Lifting shell structures in the dynamically assisted Schwinger effect in periodic fields

The dynamically assisted pair creation (Schwinger effect) is considered for the superposition of two periodic electric fields acting a finite time interval. We find a strong enhancement by orders of magnitude caused by a weak field with a frequency being a multitude of the strong-field frequency. The strong low-frequency field leads to shell structures which are lifted by the weaker high-frequency field. The resonance type amplification refers to a new, monotonously increasing mode, often hidden in some strong oscillatory transient background which disappears during the smoothly switching off the background fields, thus leaving a pronounced residual shell structure in phase space.


Introduction
For many decades the Schwinger effect [1] has been considered crucial for testing non-perturbative QED as a pillar of the standard model of particle physics in the strong-field regime. An obvious motivation for the broad interest can be seen in the formal structure and numerical smallness of the decay rate positrons which screen the original field. Schwinger's seminal formula was R ∝ E 2 0 exp (−πE c /E 0 ) in leading order, where the scale is set by the electron's mass m and charge e reading E c = m 2 /e (we use units with = c = 1) first introduced by Sauter [2]. Presently achievable long-living fields in the laboratory are weak compared to E c , E 0 E c . Accordingly, the Schwinger rate is exponentially small and has escaped an experimental verification until now.
The fields created in peripheral relativistic heavy-ion collisions are short-lived, of the order of a few fm/c [3], thus not suitable for an exploration of the original Schwinger effect which is for a spatio-temporal constant field. Nevertheless, a plethora of interesting strong-field effects are under consideration [4]. For instance, magnetars are astrophysical objects with strong fields which could serve for identifying Schwinger type effects [5,6]. One should also recall that the Schwinger effect for chromoelectric fields is employed in phenomenological models of particle production in strong interaction processes [7][8][9].
Two further aspects highlight the role of the Schwinger effect. (i) It is conceivable that QED is an effective weak-field theory which breaks down for fields of the order of E c . (ii) A long-living field O(E c ) can not be achieved due to screening processes and cascades which consume and transfer the original field energy into other degrees of freedom, as discussed in [10][11][12][13]. We mention further that the decay of a strong external field due to particle production is not a privilege of QED, but is generic. For instance, the Hawking radiation off a horizon is a famous example w.r.t. gravitational fields [14,15].
In the course of seeking set-ups which could offer the opportunity to verify the above static Schwinger effect, the idea has been explored that ultra-intense laser fields could enable the detection of the dynamical Schwinger effect [16].
For instance, in the antinodes of two counter propagating, linearly polarized laser beams we have a periodic (frequency ν), essentially electric field E(t) with spatial homogeneity length of O(1/ν) which is, for optical lasers, much larger than the Compton wave length λ C = 2π/m of the electron. The prospects of e + e − pair production in dependence on E 0 and ν have recently been analyzed [17]. While in a plane wave or null field the pair production rate is zero [1], a focused laser field provides a non-zero rate, as pointed out in [18]. However, the rate is still very small, unless such ultra-intense laser fields as envisaged at ELI [19] are at our disposal. Finally we mention Ref. [20], where the mimicking of the dynamical Schwinger effect is accomplished in an all-optics setup of a wave guide with curved optical axis.
While the Schwinger effect is originally related to a tunneling process, which escapes the standard perturbative QED described by Feynman diagrams, in the dynamically assisted Schwinger effect [21][22][23] the tunneling is combined with a multi-photon process, thus potentially enhancing the pair production rate significantly. The essence is a combination of a strong field (may be slowly varying) with a weak field which introduces, in particular, a high-frequency component.
Various combinations have recently been investigated to look for optimum parameter settings. In Refs. [24,25], the superposition of two Sauter pulses was considered; Ref. [26] analyzed the superposition of a strong Sauter pulse with various other weak-pulse shapes. The Sauter pulse has a d.c. component and can hardly be shaped with present laser technologies. It is therefore tempting to investigate the rate enhancement in the superposition of two periodic fields, e.g.
as recently done also in [27,28]. Such a situation seems to be more realistic in respect to a suitable combination of XFEL and optical laser beams. The opportunities at plain XFEL beams are considered in [29]. References [30,31] consider the frozen-out early-time population of low-momentum electrons (positrons) in various field configurations, while we consider the residual phase space occupation with a realistic (smooth) switching on/off the combined fields.
Our framework is the kinetic equation for the single-particle distribution derived in [32], see also [17,[33][34][35]. Despite the ostensible simplicity of the kinetic equation and the possibility to give a compact expression for its solution, it is fairly intransparent due to the non-linear and non-Markovian character. Therefore, it is hardly possible to read off in a simple manner the dependence of the solution on the field parameters. WKB type approaches [23,36,37], the world line formalism [38] and optimization theory [25] have been developed to gain further insights into the pair production process. We here rely on numerical solutions of the kinetic equation to elucidate parameter regions where the dynamically assisted Schwinger effect in two periodic fields, which are smoothly switched on and off, leads to a significant enhancement of the rate.
The numerical simulations (section 2) are accompanied and interpreted by analytical approximations (section 3) explaining the shell structure in phase space. This is supplemented by a systematic scan of parameter dependence (section 4). Our summary is given in section 5.

Solutions of quantum kinetic equations
The quantum-kinetic equation without back reaction for the time (t) evolution of the one-particle distribution function f (cf. [39] for a discussion of the meaning of f ) summed over spin projections is given either as an integro-differential equation [17,32,40] or equivalently as a system of three coupled differential equationṡ where u and v denote auxiliary quantities and Θ, ω and Q are defined by with A(t) and E(t) = −Ȧ(t) being the z component of the vector potential and the electric field, respectively. Our field is thus assumed spatially homogeneous, pointing along the z direction. Consequently, p denotes the momentum (e.g. of electrons) parallel to the z axis and p ⊥ the momentum perpendicular to it; ⊥ = m 2 + p 2 ⊥ is the transverse energy; p and p ⊥ are components of the three-vector p. From here on, we set t 0 = 0 and employ the initial conditions (1)  In what follows we consider the synchronized superposition of a slow strong field ("1") and a fast weak field ("2") with potential where ν = 2π/T is the frequency of the slow field and N the ratio of the frequencies chosen to be integer. We utilize a C ∞ envelope function (which is infinitely often differentiable) which is chosen as x for x > 0. The field (6) is therefore smoothly switched on and off for a suitable choice of the ramping ("ramp") interval from 0 to τ ramp and deramping interval from (6) and thus also the electric field acts for the finite duration τ pulse . We have chosen τ ramp = 5 · 2π and τ f.t. = 50 · 2π meaning five (fifty) oscillations of field "1" for ramping and deramping (the flat-top interval). Thus, the field configuration (6) is a special model for the spatial homogeneity region of a common antinode of several (at least four) pair-wise counterpropagating synchronized beams. In the present study we focus on time scales and field strengths similar to those in [24]: E 1 = 0.1E c and ν = 0.02m, E 2 = 0 . . . 0.05E c and N = 10 . . . 50. That means the individual Keldysh parameters are γ 1 = (E c /E 1 )(ν/m) = 0.2 and . While this parameter regime does not exactly match presently available XFEL and intense laser technology, it allows for an easy numerical treatment of the kinetic equations (and comparison with available literature). In [16], γ 1 is referred to as tunneling regime, while γ 1 is the multi-photon regime.
Solutions of (2a) for η = 1 (i.e. with Pauli blocking) and for p = 0 are exhibited in Fig. 1 for νt > τ pulse where, according to (2),ḟ = 0 since E(νt > τ pulse ) = 0. (That means, f (νt > τ pulse ) represents the residual phase space distribution within the considered framework.) The middle panels in Fig. 1 exhibit the residual phase space distributions in p ⊥ direction at p = 0 for the field (6), while the left (right) panels are for the strong (weak) field alone.
One observes pronounced peaks which continue (albeit at different positions) when displaying other cuts in the p ⊥ -p plane or sharp ridges in contours over the p ⊥ -p plane. These peaks or ridges are referred to as shell structures, already described, for a single periodic field, in various previous papers [17,[41][42][43], originally found in [16] and further elaborated in [44][45][46][47]. From Fig. 1 one infers that the residual phase space occupations for any one of the two field contributions that appear in (6) are much smaller than the phase space occupations for the superposition of both fields. For instance, shell ("1") = 341 (left panels in Fig. 1) with peak altitude 2.5 × 10 −10 becomes, due to the impact of the field "2", shell ("1"+"2") = 341 (middle panels in Fig. 1) with peak altitude 1.5 × 10 −4 or 2.0 × 10 −3 depending on N . The peak pattern is dominated by the slow strong field "1", where "2" lets even shells additionally appear, e.g.
shells ("1"+"2") = 342, 344 etc. for N = 25, which are not visible for the field "1" alone (cf. left panels). Due to the comparatively high frequency ν 2 = N ν 1 of the field "2", the shell numbers ("2") are much smaller and the corresponding peaks are much higher, but individual structures resembling the right panels in Fig. 1 are not evident in the middle panels. The assistance of field "2" consists obviously in lifting the pattern governed by field "1". The found non-linear amplification is huge -much larger than for the superposition of two Sauter pulses in [24]. References [30,31] also report very strong amplification effects for periodic fields, but for a very special shape function K and a different early-time mode. Other field configurations are considered in [48][49][50], where relatively strong effects in the momentum dependence and particle rate are found by modifying a Gaussian electric field by a subcycle sinusoidal field.

Shell structure and shell shape
To arrive at a qualitative understanding of the numerical results of the previous section we resort to the low-density approximation (exponentiating results in the Markovian approximation [51]) which discards the Pauli blocking by setting η = 0 in (1)

On-shell occupancy
On shell , (13) inserted in (8) delivers where G(p ( ) , t) and H(p ( ) , t) are bounded oscillating functions depending on p ( ) . The peak height of a shell at position p ( ) increases accordingly quadratically with time (first term in (16)), being periodically modulated with a linearly increasing (second term) and a constant amplitude (last term). Due to the superposition of these modes the actual transient time evolution can be quite involved but lacks a physical meaning, as recalled in [39]. We observed in our numerical simulations based on (2), however, that after smoothly switching off the field, the peak height f (p ( ) , νt > τ pulse ) coincides with the first term in (16): The numerical evaluation of F (p ( ) ) according to (12) and using as flat-top interval time agrees well with numerical results of the peak heights by integrating (2). Thus can be identified with the residual on-shell occupancy f (p ( ) ).

Shell shape
For a more detailed account of the shell shape, let us expand (13) Since the full width at half maximum (FWHM) of sin 2 (xt)/x 2 evolves as ∝ 1/t, the FWHM of f (p i.e. despite the quadratic growth of the shell height, the shrinking causes a linear increase with time of the line integrated density. In fact, the residual density is determined by (18) with t → t f.t. , as our numerical investigations based on (2) show. Neglecting the pedestrials under the sharp peaks (cf. Fig. 1) the residual density n = 2π dp dp ⊥ p ⊥ f (p) can be estimated by summing over all shells when neglecting the anisotropy in phase space by setting p

Survey on the parameter dependence
After having identified the decisive role of the Fourier coefficients F defined in (12) for shell heights and widths and residual density we proceed with a brief survey on some systematics. Figure 2 exhibits the Fourier coefficients for shells = 341 and = 342 which are the lowest allowed shells for both the field (6) (cf. middle column in Fig. 1) and the slow strong field alone (cf. left column in Fig. 1). Let us first consider shell 341. One observes for sufficiently large values of N and field strength E 2 of the fast weak field a strong increase due to the action of the faster field. The blue line is for the slow strong field alone, i.e.  = 342 (right) and various field intensities E 2 (green squares: E 2 = 0.01Ec, red triangles: E 2 = 0.02Ec, cyan diamonds: E 2 = 0.05Ec). The blue lines are for E 2 = 0, i.e. the field "1" alone. Note the ∆N = 2 staggering for the even shell (right). spectra (calculated by means of (1, 2)) for field "1" alone and for fields "1+2" look similar to the respective panels in Fig. 1 with (i) more closely spaced peaks due to smaller ν 1 and (ii) peak maxima somewhat reduced. That means our amplification is robust, as also under variations of E 1 (keeping E 1 > E 2 ), as confirmed by an analysis of the Fourier coefficients.
As anticipated in section 3, enlarging τ f.t. makes the peaks (shells) higher and sharper (cf. (16,17)), while the pedestrials (accessible by (1, 2)) hardly change. The ramping interval τ ramp must not be too short to avoid unwanted spikes bracketing the electric field; larger values of τ ramp can be accomodated in an enlarged effective τ f.t. .
Finally, we mention that non-integer values of N result in a similar (albeit nonresonant) amplification, however, with a more involved phase space distribution which is no longer accessible by the harmonic analysis in section 3.

Summary
In the present work we have considered the dynamically assisted Schwinger effect for resonant periodic fields within the framework of the quantum kinetic equation. We have isolated a non-linear parametric mechanism which increases the pair creation rate by many orders of magnitude when combining suitably a strong low-frequency field with a weak high-frequency field compared to the rates if both fields acted alone. Both fields are subcritical with respect to frequencies and field strengths. In contrast to previous work, which often deals with instantaneous switching off, the residual phase space distribution exhibits a distinct shell structure which survives the involved transiently oscillating pattern during the time-limited action of the periodic fields. The occupancy of the shells grows linearly with the flat-top time, while the shell peaks grow quadratically due to a new resonance like behaviour. The obvious motivation for such a configuration of combined two periodic fields is the superposition of the European XFEL with an ultra-intense optical laser system as envisaged in HIBEF [53]. For an easy numerical treatment, however, we have selected, in the present case study, patches in the field-strength vs. frequency space which, while located in the tunneling and multi-photon domains respectively, are quite different from more realistic values, for example those in table 1 in [29]. Based on the systematics presented here, we argue that no qualitative changes arise when moving towards parameters being more representative for an optical laser-XFEL combination.