Signatures of the Schwinger mechanism assisted by a fast-oscillating electric field

TL;DR

This paper investigates the nonperturbative Schwinger mechanism enhanced by a high-frequency electric field, deriving analytical expressions for pair production rates and proposing an experimental setup to observe the effect.

Contribution

It provides the first analytical formula for the density rate of electron-positron pairs in a dynamically-assisted Schwinger process, bridging tunneling and multi-photon regimes.

Findings

Derived asymptotic expressions for pair production distribution.

Established a nonperturbative dependence of particle density on electric fields.

Proposed an experimental setup to observe the effect.

Abstract

The spontaneous production of electron-positron pairs from the vacuum--in a field configuration composed of a high-frequency electric mode of weak intensity and a strong constant electric field--is investigated. Asymptotic expressions for the single-particle distribution function ruling this nonperturbative process are established by considering the low-density approximation in the Boltzmann-Vlasov equation. An analytical formula for the density rate of yielded particles is established which is shown to manifest a nonperturbative dependence on both the strong and weak electric fields and to interpolate between the tunneling and multi-photon regimes. It is shown that--under appropriate circumstances--the produced plasma of electrons and positrons might reach densities for which their recombinations into high-energy photons occurs copiously. On the basis of this feature, an experimental setup for observing the dynamically-assisted Schwinger effect is put forward.

Equations

$$\begin{array}{rl}\text{mathpzc}{A}^{\mu }(t)& =-\left[E_{s}t+\frac{E_w}{\omega }\sin (\omega t)\right]{♭}^{\mu }\\ & \times \Theta (t+T/2)\Theta (T/2-t),\end{array}$$

$$\begin{array}{rl}& \dot{W}(\mathbf{p};t)=Q(\mathbf{p},t){\int }_{-\infty }^{t}d\tilde{t}Q(\mathbf{p},\tilde{t})\\ & \times \left[1-W(\mathbf{p};\tilde{t})\right]\cos \left[2{\int }_{\tilde{t}}^{t}d{t}^{\prime }\text{mathpzc}{w}_{\mathbf{p}}(t^{\prime })\right]\end{array}$$

$$W_T(\mathbf{p})\approx \frac{1}{2}{\left|{\int }_{-\text{nicefrac}T2}^{\text{nicefrac}T2}d\tilde{t}{Q}_{\mathbf{p}}(\tilde{t})e^{i{\Lambda }_{\mathbf{p}}(\tilde{t})}\right|}^2$$

$$\begin{array}{rl}& {\int }_{-\text{nicefrac}T2}^{\text{nicefrac}T2}d\tilde{t}\dots ={\int }_{\frac{{\gamma }_{\parallel }}{{\gamma }_{\perp }}-\frac{\pi N}{{\gamma }_{\perp }}}^{\frac{{\gamma }_{\parallel }}{{\gamma }_{\perp }}+\frac{\pi N}{{\gamma }_{\perp }}}\frac{d\tau }{1+{\tau }^2}\exp \left[\frac{{ϵ}_{\perp }^2}{e{E}_s}\mathcal{S}(\tau )\right],\\ & \mathcal{S}(\tau )=2i{\int }_0^{\tau }d\tilde{\tau }\frac{(1+{\tilde{\tau }}^2{)}^{\text{nicefrac}12}}{1+\epsilon \cos (\omega \text{mathpzc}t)},\end{array}$$

$$\text{mathpzc}t(\tilde{\tau })\approx \frac{1}{\omega }({\gamma }_{\perp }\tilde{\tau }-{\gamma }_{\parallel })$$

$${\gamma }_{\parallel }=\gamma \frac{p_{\parallel }}{m}\mathrm{and}{\gamma }_{\perp }=\gamma \frac{{ϵ}_{\perp }}{m},$$

$${\tau }_{\pm \text{mathpzc}k}=\frac{{\gamma }_{\parallel }+(2\text{mathpzc}k-1)\pi }{{\gamma }_{\perp }}\pm i\frac{{\gamma }_{\mathrm{cr}}}{{\gamma }_{\perp }},\text{mathpzc}k\in \mathbb{Z},$$

$${\gamma }_{\mathrm{cr}}=\ln \left(\frac{2}{\epsilon }\right),\epsilon \ll 1$$

$$\begin{array}{rl}& \mathcal{S}(\tau )\approx \frac{i}{2}{\mathcal{S}}_0+\frac{1}{\sqrt{2}}|\tau -i{|}^{\text{nicefrac}32}\left|\kappa \right|e^{i\arg \kappa +i\frac{3\pi }{4}+i\frac{3}{2}{\phi }_{+}},\\ & {\mathcal{S}}_0=4{\int }_0^{i}d\tilde{\tau }\frac{(1+{\tilde{\tau }}^2{)}^{\text{nicefrac}12}}{1+\epsilon \cos ({\gamma }_{\perp }\tau -{\gamma }_{\parallel })}.\end{array}$$

$$\begin{array}{rl}{\phi }_{+1}=\frac{\pi }{6}-& \frac{2}{3}\arg \kappa ,{\phi }_{+2}=-\frac{7\pi }{6}-\frac{2}{3}\arg \kappa .\end{array}$$

$$\begin{array}{rl}& \arg \kappa ={\tan }^{-1}\left[\frac{\epsilon \sinh ({\gamma }_{\perp })\sin ({\gamma }_{\parallel })}{1+\epsilon \cosh ({\gamma }_{\perp })\cos ({\gamma }_{\parallel })}\right]+\text{mathpzc}{s}_i\end{array}$$

$$\begin{array}{rl}& {\int }_{-\infty }^{\infty }d\tau \dots =-\exp \left[\frac{i{ϵ}_{\perp }^2}{2e{E}_s}{\mathcal{S}}_0\right]\\ & \times {\int }_{\Gamma }\frac{d\tau }{2i(\tau -i)}\exp \left[\frac{{ϵ}_{\perp }^2}{\sqrt{2}e{E}_s}\kappa [i(\tau -i){]}^{\text{nicefrac}32}\right],\end{array}$$

$$\begin{array}{rl}& {\int }_{-\infty }^{\infty }d\tau \dots =\frac{2\pi i}{3}\exp \left[\frac{i{ϵ}_{\perp }^2}{2e{E}_s}{\mathcal{S}}_0\right]{\oint }_{\tilde{\Gamma }}\frac{dw}{2\pi w}{e}^w.\end{array}$$

$$W_T(\mathbf{p})\approx 2e^{-\frac{{ϵ}_{\perp }^2}{e{E}_s}\mathrm{Im}{\mathcal{S}}_0({\gamma }_{\perp },{\gamma }_{\parallel })},$$

$$\begin{array}{rl}& \mathrm{Im}{\mathcal{S}}_0({\gamma }_{\perp },{\gamma }_{\parallel })=4{\int }_0^{1}dy\frac{\sqrt{1-y^2}{\cos }^2({\varphi }_y)}{1+\epsilon \cosh ({\gamma }_{\perp }y)\cos ({\gamma }_{\parallel })}.\end{array}$$

$$\begin{array}{rl}\mathrm{Im}{\mathcal{S}}_0({\gamma }_{\perp },{\gamma }_{\parallel })& \approx 4{\int }_0^{\infty }\frac{dy}{1\pm \epsilon \cosh ({\gamma }_{\perp }y)}\\ & \pm \frac{8}{\epsilon }{\int }_{\text{mathpzc}{y}_0}^{1}dy{e}^{-{\gamma }_{\perp }y}\sqrt{1-y^2}\\ & -4{\int }_{\text{mathpzc}{y}_0}^{\infty }\frac{dy}{1\pm \epsilon \cosh ({\gamma }_{\perp }y)}.\end{array}$$

$$\begin{split}&\mathrm{Im}\;\mathcal{S}_{0}(\gamma_{\perp},\gamma_{\parallel})\approx 4\left\{\begin{array}[]{cc}\displaystyle\frac{\gamma_{\mathrm{cr}}}{\gamma_{\perp}}-\frac{1}{\gamma_{\perp}}e^{-\gamma_{\perp}}&\mathrm{for}\ \gamma_{\parallel}=0,\\ \\ \displaystyle\frac{\gamma_{\mathrm{cr}}}{\gamma_{\perp}}+\frac{1}{\gamma_{\perp}}e^{-\gamma_{\perp}}&\mathrm{for}\ \gamma_{\parallel}=\pi.\end{array}\right.\end{split}$$

$$\begin{array}{r}\mathrm{Im}{\mathcal{S}}_0({\gamma }_{\perp },{\gamma }_{\parallel })\approx \pi -\frac{2\pi }{{\gamma }_{\perp }}\epsilon \text{mathpzc}{I}_1({\gamma }_{\perp })\cos ({\gamma }_{\parallel }),\end{array}$$

$$W_T(\mathbf{p})\approx 2{\left(\frac{\epsilon }{2}\right)}^{\frac{4{ϵ}_{\perp }}{\omega }},{\gamma }_{\perp }\gg {\gamma }_{\mathrm{cr}},{\gamma }_{\perp }\gg 1.$$

$$\begin{array}{rl}\text{mathpzc}N& =\int \frac{d^{3}p}{(2\pi {)}^3}{W}_T(\mathbf{p})\\ & =\frac{e{E}_{s}T}{2{\pi }^2}{\int }_0^{\pi }d{\gamma }_{\parallel }{\int }_0^{\infty }\frac{d{p}_{\perp }}{2\pi }{p}_{\perp }{W}_T(\mathbf{p}),\end{array}$$

$$\begin{array}{rl}& \dot{\text{mathpzc}N}\equiv \frac{\text{mathpzc}N}{T}\approx \frac{(e{E}_s{)}^3}{2{\pi }^3{\omega }^2}{\int }_0^{\pi }d{\gamma }_{\parallel }{\int }_{\gamma }^{\infty }ds\\ & \times s\exp \left[-\frac{e{E}_s}{{\omega }^2}{s}^2\mathrm{Im}{\mathcal{S}}_0(s,{\gamma }_{\parallel })\right],\end{array}$$

$$\begin{array}{rl}& \dot{\text{mathpzc}N}\approx \frac{(e{E}_s{)}^2}{8{\pi }^4}\frac{\mathrm{erf}\left(\sqrt{\frac{1}{2}\frac{m}{\omega }\text{mathpzc}{h}_{\epsilon }(\gamma )}\pi \right)}{1-\text{mathpzc}{g}_{\epsilon }(\gamma )-\frac{1}{2}\gamma \text{mathpzc}{g}_{\epsilon }^{\prime }(\gamma )}\\ & \times \sqrt{\frac{2\pi \omega }{m\text{mathpzc}{h}_{\epsilon }(\gamma )}}\exp \left[-\pi \frac{m^2}{e{E}_s}\left\{1-\text{mathpzc}{g}_{\epsilon }(\gamma )\right\}\right]\end{array}$$

$$\begin{split}\mathpzc{g}_{\varepsilon}(\gamma)&=1-\frac{1}{\pi}\mathrm{Im}\;\mathcal{S}_{0}(\gamma,0)\\ &\approx\left\{\begin{array}[]{ccc}\displaystyle\frac{2\varepsilon}{\gamma}I_{1}(\gamma)&\mathrm{for}&\gamma\ll 1,\\ \displaystyle 1-\frac{4}{\pi}\frac{\gamma_{\mathrm{cr}}}{\gamma}&\mathrm{for}&\begin{array}[]{c}\gamma\gg 1,\\ \;\;\gamma\gg\gamma_{\mathrm{cr}}\end{array}\end{array}\right.\end{split}$$

$$\begin{array}{rl}\text{mathpzc}{h}_{\epsilon }(\gamma )& \equiv \gamma {\frac{{\partial }^2}{\partial {\gamma }_{\parallel }^2}\mathrm{Im}{\mathcal{S}}_0(\gamma ,{\gamma }_{\parallel })|}_{{\gamma }_{\parallel }=0}\\ & =8\epsilon \gamma {\int }_0^{1}dy\frac{\sqrt{1-y^2}}{[1+\epsilon \cosh (\gamma y){]}^3}\\ & \times \left[\frac{1}{2}\cosh (\gamma y)+\epsilon \left(1-\frac{1}{2}{\cosh }^2(\gamma y)\right)\right],\end{array}$$

$$\mathpzc{h}_{\varepsilon}(\gamma)\approx\left\{\begin{array}[]{ccc}\displaystyle 2\pi\varepsilon I_{1}(\gamma)&\mathrm{for}&\gamma\ll 1,\\ \displaystyle 4\frac{\gamma_{\mathrm{cr}}}{\gamma^{2}}&\mathrm{for}&\begin{array}[]{c}\gamma\gg 1,\\ \;\;\gamma\gg\gamma_{\mathrm{cr}}\end{array}.\end{array}\right.$$

$$\begin{array}{rl}& \dot{\text{mathpzc}N}\approx \frac{(e{E}_s{)}^2}{4{\pi }^3}\exp \left[-\pi \frac{E_{\mathrm{cr}}}{E_s}\right].\end{array}$$

$$\begin{array}{rl}& \dot{\text{mathpzc}N}\approx \frac{e{E}_s}{8{\pi }^2}\frac{m\omega }{\ln \left[2\frac{E_s}{E_w}\right]}{\left[\frac{1}{2}\frac{E_w}{E_s}\right]}^{\frac{4m}{\omega }}.\end{array}$$

$$\begin{array}{rl}\Delta \text{mathpzc}{n}_{\gamma }(t)& =\text{mathpzc}{n}_{\mathrm{total}}(t)-\text{mathpzc}{n}_{\gamma }(T)\\ & =\text{mathpzc}{n}_{e^{-}{e}^{+}}\frac{\delta t}{\tau }\frac{1}{1+\frac{\delta t}{\tau }}.\end{array}$$

$$\Delta \text{mathpzc}{n}_{\gamma }(t)\approx \text{mathpzc}{n}_{e^{-}{e}^{+}}\frac{\delta t}{\tau }.$$

$$\begin{array}{rl}& d\dot{\nu }=\frac{1}{2}{e}^4{\text{mathchar}22\text{d}}^3{k}_1{\text{mathchar}22\text{d}}^3{k}_2{\text{mathchar}22\text{d}}^3{p}_{+}{\text{mathchar}22\text{d}}^3{p}_{-}{\text{mathchar}22\delta }_{k_1+k_2,p_{+}+p_{-}}^4\\ & \times W_T({\mathbf{p}}_{+})W_T({\mathbf{p}}_{-}){\overline{{\mathcal{M}}^2}}_{e^{-}{e}^{+}\to \gamma \gamma }(k_1,k_2,p_{+},p_{-}),\end{array}$$

Full text

Vacuum Instability, Standing Wave, Pair Production, Pair Annihilation.

I Introduction

Finding suitable and controllable experimental conditions to materialize the all-permeating quantum vacuum fluctuations has been a fundamental goal in particle physics since the time when our inert vacuum perception strikly changed into a nontrivial regulatory void, responsible of mediating the interactions between elementary particles. Even before the full establishment of quantum electrodynamics (QED), it was noted that this sort of vacuum instability could be conceived by producing electron-positron pairs if a macroscopic electric field $E$ is held in vacuum Sauter:1931 ; Heisenberg:1935 ; Schwinger:1951nm . Notwithstanding, the corresponding pair production (PP) rate $\mathcal{R}\sim\exp[-\pi E_{\mathrm{cr}}/E]$ provides evidences that an experimental verification of this so-called Schwinger mechanism is far from our reach; save yet unaccessible field strengths–comparable to the critical scale of QED $E_{\mathrm{cr}}=m^{2}/e\sim 10^{16}\ \rm V/cm$–become available.111Here and henceforth the mass and the absolute charge of an electron will be denoted by $m$ and $e$, respectively. Besides, throughout the manuscript Heaviside$-$Lorentz units–with the speed of light and the Planck constant set to unity $c=\hbar=1$–are used. Although significant progresses toward high-intensity laser technology are raising our hopes of reaching the required field strengths within the focal spot of multipetawatt laser pulses, it is rather likely that an experimental verification of the Schwinger PP process remains a challenging task to achieve, at least in a near future. Mainly, because the peak field strengths $\sim 10^{-2}E_{\mathrm{cr}}$ expected at the new generation of laser systems, including the Extreme Light Infrastructure (ELI) ELI and the Exawatt Center for Extreme Light Studies (XCELS) xcels , would keep the production rate very small.

A central aspect in investigations aiming to relieve the exponential suppression of $\mathcal{R}$ is the identification of field setups which may allow us to maximize the Schwinger effect Hebenstreit:2009km ; Bulanov ; BlaschkeCPP ; Kohlfurst:2012rb ; Hebenstreit:2014lra ; Gonoskov . Perhaps the most robust configuration found so far is the one implemented in what is nowadays known as the dynamically-assisted Schwinger mechanism dgs2008 ; dgs2009 , where–in addition to a strong quasi-static electric field–a weak but high-frequency field component is superimposed. In the original papers on the subject, the combined field was composed of two Sauter pulses dgs2008 or a constant electric field and a high energy electromagnetic wave with $\omega\lesssim 2m$ dgs2009 . The latter ingredient–partially motivated by the experimental verification of the nonlinear Breit-Wheeler reaction Burke:1997ew –stimulates the creation of pairs substantially. Indeed, first estimates resulting from this assisted scenario predict an enhancement of the PP rate $\mathcal{R}\sim\exp[-\kappa E_{\rm cr}/E]$ with $0<\kappa\ll 1$, while its nonperturbative feature in the strong field strength is kept. Similar improvement has been predicted to take place in production channels other than the one described so far, provided the assisted high-frequency laser wave is present DiPiazza:2009py ; Jansen2013 ; Augustin . Qualitatively, this sort of catalysis is understood as a direct consequence of the absorption of photons from the weak field, which causes an effective reduction of the barrier width that an electron has to tunnel from negative to positive Dirac continuum. A large number of transitions are thus facilitated–pairs are created copiously–leading to increase our chances for observing a signature of the vacuum instability.

This paper is devoted to study the spontaneous production of electron-positron pairs as might occur in a dynamically-assisted setup driven by combination of a constant and a purely time-dependent electric field. Our theoretical approach relies on the quantum transport equation that dictates the time evolution of the PP process Schmidt:1998vi ; Kluger ; Schmidt:1999vi ; alkofer:2001ib . Noteworthy, several investigations of this nature have already been carried out; most of them by using numerical techniques from which valuable information and features have been extracted Orthaber2011 ; Grobe2012 ; Akal:2014eua ; Aleksandrov:2018uqb ; Panferov:2015yda ; Otto:2014ssa . Meanwhile various research have focused on deriving formulae for the created particles spectra Linder:2015vta ; Otto:2014ssa ; Panferov:2015yda ; torgrimsson2016 ; Fey . This way, illuminating the crucial aspects from which an optimized version of the aforementioned enhancement could be reached. Two recent papers went a step further by providing analytical expressions for the total probability of produced pairs Togrimsson1 ; Togrimsson2 . In these investigations a pertubative treatment in the weak field was used within the WKB and the world-line instanton methods, respectively. Particular attention was laid on weak fields with Sauter and Gaussian profiles. However, the case of a periodically oscillating mode was touched only briefly. Here we complement these analytical studies by using a quantum kinetic approach in which both the weak and strong field are treated nonperturbatively. We discuss in details the case in which the assisted mode oscillates periodically and obtain a formula for the density rate of yielded particles which is shown to manifest a nonperturbative dependence on both the strong and weak electric fields and to interpolate between the tunneling and multi-photon regimes. Besides, the outcomes of this analysis are exploited to put forward an experimental setup which aims to verify indirectly the realization of the dynamically-assisted Schwinger effect. We show that–once the field is switched off–the plasma density can be high enough as to facilitate the copious annihilation of pairs into gamma photons. The detection of these photons is the central aspect in the proposed setup.

This paper is organized as follows: In Sec. II we adopt the model to be analyzed and summarize briefly the main aspects linked to the quantum Vlasov equation and its solution within the low-density limit. The aforementioned approximation is particularized in Sec. III to the case in which the external background combines a strong static electric field and a fast-oscillating electric field. There we establish an analytic expression for the single-particle distribution function and discuss its behavior in various regimes of interest. Later on, in Sec. IV, we derive a compact asymptotic formula for the PP rate and reveal explicitly how the enhancement caused by the weak mode is closely connected with the perturbative PP rate associated with the absorption of photons. Details about some special aspects of these calculations are given in Appendices A and B. Finally, in Sec. V, we investigate the evolution of the electron-positron plasma after its creation via the assisted Schwinger mechanism. An expression for the total number of photons resulting from the annihilation of pairs at early times is derived. Some insight on a viable experimental setup aiming to detect these photons, and thus the spontaneous production of pairs, is given afterwards.

II General aspects

We consider the spontaneous production of electron-positron pairs taking place in a time-dependent but homogeneous electric field combining a strong and a weak mode with frequencies $\Omega$ and $\omega$, respectively. We will suppose this field is localized temporally between $-T/2\leqslant t\leqslant T/2$, its pulse length $T=2\pi N/\omega$ being determined by the number of cycles $N$ and $\omega$. In the following, we further restrict the model to the case in which the variation linked to the perturbative mode is much faster than the one undergone by the strong field counterpart [$\omega\gg\Omega$] and where, accordingly, $N$ is very large. Hence, the PP problem can be formulated as if the creation process was taking place in a background characterized by a constant electric field and a fast-oscillating mode generated by the four-potential

[TABLE]

where $\flat^{\mu}=(0,0,1,0)$ is the polarization four-vector and $\Theta(x)$ denotes the unit step function: $\Theta(x)=1$ at $x\geqslant 0$, $\Theta(x)=0$ at $x<0$. Here the subscripts “$s$” and “$w$” are used to identify the strong and weak field strengths [$E_{w}\ll E_{s}$], respectively.

Our investigation adopts the quantum kinetic approach as theoretical tool to describe the production of electron-positron pairs. This formulation–which is equivalent to other well-known approaches based on QED in unstable vacuum Gitman ; Fradkin –comprises the dynamical information of the PP process in the single-particle distribution function $W(\boldsymbol{p};t)$–summed over the spin projections–of electrons and positrons to which the degrees of freedom in the external field are relaxed at asymptotically large times [$t\to\pm\infty$], i.e., when the electric field is switched off $\boldsymbol{E}(\pm\infty)\to 0$. The time evolution of this quantity is dictated by a quantum Boltzmann-Vlasov equation Schmidt:1998vi ; Kluger ; Schmidt:1999vi ; alkofer:2001ib , whose integro-differential version:

[TABLE]

manifests both the nonequilibrium nature of the PP process and its non-Markovian feature.222The quantum field theoretical approach of the pair production problem–as encompassed by Eq. (2) concisely–can be formulated alternatively through a Riccati equation dumlu0 ; dumlu1 or via a representation involving three coupled ordinary differential equations. See for instance Refs. Hebenstreit:2010vz ; Otto:2014ssa ; Panferov:2015yda . The formula above assumes the vacuum initial condition $W(\boldsymbol{p},-\infty)=0$ and applies the notation $\dot{W}(\boldsymbol{p};t)\equiv\partial W(\boldsymbol{p};t)/\partial t$. Besides, it is characterized by the function $Q(\boldsymbol{p},t)\equiv eE(t)\epsilon_{\perp}/\mathpzc{w}_{\boldsymbol{p}}^{2}(t)$, which depends on the transverse energy of the Dirac fermions $\epsilon_{\perp}=\sqrt{m^{2}+p_{\perp}^{\,2}}$ and the respective total energy squared $\mathpzc{w}^{2}_{\boldsymbol{p}}(t)=\epsilon_{\perp}^{2}+[p_{\parallel}-e\mathpzc{A}(t)]^{2}$ of an electron. Here $\boldsymbol{p}_{\perp}=(p_{x},0,p_{z})$ and $\boldsymbol{p}_{\parallel}=(0,p_{y},0)$ are the components of the canonical momentum perpendicular and parallel to the direction of $\boldsymbol{E}(t)$, respectively.

It is known that Eq. (2) can only be solved exactly for a few special backgrounds, e.g., constant and Sauter-type electric fields. Finding analytic solutions beyond the aforementioned configurations is a difficult task. However, estimates can be obtained by using the low-density approximation [$W(\boldsymbol{p};t)\ll 1$] within the Boltzmann-Vlasov equation. In such a case, the single-particle distribution function at time for which the field has been switched off [$W_{T}(\boldsymbol{p})\equiv\lim_{t\to T}W(\boldsymbol{p};t)$] can be approximated by Schmidt:1999vi ; BlaschkeCPP

[TABLE]

with $\Lambda_{\boldsymbol{p}}(\tilde{t})\equiv\int_{0}^{\tilde{t}}d\mathpzc{t}\;\mathpzc{w}_{\boldsymbol{p}}(\mathpzc{t})$. We note that the integration contained in this formula is nothing but the solution of the linearized Riccati equation on which the study in Ref. dumlu1 relies. At this point it turns out to be rather illuminating to perform the change of variables $\tau=[p_{\parallel}-e\mathpzc{A}(\tilde{t})]/\epsilon_{\perp}$ and $\tilde{\tau}=[p_{\parallel}-e\mathpzc{A}(\mathpzc{t})]/\epsilon_{\perp}$. As a consequence, the integral in Eq. (3) becomes

[TABLE]

where $\varepsilon=E_{w}/E_{s}\ll 1$ parametrizes the relative weakness of the fast-oscillating mode. The expression above constitutes the starting point for further considerations. In its second line, $\mathpzc{t}$ has to be considered as a function of $\tilde{\tau}$. However, this inversion cannot be determined analytically, but only through reversion of the corresponding series abramowitz ; Kim2007 . In this case the leading order term

[TABLE]

coincides with the inverse of the function $\tau(\mathpzc{t})$ averaged over a cycle of the weak field. Here, we have introduced the dimensionless parameters

[TABLE]

Observe that, in the limit of $p_{\perp}\to 0$, $\gamma_{\perp}$ reduces to the combined Keldysh parameter [$\gamma=\omega m/(eE_{s})$]. In order to suitably fit the external parameters to current and foreseeable experimental setups, we will suppose hereafter that $m^{2}\gg(eE_{s})$ and $2m>\omega$. We note that an assisted scenario with $2m>\omega\gtrsim m$, is characterized by the restriction $\gamma_{\perp}\geqslant\gamma\gg 1$. Conversely, if $\omega\ll m^{3}/(eE_{s})$–leading to $\gamma\ll 1$–the effective reduction of the barrier width between the negative and positive continuum is expected to be almost insignificant, and the oscillating field would not play a significant role in the production of pairs.

III Properties of the particle spectrum

We wish to find closed-form analytic expressions for Eq. (3) in the case characterized by the condition $\pi N\gg\gamma_{\perp},|\gamma_{\parallel}|$. Therefore, the treatment developed in this subsection is limited to small momentum components relative to the one associated with the external field, i.e., to values $|p_{\parallel}|\ll eE_{s}T/2$ and $|p_{\perp}|\ll eE_{s}T/2$. To facilitate the mathematical treatment of the problem, we will formally extend the outer integration limits in Eq. (4) to $\pm\infty$. As a consequence, the single-particle distribution function $W_{T}(\boldsymbol{p})$ [see Eq. (3)] approaches to a $2\pi-$periodic function in $\gamma_{\parallel}$. Correspondingly, its dependence on this variable will be investigated in the interval $-\pi<\gamma_{\parallel}\leqslant\pi$.

Since the factor $\epsilon_{\perp}^{2}/(eE_{s})\geqslant m^{2}/(eE_{s})\gg 1$, the exponential in Eq. (4) oscillates very fast and the steepest-decent method represents a suitable tool to carry out its outer integration. In order to apply this method, we first extend the integration variables to the complex $\tau-$plane. As it is characteristic in problems of this nature, the poles at $\tau=\pm i$ are also branch-points of the integrand. The branch-cuts are then chosen from $\tau=i$ to $\tau=i\infty$ and from $\tau=-i$ to $\tau=-i\infty$, i.e., $\tau^{2}+1=(\tau+i)(\tau-i)=|\tau+i||\tau-i|e^{i\varphi_{+}}e^{i\varphi_{-}}$ with $-3\pi/2\leqslant\varphi_{+}<\pi/2$ and $-\pi/2\leqslant\varphi_{-}<3\pi/2$ referring to the local polar angle linked to $\tau=+i$ and $\tau=-i$, respectively. Still, in the cut $\tau$-plane there exist poles linked to the integrand of $\mathcal{S}(\tau)$:

[TABLE]

where a loss of analyticity is exhibited. While the integer value $\mathpzc{k}$ manifests the periodicity in $\gamma_{\parallel}$, the critical Keldysh parameter Linder:2015vta ; Togrimsson1 ; Togrimsson2 ; Akal:2018udh

[TABLE]

rules two different scenarios depending on whether $\gamma_{\perp}$ does or does not exceed the value of $\gamma_{\rm cr}$. Below we will describe further this point.

Now, the saddle-points linked to the exponent [$\tau=\pm i$] coincide with the branch-points found previously. In a vicinity of $\tau=i$, the pre-exponential of the integrand behaves as $\sim 1/[2i(\tau-i)]$, whereas

[TABLE]

In the first line, $\arg\kappa$ denotes the principal value of the argument of $\kappa=[1+\varepsilon\cos\left(\omega\mathpzc{t}(i)\right)]^{-1}$. Here, the directions of the steepest-descents can be locally approximated by choosing $\varphi_{+}=-\frac{\pi}{2}+\frac{2}{3}k\pi-\frac{2}{3}\arg\kappa$ with $k\in\mathbb{Z}$ such that the condition $\cos(\arg\kappa+\frac{3}{4}\pi+\frac{3}{2}\varphi_{+})<0$ holds. In connection we find

[TABLE]

Since the angles above must lie within the interval $[-\frac{3\pi}{2},\frac{\pi}{2})$ [read above Eq. (9)], $\arg\kappa$ turns out to be restricted to the region $-\frac{\pi}{2}<\arg\kappa\leqslant\frac{\pi}{2}$ with

[TABLE]

with $\mathpzc{s}_{1,4}=0$, $\mathpzc{s}_{2}=\pi$, $\mathpzc{s}_{3}=-\pi$, where the choice of the constant $\mathpzc{s}_{i}$ corresponds to suitable values in each of the four quadrants Ablowitz . Manifestly, Eq. (11) reveals that the directions of the steepest-descents change as $\gamma_{\perp}$ and $\gamma_{\parallel}$ are varied. We remark that, in the absence of the perturbative mode [$\varepsilon=0\Leftrightarrow\arg\kappa=0$], these directions are fully specified: $\varphi_{+1}=\pi/6$ and $\varphi_{+2}=-7\pi/6$. It is worth remarking that these angles do not change appreciably when the fast oscillating field is turned-on [$\varepsilon\neq 0$] and $\gamma_{\rm cr}\gg\gamma_{\perp}$; no matter the value of $\gamma_{\parallel}$. The situation does not change either if $\varepsilon\neq 0$, $\gamma_{\perp}\gg\gamma_{\rm cr}$ with $\gamma_{\perp}\gg 1$ and $|\gamma_{\parallel}|\ll 1$. Even, for $|\gamma_{\parallel}|\ll 1$ and $\gamma_{\perp}\sim\gamma_{\rm cr}>2$ it can be verified that $\arg\kappa\approx 0$ and the directions of the steepest-descents approach locally to those arising when the strong electric field is present only.

We take advantage of the described feature to deform the initial integration contour [see Eq. (4)] as depicted in Fig. 1. The portion covering the real axis deviates several times through circles $\mathpzc{c}_{\mathpzc{k}}$ with an infinitesimal radius [$\mathpzc{r}\to 0$]. At this point it should be understood that each of them join the real $\tau-$axis via parallel shortcuts with opposite directions. However, in the picture, they have been omitted for simplicity and because they do not contribute at all. Here $\mathpzc{k}$ labels a circumvented pole between the most distant extremes of the circuit [see Eq. (7)]. In the picture, the imaginary parts of the poles have been taken with $\gamma_{\perp}\gg\gamma_{\rm cr}$. For $\gamma_{\perp}\ll\gamma_{\rm cr}$, the pole locations are moved upward, and many of them could lie above the path $\Gamma=\mathpzc{C_{R}}\cup\mathpzc{C_{\epsilon}}\cup\mathpzc{C_{L}}$. In such a situation, the contour of integration is chosen similar to the previous case: those poles remaining below $\Gamma$ are then eluded. Consequently, the Cauchy’s theorem allows us to express

[TABLE]

where we have taken into account that the contribution linked to each pole’s circumvention, i.e., over $\mathpzc{c}_{\mathpzc{k}}$, vanishes as $\mathpzc{r}\to 0$ [for details read the Appendix A]. Likewise, we have considered that the integrations over $\mathpzc{C_{\pm}}$ give no contributions when $N\to\infty$. The details of this considerations are summarized in Appendix B.

We point out that Eq. (12) can also be applied to those cases in which the integration contour differs substantially from the one analyzed explicitly here, provided no contribution arises from those circuits connecting the region $-\pi N/\gamma_{\perp}\leqslant\mathrm{Re}\;\tau\leqslant\pi N/\gamma_{\perp}$ with the sectors ending in the steepest descents. Hereafter we will suppose that this is the case. Thus, by performing the map $w=\epsilon_{\perp}^{2}\kappa[i(\tau-i)]^{\nicefrac{{3}}{{2}}}/[\sqrt{2}eE_{s}]$, Eq. (12) becomes

[TABLE]

Observe that the closed path of integration $\tilde{\Gamma}$ [see Fig. 2] encloses a single pole at infinity [$w=+\infty$]. Hence, the application of the residue theorem leads to $\oint_{\tilde{\Gamma}}dwe^{w}/(2\pi w)=i$ Brezin . We substitute this outcome into Eq. (13). The resulting expression is inserted into Eq. (3) afterwards. As a consequence, the single-particle distribution function reduces to

[TABLE]

where an unessential pre-exponential factor of the order of unity has been omitted [$\pi^{2}/9\approx 1.1$]. We stress that $\mathrm{Im}\;\mathcal{S}_{0}(\gamma_{\perp},\gamma_{\parallel})$ is given in Eq. (9). By taking $\tau=iy$, it reads

[TABLE]

Here the function $\phi_{y}$ is the angle determined by the $\tan^{-1}-$function involved in Eq. (11) with $\gamma_{\perp}$ replaced by $\gamma_{\perp}y$. We remark that the expression above is an even function in $\gamma_{\parallel}$ which applies for any of the following two regimes: $\gamma_{\mathrm{cr}}\gg\gamma_{\perp}$ and $\gamma_{\perp}\gg\gamma_{\mathrm{cr}}$ with $\gamma_{\perp}\gg 1$. Hence we further restrict our investigation to $0\leqslant\gamma_{\parallel}\leqslant\pi$.

Observe that, in the limit of $\varepsilon\to 0$, $\mathrm{Im}\;\mathcal{S}_{0}(\gamma_{\perp},\gamma_{\parallel})=\pi$. Hence, $W_{T}(\boldsymbol{p})$ reduces to the known expression in a constant electric field Hebenstreit:2010vz : $W_{T}(\boldsymbol{p})\approx 2\exp[-\pi\epsilon_{\perp}^{2}/(eE_{\mathrm{cr}})]$. The situation changes when $\varepsilon\neq 0$. To show this, we investigate some analytic and asymptotic properties of Eq. (LABEL:ImS). First of all, when setting $\partial\mathrm{Im}\;\mathcal{S}_{0}(\gamma_{\perp},\gamma_{\parallel})/\partial\gamma_{\parallel}$ to zero, we find that the corresponding extreme points are located at the borders of the region encompassed by $0\leqslant\gamma_{\parallel}\leqslant\pi$. To elucidate which of them maximize and minimize Eq. (LABEL:ImS), we will establish its asymptotic formulas evaluated at $\gamma_{\parallel}=0$ and $\gamma_{\parallel}=\pi$, respectively. Let us begin with the case in which $\gamma_{\perp}\gg\gamma_{\rm cr}$ with $\gamma_{\perp}\gg 1$. We then introduce a positive splitting parameter $\mathpzc{y}_{0}$ satisfying the conditions $\gamma_{\perp}^{-1}\ll\mathpzc{y}_{0}\ll 1$ and $\gamma_{\mathrm{cr}}/\gamma_{\perp}\ll\mathpzc{y}_{0}$. Afterward, the $y$ integration is divided as follows: $\mathrm{Im}\;\mathcal{S}_{0}(\gamma_{\perp},\gamma_{\parallel})=4\int_{0}^{\mathpzc{y}_{0}}dy\ldots+4\int_{\mathpzc{y}_{0}}^{1}dy\ldots$. In the integral defined over the region $[0,\mathpzc{y}_{0}]$ the integration variable is very small [$y\ll 1$], leading to approximate the square root contained in the integrand by unity [$(1-y^{2})^{\nicefrac{{1}}{{2}}}\approx 1$]. Conversely, the main contribution to the integral defined over $[\mathpzc{y}_{0},1]$ results from those values of $y$ fulfilling the condition $y\gg\gamma_{\mathrm{cr}}/\gamma_{\perp}$, in which case the integrand can be approximated by $\sim\pm 2\exp[-\gamma_{\perp}y]/\varepsilon$. Consequently,

[TABLE]

While the positive sign corresponds to $\gamma_{\parallel}=0$, the negative one is associated with $\gamma_{\parallel}=\pi$. Now, the first integral involved in this expression can be calculated by using Eq. ($3.513.2$) in Ref. Gradshteyn . The remaining two can be combined in an integral independent of the parameter $\mathpzc{y}_{0}$. Indeed, note that as $\gamma_{\perp}\gg 1$, the integrand in $\int_{\mathpzc{y}_{0}}^{1}dy\ldots$ behaves as $\sim\exp[\gamma_{\perp}y]$, whereas the one defined in $[\mathpzc{y}_{0},\infty)$ can be approximated by $\sim\pm 2\exp[-\gamma_{\perp}y]/\varepsilon$. Keeping all these details in mind we obtain

[TABLE]

As the extreme points linked to $\gamma_{\parallel}$ coincide with the borders of the interval $[0,\pi]$, and $\mathrm{Im}\;\mathcal{S}_{0}(\gamma_{\perp},\pi)>\mathrm{Im}\;\mathcal{S}_{0}(\gamma_{\perp},0)$, the quantity $\mathrm{Im}\;\mathcal{S}_{0}(\gamma_{\perp},\gamma_{\parallel})$ as a function of $\gamma_{\parallel}$ grows monotonically in this region for any $\gamma_{\perp}$ satisfying the conditions $\gamma_{\perp}\gg\gamma_{\rm cr}$ and $\gamma_{\perp}\gg 1$. This behavior is also verified by looking at the solid curves in Fig. 3, which have been obtained by setting $\gamma_{\parallel}=0$ (black), $\gamma_{\parallel}=\pi/2$ (red) and $\gamma_{\parallel}=\pi$ (darker cyan). Furthermore, the formula above reveals that the amplitude of oscillations in $\gamma_{\parallel}$ decreases as $\sim 8\exp[-\gamma_{\perp}]/\gamma_{\perp}$ when $\gamma_{\perp}$ grows, a fact exhibited clearly in Fig. 3 [compare the solid curves in black and darker cyan at $\gamma_{\perp}=10$, for instance]. We remark that this trend changes as the condition $\gamma_{\rm cr}\gg\gamma_{\perp}$ is fulfilled. Under this circumstance all instances in Eq. (LABEL:ImS) depending on $\varepsilon$ can be Taylor expanded. After the integration,

[TABLE]

where Eq. ($3.534.1$) in Ref. Gradshteyn has been used. Here, $\mathpzc{I}_{1}(x)=\frac{x}{2}+\frac{x^{3}}{2^{2}4}+\frac{x^{5}}{2^{2}4^{2}6}+\ldots$ denotes the modified Bessel function of the first kind with order one NIST . The formula above coincides with Eq. (37) in Ref. Linder:2015vta . It manifests clearly a monotonic growing in $0\leqslant\gamma_{\parallel}\leqslant\pi$. Furthermore, in the current regime, the amplitude of oscillation in $\gamma_{\parallel}$ scales as $\sim 2\pi\varepsilon$.

Interestingly, when $\gamma_{\perp}$ moves away from $\gamma_{\rm cr}$ toward larger values, all solid curves in Fig. 3 show a significant falling as compared to the characteristic value linked to the constant field case [$\mathrm{Im}\;\mathcal{S}_{0}\approx\pi$]. Indeed, at $\gamma_{\perp}=20$, they have fallen to $\mathrm{Im}\;\mathcal{S}_{0}\approx 1$. Besides, as Eq. (17) applies for $\gamma_{\perp}\gg\gamma_{\mathrm{cr}}$ with $\gamma_{\perp}\gg 1$, the exponential contributions can be safely ignored. This fact implying that $\mathrm{Im}\;\mathcal{S}_{0}(\gamma_{\perp},\gamma_{\parallel})$ does not undergo an appreciable variation in $\gamma_{\parallel}$. Keeping all these details in mind, we find that the single-particle distribution function [see Eq. (14)] behaves as

[TABLE]

This expression deserves further comments. Firstly, at $\boldsymbol{p}=0$, the exponent associated with this asymptotic formula coincides with the minimal “number” of quanta necessary to produce a pair at rest from the weak mode solely. This observation already provides evidences that an enhacement in $W_{T}(\boldsymbol{p})$ could take place via the absorption of a quantum from the fast-oscillating field as compared with the case in wich only a constant electric field drives the vacuum instability. Indeed, let us suppose the particles are created in an assisted field setup characterized by the following parameters: $\omega=1.7\;m$, $E_{s}=10^{-1}\;E_{\mathrm{cr}}$, i.e. $\gamma\approx 17$, and $E_{w}=10^{-3}\;E_{\mathrm{cr}}$ corresponding to $\varepsilon=10^{-2}$ [$\gamma_{\rm cr}\approx 5.3$]. Under such circumstances we find that $W_{T}(\boldsymbol{0})\sim 10^{-5}$ exceeds by $9$ orders of magnitude the corresponding distribution function $W_{T}(\boldsymbol{0})\sim 10^{-14}$ of the standard Schwinger mechanism. Apparently the enhancement becomes stronger as $E_{s}$ decrease gradually. However, it is worth pointing out that such a trend is justified whenever $W_{T}(\boldsymbol{p})$ remains smaller than unity [read discussion above Eq. (3)]. Hence, at $\boldsymbol{p}=\boldsymbol{0}$, this condition translates into the restriction $E_{s}\gg\frac{1}{2}E_{w}2^{\frac{\omega}{4m}}$, and by using the parameters above, for instance, it will imply that $E_{s}\gg 7\times 10^{-4}E_{\rm cr}$.

IV Pair production rate

The density of created electron-positron pairs follows from integration of the single-particle distribution function over the three momentum components. When exploiting both the cylindrical symmetry of our problem and the $2\pi m\gamma^{-1}-$periodicity of $W_{T}(\boldsymbol{p})$ in $p_{\parallel}$ [see below Eq. (LABEL:ImS)] we can write

[TABLE]

where the even feature of $W_{T}(\boldsymbol{p})$ in $p_{\parallel}$ has also been used. To satisfy the condition under which $W_{T}(\boldsymbol{p})$ was derived [see first paragraph in Sec. III], the integral over $p_{\perp}$ must be performed over the region $[0,eE_{s}T]$. However, the fast damping of its integrand in this variable allows us to extend its upper integration limit to infinity with no appreciable error.

At this point, it turns out to be convenient to carry out the change of variable $s^{2}=\omega^{2}\epsilon_{\perp}^{2}/(eE_{s})^{2}$ in Eq (20) and to go over to the rate of created pairs. Consequently,

[TABLE]

where Eq. (14) has been inserted. Observe that the function $s^{2}\mathrm{Im}\mathcal{S}_{0}(s,\gamma_{\parallel})$ grows monotonically in both $s$ and $\gamma_{\parallel}$ and $\frac{eE_{s}}{\omega^{2}}s^{2}\mathrm{Im}\;\mathcal{S}_{0}(s,\gamma_{\parallel})\geqslant\frac{E_{\mathrm{cr}}}{E_{s}}\mathrm{Im}\;\mathcal{S}_{0}(\gamma,\gamma_{\parallel})\gg 1$.333We note that the insertion of Eq. (17) into the condition $\frac{E_{\mathrm{cr}}}{E_{s}}\mathrm{Im}\;\mathcal{S}_{0}(\gamma,\gamma_{\parallel})\gg 1$ provides a restriction $E_{s}\gg\frac{1}{2}E_{w}e^{\frac{\omega}{4m}}$, which is similar to the one given at the end of Sec. III. Therefore, we integrate by parts in $s$ and expand the resulting integrand around $\gamma_{\parallel}\sim 0$. Consequently,

[TABLE]

with $\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}dte^{-t^{2}}$ denoting the error function NIST . The formula above constitutes our main analytic result. It allows us in particular to obtain the scaling behavior of the process with the field parameters, and thus, offers genuine advantages for optimizing the pair production yield and discriminating it from undesirable backgrounds.

Notice that its pre-exponential portion contains the functions $\mathpzc{g}_{\varepsilon}(\gamma)$ and $\mathpzc{g}_{\varepsilon}^{\prime}(\gamma)\equiv\left.\partial\mathpzc{g}_{\varepsilon}(s)/\partial s\right|_{s=\gamma}$ with

[TABLE]

characterizing the decrement of the exponential function. As before, $\mathpzc{I}_{1}(x)=\frac{x}{2}+\frac{x^{3}}{2^{2}4}+\frac{x^{5}}{2^{2}4^{2}6}+\ldots$ refers to the modified Bessel function of the first kind with order one NIST . Additionally, Eq. (22) introduces the function

[TABLE]

whose behavior as a function of the parameter $\gamma$ is depicted in Fig. 4. All curves exhibited there remain below one [$\mathpzc{h}_{\varepsilon}(\gamma)<1$]. They manifest fast decaying laws for both $\gamma\gg 1$ and $\gamma\ll 1$. These trends can be understood when looking for their asymptotes, which turn out to be:

[TABLE]

We note that the expression linked to the case $\gamma\ll 1$ can even be applied to study the behavior of $\mathpzc{h}_{\varepsilon}(\gamma)$ in regions for which $\gamma\gtrsim 1$. The dotted curves in the left portion of Fig. 4 reveal us that the loss of accuracy in such a case is almost undiscernible.

In the regime characterized by the conditions $\gamma\ll 1$ and $m<\omega\lesssim 2m$, the argument of the error function is very small and its leading order contribution behaves as $\mathrm{erf}(x)\sim 2x/\sqrt{\pi}$. In such a scenario, Eq. (22) reproduces the known rate for the Schwinger mechanism [quasi-static limit]

[TABLE]

When $\gamma$ exceeds unity and the critical value $\gamma_{\mathrm{cr}}$ significantly [$\gamma\gg 1$ and $\gamma\gg\gamma_{\mathrm{cr}}$] with $m<\omega\lesssim 2m$, the small argument behavior of the error function still can be applied to Eq. (22) and the production rate approximates

[TABLE]

Interestingly, this formula manifests a nonperturbative dependence not only in $E_{s}$ but also in the field strength $E_{w}$ associated with the pertubative electric mode. Clearly, in the intermediate regime not covered by these asymptotic cases, the density rate for pair production [see Eq. (22)] mixes both tunneling and multi-photon effects.

In order to extend further our knowledge on $\dot{\mathpzc{N}}$, we show in Fig. 5 its behavior as a function of the strong [left panel] and weak [right panel] field strengths. The results exhibited in both panels have been obtained by setting the frequency of the fast-oscillating field to $\omega=1.7m$. Following the discussion at the end of Sec. III, the left panel has been generated by varying the strong field betwen $5\times 10^{-2}\leqslant E_{s}/E_{\rm cr}\leqslant 3\times 10^{-1}$. Here the dashed curve corresponds to the standard Schwinger mechanism [see Eq. (26)], whereas the dotted curves result from the case in which the pair production process is driven by the fast-oscillating mode only. The expression used to generate the curves linked to the latter scenario is given in Refs. popov ; Popov2002 ; Ringwald . Here, by increasing the weak field amplitude by an order of magnitude, the production rate grows by a factor $10^{4}$, indicating that the process occurs in the perturbative regime with absorption of two quanta $\omega$. For comparison we note that the curves for the assisted setup differ by relative factors of about $200$ each at $E_{s}\approx 6.6\times 10^{-2}E_{\rm cr}$. We observe besides that, in average, the slope linked to the rate of the standard Schwinger effect, is larger than the ones corresponding to the assisted setup. Hence, as the ratio $E_{s}/E_{\rm cr}$ grows the enhancing due to the fast-oscillating field becomes less pronounced. This fact corroborates the idea that, in an assisted scenario, there exist two channels for increasing the pair production rate, either by growing the strong field strength or via the absorption of quanta from the fast-oscillating mode. The former path rules the process as $E_{s}$ grows, while the latter dominates as the contrary condition occurs.

Fig. 6 is intended to provide insight about the trend of the rate with the change of $\omega$. It has been obtained by setting the strong field to $E_{s}=6.6\times 10^{-2}E_{\rm cr}$. As in Fig. 5, the dashed line follows from the expression associated with the Schwinger mechanism [see Eq. (26)]. We note that all solid curves closely approach to the value of the standard Schwinger mechanism when $\omega\ll m$.

We wish to put our outcomes in context. So, let us suppose an experiment driven by the strong field to be reached at the forthcoming ELI laser system. Accordingly, we take $E_{s}\approx 6.6\times 10^{-2}E_{\rm cr}$ as a reference parameter. To achieve a peak field strength of this nature a strong focusing–close to the diffraction limit–is required. Consequently, we will suppose that both the spatial extent and the temporal length of the laser pulse are $\ell\sim 1\;\rm\mu m$ and $T\sim 1\;\rm fs$, respectively. This laser is expected to operate with a central frequency $\Omega\approx 1.55\;\rm eV$. Notice that, if the fast-oscillating mode operates at $\omega\approx 1.7m$, the number of cycles it makes during the pulse length $T$ is $N\approx 2.1\times 10^{5}$ which exceeds largely the combined Keldysh parameter $\gamma\approx 26$. Under such a circumstance the number of pairs yielded from an assisted configuration characteristized by $E_{w}=10^{-3}E_{\rm cr}$ is $\mathpzc{n}_{e^{-}e^{+}}=\dot{\mathpzc{N}}\ell^{3}T\approx 7.4\times 10^{16}$ [curve in cyan in the left panel of Fig. 5] would exceed the result associated with the Schwinger mechanism [black dashed curve] $\mathpzc{n}_{e^{-}e^{+}}\sim 1$ by sixteen orders of magnitude roughly. However, when comparing this value with the dottet curve in cyan–corresponding to the case in which PP is due to the fast-oscillating mode only–we find $\mathpzc{n}_{e^{-}e^{+}}\approx 7\times 10^{9}$, which leads to an effective enhancement by seven orders of magnitude, approximately.

Some comments are in order. Firstly, we note that the discussed improvements have been obtained from a single-laser shot only. Certainly, our study provides evidences that the pair production enhancement increases significantly for frequencies $m\lesssim\omega<2m$. We note that, in such an energy regime, differences between pair production in an oscillating electric field and pair production in a standing laser wave have been pointed out, owing to the spatial dependence and magnetic component of the latter Ruf ; Dresden ; Grobe3 . Hence, the results presented here are expected to describe only qualitatively the phenomenon taking place in realistic laser fields. Notwithstanding this, the general physical conclusions drawn regarding the dependence on the electric field strengths of both the strong and the fast-oscillating fields are expected to find their counterparts in a laser-based experiment for the assisted Schwinger mechanism.

V High-energy photon emission as a probe of the dynamically-assisted Schwinger mechanism

The enhacement induced by the superposition of the fast-oscillating wave onto the strong field background might facilitate the experimental verification of the spontaneous production of pairs from the vacuum, by detecting the generated particles directly. However, the high number densities of created pairs promote their recombinations [see Fig. 7], and thus, the emission of photons. Hence, the created electron-positron plasma has a pronounced unstable nature and the number of pairs evolves according to the law $\mathpzc{n}_{e^{-}e^{+}}(t)=\mathpzc{n}_{e^{-}e^{+}}[1+\delta t/\tau]^{-1}$,444This decay law arises when assuming that the associated rate is proportional to both the number of electrons and of positrons present at time $t$, i.e. for $\dot{\mathpzc{n}}_{e^{-}e^{+}}(t)=-\frac{1}{\tau}\mathpzc{n}_{e^{-}e^{+}}^{2}(t)$. where $\mathpzc{n}_{e^{-}e^{+}}$ is the number of pairs at the moment when the external field is switched off [$\delta t\equiv t-T/2=0$], whereas $\tau$ denotes the plasma life time. An estimate for $\tau$ will be given below. It is worth remarking that the number of photons emitted simultaneously at time $t$ is

[TABLE]

While $\mathpzc{n}_{\mathrm{total}}(t)$ denotes the total number of photons present at the time $t$, $\mathpzc{n}_{\gamma}(T)$ refers to the existing amount of them when the external field is switched off.555The pair annihilation occurring during the time in which the field is on has been analyzed numerically based on generalized Vlasov equations Ruffini . Likewise, the electron-positron recombination driven by a plane-wave has been investigated in Refs. Henrich ; Ilderton . In what follows, we suppose the latter causes only a minor effect on the single-particle distribution function $W_{T}(\boldsymbol{p})$ [see Eqs. (14), (18) and (19)] and limit ourselves to the number of photons produced from the moment on when the field is switched off.

Clearly, in a time interval of the order of $\tau$ or larger–in addition to the electron-positron annihilation process–also scattering events of particles and antiparticles are very likely to take place. As a consequence, the initial particle spectrum is supposed to change significantly. In contrast, for a $\delta t\ll\tau$, i.e. for early times, neither the single-particle distribution function nor the initial number of pairs are expected to change appreciably [$\mathpzc{n}_{e^{-}e^{+}}(t)\approx\mathpzc{n}_{e^{-}e^{+}}$]. Likewise, the total number of emitted photons approaches

[TABLE]

This formula constitutes a good approximation only when $\Delta\mathpzc{n}_{\gamma}(t)\ll\mathpzc{n}_{e^{-}e^{+}}$, in which case the consequences of both annihilation and scattering processes can be treated perturbatively. Under such circumstances, the differential number of recombination events per unit volume and unit time approaches:

[TABLE]

where the $1/2$-factor arises since the emitted photons of the final state are indistinguishable. For the sake of simplicity, the shorthand notations ${\mathchar 22\relax\mkern-9.0mu\delta}_{p,q}^{4}=(2\pi)^{4}\delta^{4}(p-q)$, ${\mathchar 22\relax\mkern-12.0mu\textrm{d}}^{3}k_{1,2}=d^{3}k_{1,2}/[2\omega_{\boldsymbol{k}_{1,2}}(2\pi)^{3}]$ and ${\mathchar 22\relax\mkern-12.0mu\textrm{d}}^{3}p_{\pm}=d^{3}p_{\pm}/[2\mathpzc{w}_{\boldsymbol{p}_{\pm}}(2\pi)^{3}]$ with $p_{\pm}=(\mathpzc{w}_{\boldsymbol{p}_{\pm}},\boldsymbol{p}_{\pm})$ and $k_{1,2}=(\omega_{\boldsymbol{k}_{1,2}},\boldsymbol{k}_{1,2})$ have been used. Positive and negative subscripts identify the positron and electron momentum, respectively. Notice that $\mathpzc{w}_{\boldsymbol{p}_{\pm}}=[p_{\perp,\pm}^{2}+p_{\parallel,\pm}^{2}+m^{2}]^{\nicefrac{{1}}{{2}}}$ is the energy of the positron and the electron when the field has been switched off. Here $\overline{\mathcal{M}^{2}}_{e^{-}e^{+}\to\gamma\gamma}$ denotes the unpolarized squared invariant amplitude of the annihilation process [see Fig. 7]. The precise expression of this object can be found in text books–see for instance Ref. Greiner –and reads

[TABLE]

At this point it turns out to be convenient to take into account the identity $\int{\mathchar 22\relax\mkern-12.0mu\textrm{d}}^{3}k_{i}=\int\frac{d^{4}k_{i}}{(2\pi)^{4}}\delta(k_{i}^{2})\Theta(k_{i0})$ and perform the integrations over $k_{2}$ and $\omega_{\boldsymbol{k}_{1}}$. As a consequence,

[TABLE]

where $\alpha=1/137$ is the fine-structure constant. Here $\theta$ is the polar angle that the wave vector $\boldsymbol{n}=\boldsymbol{k}_{1}/{|\boldsymbol{k}_{1}|}$ forms with the polarization direction of the switched off field. In contrast, $\varphi$ represents the azimuthal angle. In the expression above $\overline{\mathcal{M}^{2}}_{e^{-}e^{+}\to\gamma\gamma}$ must be understood as function depending only on $p_{\pm}$ and $k_{1}^{\mu}=\omega_{\boldsymbol{k}_{1}}(1,\boldsymbol{n})$ with

[TABLE]

The integrals over $\boldsymbol{p}_{\perp,\pm}$ can also be carried out approximately. To this end, we first develop the change of variables $\gamma_{\perp,\pm}=(eE_{s})^{2}/[(m^{2}+\boldsymbol{p}_{\perp,\pm}^{2})\omega^{2}]$ and consider the fact that the exponent associated with $W_{T}(\boldsymbol{p}_{+})$ [$W_{T}(\boldsymbol{p}_{-})$] grows monotonically with $\gamma_{\perp,+}$ [$\gamma_{\perp,-}$]. Hence, after integrating over these variables by parts separately, we end up with

[TABLE]

The applied procedure manifests that the main contribution of these integrations results from the region in which $p_{\perp,\pm}/m\ll 1$. This result is somewhat expected since the single-particle distribution function [see Eqs. (14), (18) and (19)] is sharply peaked at $p_{\perp}=0$, and the typical value of momentum perpendicular to the field direction [$\langle\boldsymbol{p}_{\perp}\rangle=0$]666Here $\langle\mathcal{O}\rangle=\int d^{3}p\;\mathcal{O}(\boldsymbol{p})W_{\infty}(\boldsymbol{p})/\int d^{3}p\;W_{\infty}(\boldsymbol{p})$. For computing the integration with $\mathcal{O}(\boldsymbol{p})=p_{\perp}^{2}$, a procedure similar to the one used for obtaining Eq. (22) has been followed.

[TABLE]

remains much smaller than $m$. Hence, the photons are emitted quasi-isotropically on the polar plane while their azimutal distribution follows a nontrivial law to be determined in brief. Still, the main trend of this angular distribution can be anticipated when noting that the typical value of momentum parallel to $\boldsymbol{E}_{s}$ grows linearly with the pulse length [$\langle\boldsymbol{p}_{\parallel}\rangle=0$]:

[TABLE]

As this largely exceeds $p_{\perp}$ from Eq. (35), we find that in average, the angle between the momentum of the created particles and the external field $\theta\sim\langle p_{\perp}^{2}\rangle^{\nicefrac{{1}}{{2}}}/\langle p_{\parallel}^{2}\rangle^{\nicefrac{{1}}{{2}}}\ll 1$. Moreover, Eq. (36) provides evidences that the yielded electrons and positrons are mostly ultrarelativistic because $eE_{s}T/m=2\pi N/\gamma\gg 1$. Indeed, their average energy turns out to be

[TABLE]

with a standard deviation $\Delta\mathpzc{w}_{\boldsymbol{p}_{\pm}}\approx\pm\langle\mathpzc{w}_{\boldsymbol{p}_{\pm}}\rangle/\sqrt{3}$. When the energy momentum balance linked to the recombination process $p_{+}^{\mu}+p_{-}^{\mu}=k_{1}^{\mu}+k_{2}^{\mu}$ is considered, it turns out that–in average–the outcoming photons are emitted back-to-back with a mean energy $\langle\omega_{\boldsymbol{k}_{1}}\rangle\approx\langle\omega_{\boldsymbol{k}_{2}}\rangle\approx\langle\mathpzc{w}_{\boldsymbol{p}_{\pm}}\rangle$. With a strong field strength $E_{s}\approx 6.6\times 10^{-2}E_{\rm cr}$ and a pulse length $T\sim 1\;\rm fs$ corresponding to $N\approx 2.1\times 10^{5}$ for $\omega=1.7\;m$ [$\gamma\approx 26$] the mean photon energy is $\langle\omega_{\boldsymbol{k}_{1}}\rangle\approx 6.3\;\rm GeV$.

Returning back to Eq. (LABEL:integratedevent0), the integrals which remain there cannot be computed analytically. To approximate them, we first develop the change of variables $\gamma_{\parallel,\pm}p_{\parallel,\pm}\omega/(eE_{s})$ and decompose them in sums over integrals defined over regions with $2\pi-$extensions:

[TABLE]

where $\mathpzc{m}$ runs from $\mathpzc{m}_{\mathrm{min}}=-\lfloor(N-1)/2\rfloor$ to $\mathpzc{m}_{\mathrm{max}}=\lfloor(N-1)/2\rfloor$ with $\lfloor x\rfloor$ referring to the integer value of $x$. Observe that the main contribution of each individual integration results from the region of $\gamma_{\parallel,i}\in[(2\mathpzc{m}-1)\pi,(2\mathpzc{m}+1)\pi]$ for which the exponent is minimized. Following our discussion in Sec. III, this takes place at $\gamma_{\parallel,i}=2\mathpzc{m}\pi$. Hence, we expand each exponent up to $\sim(\gamma_{\parallel,i}-2\mathpzc{m}\pi)^{2}$ and set $\gamma_{\parallel,i}=2\mathpzc{m}\pi$ in each pre-exponent. Afterwards, $\gamma_{\parallel,\pm}$ are integrated out and we obtain

[TABLE]

where Eq. (22) has been inserted. Here the function $\mathfrak{F}(\gamma,\theta)$ encodes the angular distribution in the polar plane and reads

[TABLE]

Other functions contained in the expression above, are given by

[TABLE]

An estimate for the number of emitted photons $\Delta\mathpzc{n}_{\gamma}(t)=\Delta\nu\delta t\ell^{3}$ in a volume $\ell^{3}$ can be established from Eq. (LABEL:integratedevent) and reads

[TABLE]

where the change of variable $x=\cos\theta$ has been carried out and

[TABLE]

Here the function $\mathpzc{G}_{\mathpzc{m},\mathpzc{m}^{\prime}}$ reads

[TABLE]

It is worth remarking that, to be consistent with our early-time requirement [see below Eq. (29)], the relation $\Delta\mathpzc{n}_{\gamma}(t)\ll\mathpzc{n}_{e^{-}e^{+}}$ has to be satisfied. Observe that such a condition translates into $\delta t\ll\tau$, where the characteristic time scale ruling the perturbative treatment is given by:

[TABLE]

This expression can be interpreted as a relaxation time of an electron (positron) till annihilation occurs. We point out that analogous expressions are obtained when elastic scattering processes are considered instead. However, as the differential cross sections linked to such phenomena are comparable with the one associated with the annihilation process, the corresponding time scales should not differ significantly from the one given above. Observe that the $\dot{\mathpzc{N}}^{-1}$ dependence exhibited in Eq. (45) allows us to infer qualitatively the behavior of $\tau$ when the external parameters are variated. For instance, by taking into account the results shown in the left panel of Fig. 5, we note that this time scale becomes shorter as the field strength $E_{s}$ increases. Moreover, for a field configuration characterized by $E_{s}\approx 6.6\times 10^{-2}E_{\rm cr}$, $E_{w}=10^{-3}E_{\rm cr}$, a pulse length $T\sim 1\;\rm fs$–corresponding to $N\approx 2.1\times 10^{5}$ for $\omega=1.7m$–the combined Keldysh parameter is $\gamma\approx 26$, $\int_{0}^{1}dx\;\mathfrak{F}(\gamma,x)\approx 5.5\times 10^{4}$ and $\dot{\mathpzc{N}}\approx 7.4\times 10^{16}\;\mathrm{fs}^{-1}\mu\mathrm{m}^{-3}$. Under such a circumstance, the relaxation time amounts to $\tau\approx 1.3\;\rm ns$. This number exceeds–to put it into relation–the para-positronium lifetime [$\tau_{0}\approx 0.12\;\rm ns$] by an order of magnitude.

Now, let us put forward a plausible experimental setup for verifying the dynamically-assisted Schwinger mechanism via the detection of photons resulting from the recombination of yielded electron-positron pairs. We propose a scenario in which the high-density electron-positron plasma is generated between two photon detectors, both placed perpendicular to the polarization direction of the strong field [$\theta=\pi/2$]. We suppose both of them equidistant from the plasma region at a distance $L\approx 2\times 10^{2}\;\rm cm$ with their centers forming a right angle with respect to the propagation direction of the colliding pulses [see Fig. 8]. In addition, we will assume both detectors characteristized by a length scale $\mathpzc{D}\approx 1\;\rm cm$ so that the angular openings on the polar and azimuthal planes are very small [$\delta\varphi,\delta\vartheta\approx 10^{-2}\;\rm rad$]. With all these details in mind, we can proceed–starting from Eq. (LABEL:integratedevent) and taking into account Eq. (45)–to estimate the amount of photons reaching a detector due to the recombination after the field has been switched off:

[TABLE]

where the parameter $\beta=\delta t/\tau$ accounts for the smallness of $\delta t$ relative to $\tau$ [see Eq. (45)]. In order to satisfy safely the early time restriction, we take $\beta=10^{-2}$, i.e. the measurement time should be $\delta t\approx 10\;\rm ps$. It is worth remarking that for our reference parameters–see paragraph that follows Eq. (45)–the form factor attains the value $\mathfrak{F}(\gamma,x=0)\approx 6.9\times 10^{3}$. With this details in mind we find that $\mathpzc{n}_{\mathrm{det}}\approx 7.4\times 10^{8}$ photons should be detected. Notably, if the weak field is reduced by two orders of magnitude while the remaining parameters keep their values, the initial number of pairs changes to $\mathpzc{n}_{e^{-}e^{+}}\approx 1.2\times 10^{12}$ [see red curve in Fig. 5] and the number of photons to be detected $\mathpzc{n}_{\mathrm{det}}\approx 1.2\times 10^{4}$ appears still viable experimentally.

VI Conclusions and outlook

An analytical investigation of the assisted Schwinger mechanism has been carried out starting from the low density approximation to the single-particle distribution function. We have revealed fundamental aspects associated with this process when a weak oscillating field mode is superimposed onto a strong constant field background. It has been shown that the particle spectrum is characterized by tiny oscillations along the external field direction, whereas perpendicular to it, the spectrum falls with the growing of $p_{\perp}$ significantly more slowly than in the case where the production process is driven by the strong field only. Once the field has been switched off, the mean-squared values of the momentum depend nontrivialy on the external field parameters, and the quantity linked to the parallel component to the field exceeds largely the one associated with the perpendicular momentum. As a consequence, most of the particles yielded at that time move approximately parallel to the field direction. Likewise, we have found that the created plasma is composed mostly of ultrarelativistic particles and antiparticles.

Both the single-particle distribution function and the density rate of yielded pairs depend on the strong and weak field strenghts in a nonperturbative way. While the pair production is predicted to increase significantly in a dynamically-assisted setup, the yielded electron-positron plasma has a pronounced unstable nature. This feature demanding to carry out experimental measurement in time intervals significantly smaller than the plasma life time, otherwise, the beneficial aspect conceded by the assisted setup is lost. In connection, we have argued that–under the early-time circumstance–the number of photons emitted as a result of electron-positron recombinations could be large enough as to constitute an indirect signal of the spontaneous production of pairs from the vacuum. Based on in this effect, a plausible experimental setup for their observation has been put forward. The robustness of our estimates for the number of photons to be detected supports the viability of the proposed setup as a genuine channel for verifying the Schwinger mechanism, provided strong field strenghts comparable to those to be reached at ELI and XCELS lasers are exploited.

Acknowledgments

This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Grant No. 388720772 (MU 3149/5-1).

Appendix A Sum of integrals over $\boldsymbol{\mathpzc{c}_{\mathpzc{k}}}$

This appendix is devoted to determine the contribution due to the poles circumvented by the chosen integration path [see Fig. 1]. Explicitly,

[TABLE]

where the function involved in the exponent is

[TABLE]

Over each circle $\mathpzc{c_{k}}$, we have $\tau=\tau_{+\mathpzc{k}}+\mathpzc{r}/\gamma_{\perp}e^{i\varphi}$ with $3\pi/2\leqslant\varphi<-\pi/2$ and $\mathpzc{r}$ is an infinitesimal quantity [$\mathpzc{r}\to 0$]. Consequently,

[TABLE]

In the following we will show that the dependence on $\mathpzc{r}$ of the integral over $\varphi$ guarantees that, at $\mathpzc{r}\to 0$, the right-hand side of Eq. (49) vanishes. To this end, we will focus on determining the leading order dependence of the integral involved in the exponent. While our exposition will center on the sector in wich $\mathrm{Re}\;\tau_{+\mathpzc{k}}>0$, an extension to the remaining case, i.e., $\mathrm{Re}\;\tau_{+\mathpzc{k}}<0$ is straightforward.

The first step toward our aim is to deform appropriately the integration contour of the integral involved in the exponent [see Eq. (49)]. However, this procedure depends on the value of $\varphi$. Indeed, if $\pi/2<\varphi\leqslant 3\pi/2$ or $-\pi/2<\varphi<\arg\tau_{+\mathpzc{k}}$, the path can be chosen without enclosing the pole $\tau_{+\mathpzc{k}}$. Two plausible circuits, covering the described situation are shown Fig. 9 in dotted and dotdashed styles. Conversely, if $\arg\tau_{+\mathpzc{k}}\leqslant\varphi\leqslant\pi/2$, the contour will be chosen containing the pole inside the trajectory. Observe that those cases in which $\varphi=\arg\tau_{+\mathpzc{k}}$ or $\varphi=\pi/2$ demand to surround the pole via a small arc of radius $\mathpzc{r}/\gamma_{\perp}$ [see Fig. 9]. Obviously, the situation described previously implies that the integral over $\varphi$ has to be splitted into three portions; each of which covering one of the described $\varphi$-sectors:

[TABLE]

Let us consider the integration over the path enclosing the pole via the arc [see Fig. 9]. As a consequence of the residue theorem [$\mathrm{Re}\;\tau_{+\mathpzc{k}}>0$]:

[TABLE]

In those cases in which the integration contour avoids the pole, the Cauchy theorem applies and the integral of interest, i.e. the left-hand side in the first line of Eq. (LABEL:residueapplication) with $\pi+\arg\tau_{+\mathpzc{k}}\to\varphi$, is determined by two contributions similar to $\mathpzc{J}_{\mathrm{Re}\;\tau}(\mathpzc{r})$ and $\mathpzc{J}_{\mathrm{ver}}(\mathpzc{r})$, with $\arg\tau_{+\mathpzc{k}}\to\varphi$ and $\varphi$ taking values within the respective sector [see below Eq. (49)]. Conversely, if the inclusion of the pole is required without the necessity of a circumventing arc, the expression for the integral involved in the exponent will coincide with Eq. (LABEL:residueapplication) upto $\mathpzc{J}_{\mathrm{arc}}(\mathpzc{r})$, provided the replacement $\pi+\arg\tau_{+\mathpzc{k}}\to\varphi$ in the first line of Eq. (LABEL:residueapplication) is carried out. Likewise, one will be forced to replace $\arg\tau_{+\mathpzc{k}}\to\varphi$ in both $\mathpzc{J}_{\mathrm{Re}\;\tau}(\mathpzc{r})$ and $\mathpzc{J}_{\mathrm{ver}}(\mathpzc{r})$. Thus, the analysis of the expression above allows us to infer the outcomes related to the diverse $\varphi$ values.

Both the residue of the function $\mathpzc{f}(\tilde{\tau})$ at $\tau_{+\mathpzc{k}}$ as well as the leading order contribution linked to the tiny arc in Fig. 9:

[TABLE]

are independent of $\mathpzc{r}$. Regarding the behavior of the integral $\mathpzc{J}_{\mathrm{Re}\;\tau}(\mathpzc{r})$, i.e., the third contribution in the right-hand side of Eq. (LABEL:residueapplication); here the oscillatory contribution present in its integrand is always much smaller than unity. As a consequence, one can ignore its effect by approaching $[1+\varepsilon\cos(\gamma_{\perp}x-\gamma_{\parallel})]^{-1}\approx 1$ and the integral

[TABLE]

with $\sqrt{1+[\mathrm{Re}\;\tau_{+\mathpzc{k}}]^{2}}\geqslant 0$ becomes independent of $\mathpzc{r}$ as $\mathpzc{r}\to 0$.

Now we focus on the last integral $\mathpzc{J}_{\mathrm{ver}}(\mathpzc{r})$ in Eq. (LABEL:residueapplication), the calculation of which requires a somewhat elaborate procedure. Mainly, because its dependence on $\mathrm{Im}\;\tau_{+\mathpzc{k}}$ leads to analyze the regimes $\mathrm{Im}\;\tau_{+\mathpzc{k}}\ll 1$ and $\mathrm{Im}\;\tau_{+\mathpzc{k}}\gg 1$ separately. As this is the only plausible contribution which may provide a nontrivial dependence on $\mathpzc{r}$, we will replace in it $\arg\tau_{+\mathpzc{k}}$ by $\varphi$, and consider $\varphi$ taking values within the integration region in Eq. (49).

A.1 The case of strong enhancement: $\boldsymbol{\mathrm{Im}\;\tau_{+\mathpzc{k}}\ll 1}$

Let us consider first the situation in which $\mathrm{Im}\;\tau_{+\mathpzc{k}}\ll 1$.777Note that $\mathrm{Im}\;\tau_{+\mathpzc{k}}=\gamma_{\rm cr}/\gamma_{\perp}$ is actually independent of $\mathpzc{k}$. Because of this, the square root involved in the integrand of $\mathpzc{J}_{\mathrm{ver}}(\mathpzc{r})$ is approximately independent of $y\equiv\mathrm{Im}\;\tilde{\tau}$: $(1+[\mathrm{Re}\;\tau_{+\mathpzc{k}}+iy]^{2})^{\nicefrac{{1}}{{2}}}\approx\sqrt{1+[\mathrm{Re}\;\tau_{+\mathpzc{k}}]^{2}}$. Indeed, when the condition $\mathrm{Im}\;\tau_{+\mathpzc{k}}\ll 1$ holds, one can exploit the fact that $y$ is much smaller than unity [$1\gg y$]. Observe that this also implies that $\arg(1+[\mathrm{Re}\;\tau_{+\mathpzc{k}}+iy]^{2})\approx 0$, for all allowed values of $\mathrm{Re}\;\tau_{+\mathpzc{k}}$. Consequently,

[TABLE]

At this point it turns out to be very convenient to perform the change of variable $s=\exp[\gamma_{\perp}y]$. The calculation of the resulting integral is simplified once its integrand is decomposed into partial fractions. The described procedure leads to write

[TABLE]

where an unessential imaginary term independent of $\mathpzc{r}$ and $\varphi$ has been ommited. Inserting Eq. (55) into Eq. (LABEL:jintegration), we find that the essential contribution of the integral involved in the exponent is

[TABLE]

Correspondingly, the outer integration in Eq. (49) [see also Eq. (LABEL:splittingouterintegra1)] behaves as

[TABLE]

which guarantees that–in the regime particularized by the strong enhancement [$\mathrm{Im}\;\tau_{+\mathpzc{k}}\ll 1$]–the sum over the circles eluding the poles gives no contribution to the single-particle distribution function $W_{T}(\boldsymbol{p})$, provided the appropriate limit $\mathpzc{r}\to 0$ is taken.

A.2 The case of weak enhancement: $\boldsymbol{\mathrm{Im}\;\tau_{+\mathpzc{k}}\gg 1}$

The pole always lies below the line $\mathpzc{C_{R}}$ and to the left of $\mathpzc{C_{+}}$ [read also discussion above Eq. (12)]. This provides the following condition for the real and imaginary parts of the pole $\tau_{+\mathpzc{k}}$:

[TABLE]

However, here we will restrict ourselves to the case in which $\mathrm{Im}\;\tau_{+\mathpzc{k}}\gg 1$. As a consequence, $\mathrm{Re}\;\tau_{+\mathpzc{k}}\gg 1$ and the square root involved in the integrand of $\mathpzc{J}_{\mathrm{ver}}(\mathpzc{r})$ [see Eq. (LABEL:residueapplication)] behaves as $(1+[\mathrm{Re}\;\tau_{+\mathpzc{k}}+iy]^{2})^{\nicefrac{{1}}{{2}}}\approx\mathrm{Re}\;\tau_{+\mathpzc{k}}+iy$ and

[TABLE]

The part of the integration that contains $\mathrm{Re}\;\tau_{+\mathpzc{k}}$ can be calculated following the procedure described to determine the integral in Eq. (LABEL:jintegration). Taking into account Eq. (55),

[TABLE]

where the change of variable $s=\exp[\gamma_{\perp}y]$ has been used. Here $s_{+}=\frac{2}{\varepsilon}\exp[i\mathpzc{r}\cos\varphi]$ and $s_{-}=\frac{\varepsilon}{2}\exp[i\mathpzc{r}\cos\varphi]$.

The main contribution to the first integral in Eq. (LABEL:sssz) results from $s\sim s_{+}$. Therefore, up to an unessential imaginary term independent of $\mathpzc{r}$ and $\varphi$, we find

[TABLE]

Conversely, the integration variable linked to the last integral in Eq. (LABEL:sssz) satisfies the condition $s\geqslant 1\gg s_{-}$. Hence, by safely ignoring $s_{-}$ in the denominator, we end up with

[TABLE]

Combining the outcomes in Eq. (LABEL:sssz1) and (61) into Eq. (LABEL:sssz), we obtain

[TABLE]

Inserting this result into Eq. (LABEL:residueapplication), it is straightforward to verify that the outer integration in Eq. (49) behaves as

[TABLE]

with $\vartheta=2\epsilon_{\perp}\omega^{-1}\ln|\mathpzc{r}|\mathrm{Im}\tau_{+\mathpzc{k}}$. Therefore, in the limit of $\mathpzc{r}\to 0$ and under the condition $\mathrm{Im}\;\tau_{+\mathpzc{k}}\gg 1$, the sum over the circles $\mathpzc{c_{k}}$ [see Eq. (49)] provides no contribution either to the single-particle distribution function $W_{T}(\boldsymbol{p})$.

Appendix B No contribution over $\boldsymbol{\mathpzc{C}_{\pm}}$

In this appendix we show that the integrals over the vertical segments $\mathpzc{C}_{\pm}$ [see Fig. 1] give no contribution in the limit $N\to\infty$. Let us first denote

[TABLE]

The function contained in the exponent can be found in Eq. (48). Along $\mathpzc{C_{+}}$, the complex integration variable is characterized by $\tau=\pi N/\gamma_{\perp}+iy$, whereas over $\mathpzc{C_{-}}$ by $\tau=-\pi N/\gamma_{\perp}+iy$. Over the former trajectory the minimum and maximum values of $y$ are $y_{\mathrm{min}}=0$ and $y_{\mathrm{max}}=\pi N/(\sqrt{3}\gamma_{\perp})+1$. Conversely, over the latter these values exchange their roles. For very large value of $N\gg 1$,

[TABLE]

In the following the integrals $\mathcal{J}_{\pm}$ are evaluated by modifying the corresponding paths. However, these defomartions depend upon whether the outer integration variable $y$ is smaller or greater than $\mathrm{Im}\;\tau_{+\mathpzc{k}}$. Because of this, it turns out to be beneficial to introduce a positive parameter $y_{0}\to 0^{+}$ and split the integral over $y$ accordingly

[TABLE]

Within the sector $\mathrm{Im}\;\tau_{+\mathpzc{k}}+y_{0}\leqslant y\leqslant y_{\mathrm{max}}$–covered by the second integral in the right-hand side of Eq. (LABEL:splittingouterintegra)–the chosen integration circuit exhibited in Fig. 10 avoids the poles at $\tau_{+\mathpzc{k}}$ with $\mathpzc{k}\in\mathbb{Z}$ [see Eq. (7)]. Conversely, if the outer integration variable lies within $0\leqslant y\leqslant\tau_{+\mathpzc{k}}-y_{0}$, we can integrate $\mathcal{J}_{1,4}$–inside the first contribution of the right-hand side in Eq. (LABEL:splittingouterintegra)–via a similar trajectory, with the particularization that its vertical path ends at $y\leqslant\tau_{+\mathpzc{k}}-y_{0}$, returning to the origin afterward [dotted line in Fig. 10]. As all poles are located above the dotted line, no circumvention of them is required in this case.

We will focus on the result coming from the path in which the poles are eluded. However, we will extract parallely the outcomes linked to the contour in which these circumventions are not necessary. The next step in our analysis is the application of the Cauchy’s theorem to the problem under consideration. Correspondingly,

[TABLE]

When $0\leqslant y\leqslant\tau_{+\mathpzc{k}}-y_{0}$ the expression for $\mathcal{J}_{\pm}$ is only given by the first line of this formula. We note that the integral over the $\mathrm{Re}\;\tilde{\tau}-$axis is purely real as well as the leading order contribution of each integration over the small circle $\sigma_{\mathpzc{k}}$ [$\tau=\tau_{+\mathpzc{k}}+\mathpzc{r}/\gamma_{\perp}e^{i\chi}$ with $3\pi/2\leqslant\chi<-\pi/2$]:

[TABLE]

Therefore, neither $\int_{0}^{\pm\frac{\pi N}{\gamma_{\perp}}}dx\;\mathpzc{f}(x)$ nor the involved sum contribute to $\mathrm{Im}\;\mathcal{J}_{\pm}$, which is precisely what we need [see Eq. (LABEL:sszh10)]. In the second integration, we can exploit the condition $\pi N/\gamma_{\perp}\gg 1$ to approximate the multivalued function $\left[1+(\pi N/\gamma_{\perp}\pm iy)^{2}\right]^{\nicefrac{{1}}{{2}}}\approx\pi N/\gamma_{\perp}\pm iy$. Consequently,

[TABLE]

The integral involved in this expression belongs to the class of integrals treated previously in appendix A and can be calculated straightforwardly. This leads to write

[TABLE]

With this expression to our disposal, the estimation of the integral defined over the sector $0\leqslant y\leqslant\mathrm{Im}\;\tau_{+\mathpzc{k}}-y_{0}$ [see Eq. (LABEL:splittingouterintegra)] can be carried out without much difficulties. At this point, it is worth mentioning that its integrand has an exponent whose absolute value grows monotonically with $y$ [see Fig. 11]. Hence, the main contribution of $\int_{0}^{\tau_{+\mathpzc{k}}-y_{0}}dy\ldots$ results from those values $y\ll\mathrm{Im}\;\tau_{+\mathpzc{k}}-y_{0}$. Accordingly, $\mathrm{Im}\;\mathcal{J}_{\pm}\approx\pi Ny/\gamma_{\perp}$ and

[TABLE]

Contrary to the previous case, the integral defined over $\mathrm{Im}\;\tau_{+\mathpzc{k}}+y_{0}\leqslant y\leqslant y_{\mathrm{max}}$ depends on whether the number of cycles is odd or even. In the former case the absolute value of the exponent decreases sharply with the increasing of $y$ toward the value $\sim\epsilon_{\perp}^{2}\mathcal{J}_{\pm}/(2eE_{s})$ with $\mathcal{J}_{\pm}\approx\pi N\gamma_{\mathrm{cr}}/\gamma_{\perp}^{2}$. However, in the latter situation [$N$ even] there exist an almost unappreciable growing toward the same value as the integration variable $y$ increases [see Fig. 11]. Hence, the area below both curves approaches

[TABLE]

Observe that the insertion of Eqs. (LABEL:szi) and (LABEL:szf) into Eq. (LABEL:splittingouterintegra) ensures the convergence to zero of the right-hand side of the first line in Eq. (LABEL:sszh10) as $N\to\infty$. Therefore, no contribution over $\mathpzc{C_{\pm}}$ arises to the single-particle distribution function $W_{T}(\boldsymbol{p})$. We remark that this statement applies independently of the value of $\mathrm{Im}\;\tau_{+\mathpzc{k}}$.