is just the result in the static case [5] and is valid below and at threshold γ ω ≤ γ crit ω . For γ k = 0, we recover Schwinger's result [4] for a constant field (1). Above threshold γ ω > γ crit ω on the other hand, the action is reduced by the scond-order term ∝ [γ ω − γ crit ω ] 2 .

V. CONCLUSIONS
Via the worldline instanton technique, we derived an analytical estimate (14) for the electron-positron pair creation probability (3) induced by an electric field (6) which geniunely depends both on space and on time. Superimposing a strong spatial pulse by a weak temporal pulse (6), we found that the weak pulse is negligible for small frequencies γ ω ≤ γ crit ω but can enhance the pair creation probability significantly (dyamically assisted Sauter-Schwinger effect) for larger frequencies γ ω > γ crit ω with the threshold (18) depending on the spatial Keldysh parameter (7).
In the homogeneous limit γ k ↓ 0, this threshold γ crit ω converges to π/2 in accordance with [14]. If the spatial Keldysh parameter approaches unity γ k ↑ 1, on the other hand, the threshold γ crit ω goes to zero. In this case γ k ↑ 1, the size of the spatial Sauter pulse is barely enough to produce electron-positron pairs (since the electrostatic potential difference is just above the gap of 2mc 2 ) and the instanton loop becomes very large, cf. x * 1 = artanh(γ k )/k for b = 0. Quite intuitively, even comparably small frequencies (leading to poles at large distances to the origin) can have an impact in this limit.