In intense electromagnetic fields the vacuum state is unstable and spontaneously generates electron-positron ( ${e}^{-}{e}^{+}$ ) pairs. This is known as the Schwinger effect, which is one of the highly nontrivial predictions in quantum electrodynamics (QED)^[1–3]. Because of the tunneling nature of the Schwinger effect, this interesting phenomenon is exponentially suppressed and the pair production rate is proportional to $\exp \left(-\pi {E}_{\mathrm{cr}}/E\right)$ , where the corresponding Schwinger critical field strength ${E}_{\mathrm{cr}}={m}_e^2{c}^3/e\mathrm{\hslash}=1.3\times {10}^{18}\ \mathrm{V}/\mathrm{m}$ . The associated laser intensity, e.g., $I=4.3\times {10}^{29}$ W/cm², is too high and beyond current technological possibilities. Its detection has therefore remained a challenge for many decades^[4]. However, current advances in high-power laser technology^[5–7] and forthcoming experimental facilities (for example, the Extreme Light Infrastructure, the Exawatt Center for Extreme Light Studies, and the Station of Extreme Light at the Shanghai Coherent Light Source) have led to the hope of QED predictions entering the realm of observation. On the other hand, using X-ray free-electron laser facilities can in principle yield a strong field at about $E=0.1{E}_{\mathrm{cr}}=1.3\times {10}^{17}\ \mathrm{V}/\mathrm{m}$ ^[8] and drive interest in studying pair production under super strong fields.

The Schwinger effect is one of the nonperturbative phenomena in QED, while the understanding of it is still far from complete. Therefore, studying pair production in the nonperturbative regime would deepen our knowledge about the relatively less tested branch of QED. Motivated by this, many exploratory studies of the Schwinger effect based on a number of different theoretical techniques have been undertaken, for example, within the quantum kinetic approach^[9–11] and the real time Dirac–Heisenberg–Wigner (DHW) formalism^[12–18], the WKB approximation^{[19, 20]} as well as the worldline instanton technique^[21]. Schützhold et al.^[21] found that the pair production rate can be strongly enhanced by superimposing the slowly varying strong field with the rapid oscillating weak field, which is now called the dynamically assisted Schwinger effect. In Ref. [22], by using the quantum kinetic approach, the momentum spectrum of the produced pairs has been computed, and the spectrum was found to be extremely sensitive to these physical pulse parameters. The pair production in a pulsed electric field^[23] is to present the signature of the effective mass of the created particles in the strong oscillating electric field. In Refs. [24–26] researchers have shown the importance of pulse shape effects on the pair creation process in different situations. For a concrete description of the various approaches and relevant publications, see our review of pair production^[27].

In this paper we further investigate the Schwinger effect by considering the asymmetric pulse shape with Gaussian envelope and different polarizations. We mainly consider asymmetric pulse shape effects on pair production in different polarizations, e.g., linear, elliptic and circular polarizations. We reveal some novel features of the momentum spectra of created pairs for differently polarized electric fields. In this study the real-time DHW formalism is employed as it leads to efficient calculations in the case of a circularly^[28–30] or elliptically polarized electric field^{[31, 32]}.

This paper is organized as follows. In Section 2 we introduce the model of a background field. In Section 3 we briefly introduce the DHW formalism that is used in our calculations for completeness. In Section 4 we show the numerical results for momentum spectra and analyze the underlying physics. In Section 5 we give the numerical results for the pair number density. We end the paper with a brief summary and discussion in Section 6.

We focus on the study of ${e}^{-}{e}^{+}$ pair production in differently polarized and time-dependent asymmetric electric fields. The explicit form of the external field is given as

 (1)

where ${E}_0/\sqrt{1+{\delta}^2}$ is the field amplitude, ${\tau}_1$ and ${\tau}_2$ are the rising and falling pulse durations, respectively, $\theta (t)$ is the Heaviside step function, $\omega$ is the oscillation frequency, $\phi$ is the carrier phase and $\delta$ represents the field polarization (or the ellipticity). The field parameters are chosen as ${E}_0=0.1\sqrt{2}{E}_{\rm cr}$ , $\omega =0.6m$ , ${\tau}_1=10/m$ and $\phi =0$ , where $m$ is the electron mass. For the falling pulse length, we set the parameter as ${\tau}_2=k{\tau}_1$ , where $k$ is the ratio of the falling to rising pulse length. Throughout this paper, we use natural units $\mathrm{\hslash}=c=1$ .

The main interest in this study is asymmetric pulse duration effects on pair production in differently polarized and time-dependent asymmetric electric fields. We mainly consider two different situations when the rising pulse length ${\tau}_1$ is fixed. One in which the falling pulse length ${\tau}_2=k{\tau}_1$ becomes shorter with $0, and another in which the falling pulse length ${\tau}_2=k{\tau}_1$ becomes longer with $k\ge 1$ .

The DHW formalism is an approach used to describe the quantum phenomena of a system by a Wigner function as the relativistic phase space distribution. It has also been further adopted in the studies of Sauter-Schwinger QED vacuum pair production^[12–16]. The DHW formalism automatically combines quantum electrodynamics with notions familiar from statistical physics^{[19, 22]}, and it allows one to incorporate temporal as well as spatial inhomogeneities^[12–18]. Most importantly, the DHW formalism gives access to the relativistic phase-space distribution of the produced particles. In the following, we present a brief outline of the DHW formalism for completeness.

A convenient starting point is the gauge-invariant density operator of two Dirac field operators in the Heisenberg picture

 (2)

 (3)

The important quantity in the DHW method is the covariant Wigner operator, given as the Fourier transform of the density operator (Equation (2)):

 (4)

By taking the vacuum expectation value of the Wigner operator, we obtain the Wigner function

 (5)

For numerical convenience, the Wigner function can be decomposed into a complete basis set $\{\mathbb{1},{\gamma}_5,{\gamma}^{\mu },{\gamma}_5{\gamma}^{\mu },{\sigma}^{\mu \nu}$ $:= \mathit{i}[{\gamma}^{\mu },{\gamma}^{\nu}]/2\}$ . Then we obtain the $16$ covariant real Wigner components

 (6)

Here ${\mathbb{S}}$ , ${\mathbb{P}}$ , ${\mathbb{V}}_{\mu }$ , ${\mathbb{A}}_{\mu }$ and ${\mathbb{T}}_{\mu \nu}$ are scalar, pseudoscalar, vector, axial vector and tensor, respectively. According to Refs. [12–16] and by using the equations of motion for the fermionic Heisenberg operators, the dynamical equation for the Wigner function is

 (7)

 (8a)

 (8b)

 (8c)

Inserting decomposition (Equation (6)) into the equation of motion, Equation (7), for the Wigner function, we obtain a set of partial differential equations (PDEs) for the 16 Wigner components. Furthermore, for spatially homogeneous electric fields like Equation (1), by using the characteristic method^{[28, 29]} and replacing the kinetic momentum $\mathbf{p}$ with the canonical momentum $\mathbf{q}$ via $\mathbf{q}-e\mathbf{A}(t)$ , the PDEs for the 16 Wigner components can be reduced to 10 ordinary differential equations of the nonvanishing Wigner coefficients

 (9)

For detailed derivations and explicit forms of these 10 equations, we refer the reader to Refs. [16, 17, 33]. Note that the corresponding vacuum nonvanishing initial values are

 (10)

In the following, we express the scalar Wigner coefficient by the one-particle momentum distribution function

 (11)

 (12)

So the one-particle momentum distribution function $f\left(\mathbf{q},t\right)$ can be obtained by solving the following ordinary differential equations, including it as well as the other nine auxiliary quantities:

 (13a)

 (13b)

 (13c)

 (13d)

Finally, by integrating the distribution function $f\left(\mathbf{q},t\right)$ over the full momentum space, we obtain the number density of created pairs defined at asymptotic times $t\to +\infty$ :

 (14)

In this section we report some interesting results for the momentum spectra of the produced particles with several pulse parameters under typical cases of the polarization field, such as linear ( $\delta =0$ ), elliptical ( $\delta =0.5$ ) and circular ( $\delta =1$ ) fields.

It can be seen that the momentum spectrum of the created pair is very sensitive to the asymmetry of the electric field. When the ratio parameter $k$ is changed to $k=0.5$ , the main peak of the momentum spectrum is shifted to the positive ${q}_x$ and the symmetry distribution of the momentum spectrum is destroyed. This effect is similar to the effect of carrier phase studied in Ref. [22]. Considering this fact that, for small $k$ , pulse asymmetry plays the carrier-phase-like role, the physical explanation of momentum spectrum distribution due to pulse asymmetry can be understood by assuming that particles are created with vanishing initial longitudinal momentum. In the presence of $\mathbf{E}(t)$ , the produced pairs are continuously accelerated, and particle momentum is mainly determined by its creation time^{[11, 18]}. At the earlier time ${t}_0$ it is created; after ${t}_0$ , it has to be accelerated at the longer time and finally it gets the higher longitudinal momentum. In general, most pairs are expected to appear at those times corresponding to the local maxima of the field. Then those produced pairs are subject to acceleration by the electric field, and the gained momenta are^[32]

 (15)

Because of the fact that the vector potential vanishes at asymptotic times $t\to \infty$ , the final particle momentum solely depends on the vector potential at the time when the particle was created. For example, the peak in Figure1 when $k=1$ at $\mathbf{q}\left({t}_0\right)=0$ is due to the dominant peak in the electric field at $t=0$ . As $\mathbf{E}(t)$ decreases at later times, less particles are produced. However, as $\mathbf{A}(t)$ increases at the same time, these particles are effectively accelerated more strongly. So the peak positions and/or momentum spectrum patterns depend on the pulse shape. Furthermore, when $k=0.3$ , the main parts of the momentum spectrum also appear for negative ${q}_x$ beside the positive ${q}_x$ peak, splitting the momentum spectrum. Therefore, two peaks are observed. This result is similar to the effect introduced by the frequencies chirp in Ref. [33]. For the very asymmetric case of $k=0.1$ , the momentum spectrum of the particle is again concentrated in the center but the oscillation of the momentum spectrum disappears. Finally, we note that the peak value of the momentum spectrum of the pairs is increased from $2.94\times {10}^{-5}$ ( $k=1$ ) to $8.28\times {10}^{-4}$ ( $k=0.1$ ).

We find that the center maximum value of the momentum spectrum decreases until $k\le 5$ . For the pulse length $k=10$ , the maximum value at the ring is slightly larger than that of symmetrical pulse when $k=1$ . Note that the ring structure in the momentum spectrum is a typical feature of the multiphoton pair production mechanism. Because the produced pairs momentum spectrum lines up exactly with the prediction that

 (16)

after the considering of effective mass^[23], the Keldysh parameters are large enough to support multiphoton pair production. For example, the inner ring is formed by absorbing four photons and the outermost obscured structure corresponds to the absorption of five photons.

With decreasing $k$ , the peaks of the momentum spectra display quite a rich structure and the interference effects gradually vanish. When $k=0.7$ , the peak appears in the upper-left side of the momentum spectrum space. When $k=0.3$ , the partial ring structure vanishes and the momentum spectrum becomes distorted. For the very asymmetric case of $k=0.1$ , the peak position is located at the near central region. Note that, for the circular polarization, the peak value of the momentum spectrum is remarkably enhanced by two orders of magnitude compared to that in the symmetric case, $k=1$ .

We find that, when the falling pulse width is shorter, i.e., $0, the number density of created pairs decreases with the increasing electric field polarization. We also find that the number density of ${e}^{-}{e}^{+}$ pairs in different polarizations increases with decreasing pulse length ratio value $k$ . For the larger compression, it is more obvious, especially for $k=0.4$ to $k=0.1$ . When the pulse length is shorter, the number density increases by more than two orders of magnitude for each polarization. As $k$ decreases, the electric field comprises a strong pulse (when $t<0$ ) superimposed with a weak pulse (when $t>0$ , but having wider frequency components in the sense of Fourier decomposition); therefore, these two half pulses with different time scales act as an effective dynamically assisted mechanism. Thus, this results in an enhancement of the number density of produced pairs.

Concretely, for linear polarization, the number density increases from $1.20\times {10}^{-7}$ when $k=1$ to $1.853\times {10}^{-5}$ when $k=0.1$ . For elliptical polarization, it increases from $8.799\times {10}^{-8}$ when $k=1$ to $1.673\times {10}^{-5}$ when $k=0.1$ . For circle polarization, it increases from $7.237\times {10}^{-8}$ when $k=1$ to $1.414\times {10}^{-5}$ when $k=0.1$ .

On the other hand, in the case of the elongated falling pulse, the number density of created pairs increases almost linearly with the field polarization parameter $\delta$ as well as the pulse elongation parameter $k$ except that for linear polarization, it decreases little when the falling pulse elongation is not large, but still increases with larger $k$ . This is mainly attributed to effect of field asymmetry due to the puse length on the pair production processes. These results therefore show that the degree of pulse asymmetry is an important parameter in the pair production process for polarized electric fields. Note that our results are similar to the findings of Kohlfürst et al.^[24], who considered the single Sauter pulse. They found that the particle number increases first with increasing pulse length until it reaches $\tau =0.5{m}^{-1}$ , then decreases and reaches its minimum at $\tau =30{m}^{-1}$ and finally increases again slowly; see Figure 4 of Ref. [24].

From Figures 7 and 8, we can infer that the number density exhibits polarization dependence for shorter pulse asymmetry and elongated pulse asymmetry of the field. For shorter pulse asymmetry cases, the number density decreases with increasing field polarization, while for elongated pulse cases, the number density increases with increasing field polarization except for the case in which $k=1$ . However, the number density has a minimum at some $k$ for each polarization field, for example, $k=2$ and $k=0.8$ correspond to linear ( $\delta =0$ ) and circular ( $\delta =1$ ) polarization fields, respectively. The number density exhibiting this nonlinear behavior with $k$ can be attributed to the following two features. For larger $k$ , because the duration is elongated, the number increases almost linearly with $k$ for an almost constant pair production rate. On the other hand, for progressively smaller $k$ , progressively wider frequencies make the dynamically assisted mechanism progressively more effective so that it enhances the number density.

In summary, when the falling pulse length is shortened, the number density can be increased by two orders of magnitude; however, when the falling pulse length is extended, the number density is enhanced to within only half orders of magnitude. Therefore, for asymmetric electric fields with different polarizations, in order to effectively increase the number density of the produced electron-positron pairs, it is better to shorten the falling pulse. Note that in our previous work on the linearly polarized case^[25], in which we used the quantum Vlasov equation approach, a similar finding was qualitatively presented.

In this study we have investigated the effects of the asymmetric pulse shape on the momentum spectrum of created ${e}^{-}{e}^{+}$ pairs in strong electric fields for three different polarization fields, linearly, middle elliptically and circularly polarized fields, on the momentum spectrum of created particles by applying the DHW formalism. The main results for the spectra of produced pairs can be summarized as follows.

When the falling pulse length is shorter, for linear polarization, the spectra of the produced pairs exhibit a shift and split of the peaks. For middle elliptic polarization as well as circular polarization, the momentum spectra become distorted and exhibit a shift of the peaks. Finally, for each different polarization, the peaks shifted to the central region in the momentum plane; therefore, peak values were enhanced by two orders of magnitude compared to the symmetric situation. When the falling pulse length is elongated, ring structures appear for different polarizations. We also noted that, for this asymmetric situation, the peak values increased with the field polarization compared to the symmetric case, but were smaller than in the shorter pulse cases. Some phenomena of the momentum spectra are consistent with the effect of frequency chirp^[33].

We also studied the effect of asymmetric falling pulse on the obtained number density. We found that the number density decreases and/or increases with polarization for shorter and/or elongated falling pulses. It is important to note that, when the falling pulse is shorter, the number density of the produced pairs can be significantly enhanced by more than two orders of magnitude.

The results are helpful to understand the influence of the pulse duration, which is an important parameter of the external field, and to deepen our understanding of the external pulse structure. Although these results reveal some useful information about the production of ${e}^{+}{e}^{-}$ pairs in different elliptical polarization cases, in this study we restricted ourselves to multiphoton pair creation, so asymmetric pulse shape effects for pair creation under the Schwinger mechanism need to be studied further for different polarized fields.

To understand why the multiphoton process is not obvious for shorter pulse cases $k<1$ , we note that the traditional standard multiphoton pair production is weaker because the Keldysh parameter $\gamma = m\omega /{eE}_0$ is modified by the inverse of another time scale $\tau$ of the pulse duration. For very small $\tau$ , this means that the oscillation number of the field includes fewer cycles and/or subcycles so that it is strictly not a complete multiphoton process. However, in this case, the number density of pairs can be increased remarkably due to the dynamically assisted mechanism. On the other hand, for the elongation pulse $k>1$ , as $k$ increases, pair creation is dominated by the multiphoton mechanism; at this time for $\omega =0.6m$ , the corresponding number density for the circular polarization is greater than that for the middle elliptical polarization, with the latter greater than that for the linear polarization case (see also Figure 4 of Ref. [31]).

The other important phenomenon observed in our numerical results is the spiral structure in the momentum spectrum that has an intrinsic connection with the spin and/or orbital angular momentum of field photons as well as the produced ${e}^{-}{e}^{+}$ particles. The theoretical analysis for this characteristic is not easy and almost completely ignored in the present study. However, its abundant information about the rotation degree is very important and helpful in understanding the involved strong external field interaction with a vacuum and possible applications in future real experiments.