# Phase effect and symmetry on pair production in spatially inhomogeneous
frequency chirping electric fields

Melike Mohamedsedik  Key Laboratory of Beam Technology of the Ministry of
Education, and College of Nuclear Science and Technology, Beijing Normal
University, Beijing 100875, China  Lie-Juan Li  Key Laboratory of Beam
Technology of the Ministry of Education, and College of Nuclear Science and
Technology, Beijing Normal University, Beijing 100875, China  Li Wang
Institute of Radiation Technology, Beijing Academy of Science and Technology,
Beijing 100875, China  Orkash Amat  Key Laboratory of Beam Technology of the
Ministry of Education, and College of Nuclear Science and Technology, Beijing
Normal University, Beijing 100875, China  Li-Na Hu  Key Laboratory of Beam
Technology of the Ministry of Education, and College of Nuclear Science and
Technology, Beijing Normal University, Beijing 100875, China  B. S. Xie
<EMAIL_ADDRESS>Key Laboratory of Beam Technology of the Ministry of
Education, and College of Nuclear Science and Technology, Beijing Normal
University, Beijing 100875, China  Institute of Radiation Technology, Beijing
Academy of Science and Technology, Beijing 100875, China

###### Abstract

Effect of the carrier envelop phase on the electron-positron pair production
is studied in spatially inhomogeneous electric field with symmetrical
frequency chirping. In high or low original frequency field without chirping
as well as one with chirping, we find that the strength of interference effect
of the momentum spectrum and the reduced particle number are all changeable
periodically with phase, in particular, these periodical changes are more
sensitive to the applied parameters in case of low frequency field. At the
small spatial scale, the reduced particle number change is over one order
magnitude by phase in small chirping. For the reduced particle number, the
different optimal phases are obtained at different spatial scales, however,
the larger the chirping is applied, the higher the created pair number is got.
Interestingly, some different types of symmetries, i.e., the mutual symmetry
of mirror/coincidence for two correlated phases and the individual self
symmetry for single phase, are unfolded on the momentum spectrum. The physical
reason of the mutual symmetry between two correlated phases and also the
individual symmetry for two fixed specific phases are examined and discussed
analytically in detail. The combined roles by phase and chirping on the
periodic and symmetrical behaviors of the momentum spectrum and the reduced
particle number are expected to have the potential extension to more fields
such as that with multidimensional spatial coordinate.

###### pacs:

12.20.Ds, 03.65.Pm, 02.60.-x

##  I Introduction

A strong background field ensures the vacuum to produce real particles, for
example electron-positron ($e^{-}e^{+}$) pair, that is a typical
nonpurterbative prediction known as Schwinger effect in quantum
electrodynamics (QED) Sauter:1931zz ; Heisenberg:1935qt ; Schwinger:1951nm .
It still waits an experimental verification Gelis:2015kya  because the
electric field strength afforded by the current laser technology for pair
production is rather smaller than the Schwinger critical field strength
$E_{cr}={m_{e}^{2}c^{3}}/{e\hbar}\approx 1.3\times 10^{18}{\rm V}/{\rm m}$
(corresponding laser intensity $I=4.3\times 10^{29}{\rm W}/{\rm cm^{2}}$).
However, the results of recent theoretical investigations suggest that vacuum
$e^{-}e^{+}$ pair production is possible through careful combining and shaping
of the laser pulse field even though the external field strength is one or two
orders smaller than $E_{cr}$ Schutzhold:2008pz ; Bell:2008zzb ;
DiPiazza:2009py ; Bulanov:2010ei . One may expect such schemes about the laser
pulses will make experimental tests possible with the advanced laser
facilities in the near future Ringwald:2001ib ; Heinzl:2008an ;
Marklund:2008gj ; Pike:2014wha .

At about 50 years ago, vacuum pair production has been investigated in many
original works Nikishov ; Marinov ; breyzin ; Ritus ; Nikishov1985  by using
simple but very important fields such as constant electric and magnetic fields
Nikishov , alternating electric field Marinov ; breyzin  and so on. Recent
years, many theoretical analysis and numerical calculations have been
performed based on previous researches, see Refs. Hebenstreit:2009km ;
Dumlu:2010vv ; Hebenstreit:2011wk ; Kohlfurst:2017hbd ; Ababekri:2019dkl ;
Ababekri:2020 ; Kohlfurst:2017git ; Kohlfurst:2020 ; Akkermans:2011yn ;
Dumlu:2010ua ; Dumlu:2011rr ; Xie:2017 ; Li:2015cea ; Li:2017qwd ;
Olugh:2018seh ; Wang:2019 ; Olugh:2019nej ; Abdukerim:2013vsa ;
Mohamedsedik:2021 ; Li:2021vjf ; Wang . The momentum spectrum and number
density of created particles are examined analytically or/and numerically
under both spatially homogeneous Hebenstreit:2009km ; Dumlu:2010vv ;
Akkermans:2011yn ; Dumlu:2010ua ; Dumlu:2011rr ; Xie:2017 ; Li:2015cea ;
Li:2017qwd ; Olugh:2018seh ; Wang:2019 ; Olugh:2019nej ; Abdukerim:2013vsa ;
Wang  as well as inhomogeneous electromagnetic fields Hebenstreit:2011wk ;
Kohlfurst:2017hbd ; Ababekri:2019dkl ; Ababekri:2020 ; Kohlfurst:2020 ;
Mohamedsedik:2021 ; Li:2021vjf ; Hu:2022  with many different field shapes.
Recently, Fedotov _et al._ have reviewed and summarized the key theories and
progress in strong field QED in the past decade Fedotov:2022 . These
investigations enable us to understand new and challenging phenomena occurred
in pair production process.

It is well known that the pair creation process is very sensitive to external
field parameters such as field frequency, pulse duration, carrier envelop
phase (CEP), pulse number, spatial scale and so on. The momentum spectrum
or/and particle momentum distribution function display some different
interesting structure even if these field parameters change small. In
particular, pair production process has significant dependence on the CEP
which is the phase difference between carrier wave and envelop of the pulse
profile Mackenroth ; Banerjee . Many investigations have been performed and
the different nonlinear features of vacuum decay are observed for various
oscillating forms of external fields Dumlu:2010vv ; Abdukerim:2013vsa ;
Wang:2019 ; Banerjee ; Aleksandrov ; Bra ; Ababekri:2020 ; Mohamedsedik:2021 .
Nevertheless, to our knowledge, most of them are merely concentrated on the
role of CEP in time dependent background fields except Refs. Ababekri:2020 ;
Mohamedsedik:2021 , where only two typical cases of CEP have been considered
with spatial field inhomogeneity. Therefore, it is necessary to study the CEP
effect on pair production in spatially inhomogeneous electric fields
systematically, and some important features of the momentum spectrum or/and
reduced particle number with spatial inhomogeneity should be unfolded.

In this paper, we study the CEP effect on $e^{-}e^{+}$ pair production in
spatially inhomogeneous fields with symmetrical frequency chirp. Field model
is given as

$\displaystyle E\left(x,t\right)$
$\displaystyle=E_{0}f\left(x\right)g\left(t\right)$ (1)
$\displaystyle=\epsilon\,E_{cr}\exp\left(-\frac{x^{2}}{2\lambda^{2}}\right)\exp\left(-\frac{t^{2}}{2\tau^{2}}\right)\cos(b|t|t+\omega
t+\varphi),$

where $\epsilon E_{cr}$ is the field strength, $\omega$ is the original center
frequency, $\lambda$ and $\tau$ are the spatial and temporal scales, $b$ is
symmetrical chirp parameter in the form $b=\alpha\omega/\tau$ ($\alpha\geq 0$)
Ababekri:2020 ; Mohamedsedik:2021 , and $\varphi$ is the CEP that in unit of
$\varphi_{unit}=\pi/8$ for the numerical simplicity and convenience.

Figure 1: (color online). Temporal electric field shape for
$E\left(t\right)=0.5E_{cr}\exp\left(-\frac{t^{2}}{2\tau^{2}}\right)\cos(b|t|t+\omega
t+\varphi)$. The chirp values are $b=0$, $b=0.5\omega/\tau=0.002m^{2}$,
$b=1.0\omega/\tau=0.004m^{2}$, $b=1.5\omega/\tau=0.006m^{2}$. Other field
parameters are $\tau=25m^{-1}$, $\omega=0.1m$.

For a convenience to study the CEP effect in low frequency field, we plot the
temporal electric field shape for various CEP with different chirp values in
Fig. 1. We can see that the electric field changes from positive peaked
electric field shape when $\varphi=0$ to two opposite signed peaked electric
field when $\varphi=4\pi/8$, and it changes sign oppositely by the negative
peaked electric field shape when $\varphi=\pi$. We note that the oscillation
on the field shape also intensify with chirping. In the low frequency field,
we will explain some interesting phenomena on the momentum spectrum by using
this electric field shape.

In our study, we examine the significant influence of CEP on the momentum
spectrum and the reduced particle number numerically by using the Dirac-
Heisenberg-Wigner (DHW) formalism. We find that the interference effect of
created particles reinforce with CEP. More importantly, in both of high and
low- frequency fields, we have observed several different types of symmetries
which named as exact mirror symmetry, approximated coincidence or/and mirror
symmetry and approximated self symmetry, on the momentum spectrum for the
different correlated translations of CEP, and analyzed by the field model as
well as the equations of motion. The reduced particle number is enhanced
periodically with increasing CEP and has symmetry in both of high and low-
frequency electric fields. Moreover, we obtain the combined effect of CEP and
symmetrical chirp on the momentum spectrum at different spatial scales.
Meanwhile, we show the best optimal values of them which correspond to the
maximum reduced particle number.

Note that the natural units ($\hbar=c=1$) are used throughout this paper and
all quantities are expressed in terms of the electron mass $m$. Since the
direction of the electric field is only one dimensional along the $x$-axis so
that the study is limited to this direction by setting $p_{\perp}=0$.

##  II The DHW formalism

The DHW formalism is one of the quantum kinetic approaches developed from
fermion density operator that is capable of describing many phase-space
features of Dirac vacuum in QED Bialynicki-Birula . This formalism not only
gives us the description of field-theoretic vacuum for $e^{-}e^{+}$ pair based
on the phase-space concepts, but also enables us to perform the calculation
within the standard formulation Bialynicki-Birula ; Vasak:1987um  in both
temporal Olugh:2018seh ; Olugh:2019nej ; Wang  as well as spatial
inhomogeneous electromagnetic fields Hebenstreit:2011wk ; Kohlfurst:2017hbd ;
Ababekri:2019dkl ; Ababekri:2020 ; Kohlfurst:2020 ; Mohamedsedik:2021 ;
Li:2021vjf . Since the details of the derivation of DHW formalism could be
obtained in Refs. Bialynicki-Birula ; Vasak:1987um ; Kohlfurst:2015zxi ;
Hebenstreit:2011pm , we just give a brief presentation of DHW formalism
Kohlfurst:2015zxi ; Hebenstreit:2011pm  for the sake of completeness of the
paper.

Our study is focused on the pair production in time dependent spatially
inhomogeneous electric field which is inhomogeneous on $x$-axis. So that the
set of equations of motion for Wigner components are reduced to four partial
differential equations Kohlfurst:2015zxi  as

$\displaystyle D_{t}\mathbbm{s}-2p_{x}\mathbbm{p}=0,$ (2) $\displaystyle
D_{t}\mathbbm{v}_{0}+\partial_{x}\mathbbm{v}_{1}=0,$ (3) $\displaystyle
D_{t}\mathbbm{v}_{1}+\partial_{x}\mathbbm{v}_{0}=-2m\mathbbm{p},$ (4)
$\displaystyle D_{t}\mathbbm{p}+2p_{x}\mathbbm{s}=2m\mathbbm{v}_{1},$ (5)

with pseudodifferential operator

$D_{t}=\partial_{t}+e\int_{-1/2}^{1/2}d\xi\,\,\,E_{x}\left(x+i\xi\partial_{p_{x}}\,,t\right)\partial_{p_{x}}.$
(6)

Here we define Wigner components $\mathbbm{w}_{0}=\mathbbm{s}$ ,
$\mathbbm{w}_{1}=\mathbbm{v}_{0}$ , $\mathbbm{w}_{2}=\mathbbm{v}_{1}$ and
$\mathbbm{w}_{3}=\mathbbm{p}$. In order to perform the calculation, we employ
the vacuum initial conditions Kohlfurst:2015zxi

${\mathbbm{w}_{0}}_{vac}=-\frac{2m}{\Omega}\,,\quad{\mathbbm{w}}_{2\,vac}=-\frac{2{p_{x}}}{\Omega}\,,$
(7)

where $\Omega$ is the energy of single particle
$\Omega=\sqrt{p_{x}^{2}+m^{2}}$. By subtracting initial vacuum terms in Eq.
(7), we can obtain modified Wigner components $\mathbbm{w}_{k}^{v}$

$\mathbbm{w}_{k}^{v}=\mathbbm{w}_{k}-\mathbbm{w}_{vac},$ (8)

where $\mathbbm{w}_{k}$ is the Wigner component in Eqs. (2) - (5).

The particle number density in phase space can be written as dividing the
total particle energy with the individual particle energyHebenstreit:2011wk

$n\left(x,p_{x},t\right)=\frac{m\mathbbm{s}^{v}\left(x,p_{x},t\right)+p_{x}\mathbbm{v}_{1}^{v}\left(x,p_{x},t\right)}{\Omega\left(p_{x}\right)}.$
(9)

In this way, $n\left(p_{x},t\right)$ or $n\left(x,t\right)$ could be obtained
from $n\left(x,p_{x},t\right)$ by integrating out $x$ or $p_{x}$,
respectively. The total number of created particles could be written

$N\left(t\right)=\int dxdp_{x}n\left(x,p_{x},t\right).$ (10)

Moreover, in order to obtain the nontrivial spatial dependence of results on
$\lambda$, we calculate the reduced quantities
$n\left(p_{x},t\rightarrow\infty\right)/\lambda\equiv\bar{n}\left(p_{x},t\rightarrow\infty\right)$
and
$N\left(t\rightarrow\infty\right)/{\lambda}\equiv\bar{N}\left(t\rightarrow\infty\right)\equiv\bar{N}$.

##  III CEP effect on the momentum spectrum

In this section, we study the CEP effect on the momentum spectrum in both high
and low- frequency inhomogeneous fields with symmetrical frequency chirping.
The momentum spectrum for various CEP at different spatial scales are shown
and analyzed.

###  III.1 High frequency field

In high frequency field, we set $\omega=0.7m$, $\tau=45m^{-1}$ and the pair is
predominantly produced by multiphoton absorbtion. The CEP effect on the
momentum spectrum at different spatial scales when $b=0.2\omega/\tau\approx
0.0031m^{2}$ are shown in Fig. 2.

Figure 2: (color online). The reduced momentum spectrum for various CEP in
high frequency field at different spatial scales. The field parameters are
$\epsilon=0.5$, $\omega=0.7m$, $\tau=45m^{-1}$ and $b=0.2\omega/\tau\approx
0.0031m^{2}$.

At large spatial scale, $\lambda=1000m^{-1}$, obvious oscillation is observed
on the momentum spectrum, and it has periodically changed with increasing CEP.
These oscillations are understood as interference effect of created particles
by multiphoton process. The main peak corresponding to $p=0$ splits first and
then recovers to the original shape with increase of CEP, see Fig. 2. Note
that the momentum peak amplitude is also periodically changeable with CEP.

When spatial scale decreases to $\lambda=10m^{-1}$, we can see the weak
oscillation on the momentum spectrum and it is almost unchangeable with CEP.
However, the peak value on the left side of momentum spectrum decreases while
the right peak increases gradually. Finally, the shape of left peak in Fig. 2
(k) becomes the right one in Fig. $1$ (c) of Ref. Mohamedsedik:2021  and vise
versa, which reveals an exact mirror image symmetry holds for two correlated
phases as $\varphi\leftrightarrow\pi+\varphi$, which will be discussed in the
next section.

At very small spatial scale, $\lambda=2.5m^{-1}$, we still see two momentum
peaks which are caused by the ponderomotive force in the cases of
without/small chirping Kohlfurst:2017hbd ; Ababekri:2020 ; Mohamedsedik:2021 .
The weak dependence of the momentum spectrum on CEP can be attributed to a
fact that the spatial scale and frequency chirping are too small to affect the
role of the ponderomotive force that determines mainly the momentum spectrum.

Table 1: Peak values of the reduced momentum spectrum for various CEP in high frequency field with different spatial scales when $b=0.2\omega/\tau\approx 0.0031m^{2}$. Here $\bar{n}_{main}~{}(\lambda~{}[m^{-1}])$ is main peak value, $\bar{n}_{l}~{}(\lambda~{}[m^{-1}])$ and $\bar{n}_{r}~{}(\lambda~{}[m^{-1}])$ are the peak values on the left and right sides of momentum spectrum, respectively. Other field parameters are $\epsilon=0.5$, $\omega=0.7m$ and $\tau=45m^{-1}$. $\varphi$ | $\bar{n}_{l}(2.5)$ | $\bar{n}_{r}(2.5)$ | $\bar{n}_{l}(10)$ | $\bar{n}_{r}(10)$ | $\bar{n}_{main}(1000)$
---|---|---|---|---|---
$0$ | $0.200$ | $0.196$ | $\mathbf{0.362}$ | $0.299$ | $0.445$
$\pi/8$ | $0.203$ | $0.196$ | $0.345$ | $0.274$ | $0.441$
$2\pi/8$ | $\mathbf{0.204}$ | $0.194$ | $0.313$ | $0.257$ | $0.445$
$3\pi/8$ | $\mathbf{0.204}$ | $0.194$ | $0.293$ | $0.276$ | $0.482$
$4\pi/8$ | $0.199$ | $0.193$ | $0.315$ | $0.305$ | $\mathbf{0.500}$
$5\pi/8$ | $0.197$ | $0.192$ | $0.333$ | $0.333$ | $0.484$
$6\pi/8$ | $0.195$ | $0.194$ | $0.334$ | $0.353$ | $0.445$
$7\pi/8$ | $0.196$ | $0.197$ | $0.321$ | $0.359$ | $0.442$
$\pi$ | $0.196$ | $\mathbf{0.200}$ | $0.299$ | $\mathbf{0.362}$ | $0.445$

In order to see the changes of peak values intuitively, we present the
momentum peak values for various CEP at different spatial scales in Table 1.
It is noticed that the maximum peak values mark with bold for the convenience
reading in the table.

We can see that at large spatial scale, $\lambda=1000m^{-1}$, the largest peak
value occurs for $\varphi=4\pi/8$. When $\lambda=10m^{-1}$, two peak values
exchange places with each other for the left one at $\varphi=0$
Mohamedsedik:2021  and the right one at $\varphi=\pi$, just as we observed on
the momentum spectrum. At very small spatial scale, $\lambda=2.5m^{-1}$, the
peak value $\bar{n}_{l}$ decreases while $\bar{n}_{r}$ increases with CEP,
finally two peaks exchange position with each other. However, the change on
the peak value is very small for increasing CEP.

Figure 3: (color online). The reduced momentum spectrum for various CEP in
high frequency field at different spatial scales when $b=0.5\omega/\tau\approx
0.0078m^{2}$. Other field parameters are the same as in Fig. 2.

Now let us to see the CEP effect on the momentum spectrum at different spatial
scales when the frequency chirping becomes large as $b=0.5\omega/\tau\approx
0.0078m^{2}$, which are shown in Fig. 3.

At large spatial scale, $\lambda=1000m^{-1}$, a strong oscillation is observed
on the momentum spectrum and it is more sensitive to CEP compared with the
case in Fig. 2. The different oscillation characteristic are the combined
results of interference effect of created particles by the chirping
Mohamedsedik:2021  and CEP. When large chirping is fixed, the oscillation is
found intensifying with increasing CEP, and it reaches the maximal at
$\varphi=4\pi/8$, however, it is weaken as increase of CEP when
$\varphi>4\pi/8$. This combination of chirping and CEP results in the periodic
variation of the interference effect between strong and weak. It implies that
the momentum pattern can be also controlled by adjusting the CEP beside the
frequency chirping.

At small spatial scales, $\lambda=10m^{-1}$ and $\lambda=2.5m^{-1}$, stronger
oscillations also occur for increasing CEP compared to the case in Fig. 2,
especially when $\lambda=2.5m^{-1}$, see dashed blue line in Fig. 3. It
demonstrates that CEP plays a more remarkable role on momentum spectrum when
chirp is large enough, where the CEP effect is suppressed greatly by the small
spatial scale. The periodic changes of oscillation pattern and momentum peak
values as well as the exchange phenomena of left- and right- peak are still
observed the on momentum spectrum. For example, when $\lambda=10m^{-1}$, the
peak value on the left side reaches $\bar{n}(t\rightarrow\infty,p)=1.936$ at
$\varphi=5\pi/8$, see Fig. 3(e), however, when $\varphi=6\pi/8$, it starts to
decrease again.

It is noted that when $\varphi=\pi$, see Fig. 2(k) and Fig. 3(k), at all
spatial scales, the momentum spectrum is just the inverse momentum spectrum of
the case when $\varphi=0$, refer to Fig. 1 (c) and (d) in Ref.
Mohamedsedik:2021 . We find it as an exact symmetry of the mirror image
momentum spectrum, which will be examined in the next section in detail.

Table 2: Peak values of the reduced momentum spectrum for various CEP in high frequency field at different spatial scales when $b=0.5\omega/\tau\approx 0.0078m^{2}$. Other field parameters are the same as Table 1.  $\varphi$ | $\bar{n}_{l}(2.5)$ | $\bar{n}_{r}(2.5)$ | $\bar{n}_{l}(10)$ | $\bar{n}_{r}(10)$ | $\bar{n}_{mean}(1000)$
---|---|---|---|---|---
$0$ | $1.383$ | $1.425$ | $1.770$ | $1.583$ | $1.045$
$\pi/8$ | $1.386$ | $1.458$ | $1.749$ | $1.498$ | $1.194$
$2\pi/8$ | $1.375$ | $\mathbf{1.492}$ | $1.643$ | $1.545$ | $1.530$
$3\pi/8$ | $1.397$ | $1.485$ | $1.551$ | $1.623$ | $1.815$
$4\pi/8$ | $1.372$ | $1.441$ | $1.740$ | $1.672$ | $\mathbf{1.861}$
$5\pi/8$ | $1.388$ | $1.415$ | $\mathbf{1.936}$ | $1.710$ | $1.820$
$6\pi/8$ | $1.400$ | $1.387$ | $1.927$ | $1.722$ | $1.526$
$7\pi/8$ | $1.412$ | $1.372$ | $1.777$ | $1.761$ | $1.191$
$\pi$ | $\mathbf{1.425}$ | $1.383$ | $1.583$ | $\mathbf{1.770}$ | $1.045$

In Table 2 we also present the momentum peak values for various CEP when
$b=0.5\omega/\tau\approx 0.0078m^{2}$. We can see that momentum peak values
increase significantly in comparison with the case in Table 1, it is resulted
by the combined effect of CEP and largest chirp. The periodical changes of the
peak values and peak exchangeable in Table 1 still exist. The changes of peak
values at $\lambda=2.5m^{-1}$ is more obvious in comparison with that in the
case of Table 1.

###  III.2 Low frequency field

In low frequency field, we choose $\omega=0.1m$ and $\tau=25m^{-1}$ and the
pair is predominantly produced by tunneling process. The CEP effect on
momentum spectrum at different spatial scales when $b=0$ are shown in Fig. 4.

Figure 4: (color online). The reduced momentum spectrum for various CEP in low
frequency field at different spatial scales when $b=0$. Other field parameters
are $\epsilon=0.5$, $\omega=0.1m$, $\tau=25m^{-1}$.

At large spatial scale, $\lambda=500m^{-1}$, momentum spectrum shrinks first
when $\pi/8\leq\varphi\leq 3\pi/8$, then broadens and shifts towards negative
direction when $4\pi/8\leq\varphi\leq\pi$. We think the reason can be
explained by using the temporal electric field shape in Fig. 1 (a). The
positive peaked electric field shape when $\varphi=0$ changes to two opposite
signed electric field when $\varphi=4\pi/8$, and it changes sign oppositely by
the negative peaked electric field shape when $\varphi=\pi$ finally. Such
change behavior on the field shape induces the changes of motion of created
particles in field region and it is reflected by momentum spectrum. We also
note that the oscillations on the momentum spectrum are intensifying
periodically with increasing CEP. They can be understood as the interference
effect of particles created by the electric field.

Figure 5: (color online). The reduced momentum spectrum for various CEP
parameters in the low frequency field at different spatial scales when
$b=0.5\omega/\tau=0.002m^{2}$. Other field parameters are the same as in Fig.
4.

Moreover, we can also find the periodical changes on the momentum peak values
with CEP. For example, the peak values decrease for small CEP in Fig. 4 (a) -
(d), however, they increase again for large CEP in Fig. 4 (e) - (k). These
periodical changes on momentum spectrum are similar with that in the cases of
high frequency field we have studied in the previous subsection.

At small spatial scales, $\lambda=10m^{-1}$ and $\lambda=2m^{-1}$, momentum
spectrum also broadens and shifts towards negative field region with
increasing CEP. It is because that the negative signed electric field is
stronger and closer to negative peak value with increasing CEP. At the end,
electric field changes sign and turns into negative peaked electric field when
$\varphi=\pi$, see dashed-dotted red line in Fig. 1 (a). Therefore, the
momentum spectrum also changes sign correspondingly, see Figs. 4 (a) - (k)
when $\lambda=10m^{-1}$ and $\lambda=2m^{-1}$, respectively (dashed-dotted red
and dashed blue lines).

Figure 6: (color online). The reduced momentum spectrum for various CEP in the
low frequency field at different spatial scales when
$b=1.0\omega/\tau=0.004m^{2}$. Other field parameters are the same as in Fig.
4.

Furthermore, the momentum peak value decrease gradually with increasing CEP
until $\varphi=4\pi/8$. The two opposite positioned peaks (left and right) are
observed when $\varphi=4\pi/8$. The reason is also that the field has two
opposite peaks when $\varphi=4\pi/8$, see dashed blue line in Fig. 1(a),
therefore, it leads to two opposite positioned momentum peaks correspondingly.
We will give further discussion on this problem in Section. VII.

In Figs. 5 \- 7, we also plot momentum spectrum for various CEP at different
spatial scales with chirp values $b=0.5\omega/\tau=0.002m^{2}$,
$b=1.0\omega/\tau=0.004m^{2}$ and $b=1.5\omega/\tau=0.006m^{2}$, respectively.

Figure 7: (color online). The reduced momentum spectrum for various CEP in the
low frequency field with different spatial scales when
$b=1.5\omega/\tau=0.006m^{2}$. Other field parameters are the same as in Fig.
4.

At large spatial scale, $\lambda=500m^{-1}$, the periodically strengthened
oscillations with increasing CEP still exist and are more significant for the
largest chirp, see solid black lines in Figs. 5 \- 7. These oscillations are
also observed even at small spatial scale, $\lambda=10m^{-1}$, when chirp is
large enough, also see dashed-dotted red lines in Figs. 5 \- 7. For example,
the opposite positioned peaks are still observed for small chirping case when
$\varphi=4\pi/8$ and they are replaced by strong oscillations for the largest
chirp, see Fig. 5(d) - Fig. 7(d). These oscillations are caused by the
different electric field shapes with various CEP which oscillates faster and
faster with increasing chirp, see Fig. 1 (b) - (d), and it can also be
understood as the interference effect of created particles. Again it is caused
by the combined effect of CEP and chirping as the enforced interference of
created particles in the view of Wentzel-Kramers-Brillouin (WKB) and turning
points structure Mohamedsedik:2021 . We also note that at the very small
spatial scale, $\lambda=2m^{-1}$, we could not find obvious oscillation on the
momentum spectrum with increasing CEP even if the chirp becomes the largest,
see the dashed blue lines in Figs. 5 \- 7. Because that the field exists in
very small spatial scale, the field energy is also very small correspondingly.
By the way, the work done by the electric field would be reduced
significantly. Consequently, very few particles are created from tunneling and
unable to lead obvious interference effect on the momentum spectrum.

At all spatial scales, the momentum peak values also change periodically with
CEP, meanwhile increase significantly with increasing chirp. For example, the
maximum peak value $\bar{n}_{max}(500m^{-1})=0.012$ is observed when
$\varphi=4\pi/8$ for the largest chirp $b=1.5\omega/\tau=0.006m^{2}$.
Moreover, it is similar to that of high frequency field, we obtain exact
mirror image spectrum in the low frequency field. For example, one can see the
symmetry between Fig. 5 of Ref. Mohamedsedik:2021  when $\varphi=0$ and Fig.
4(k) - Fig. 7(k) when $\varphi=\pi$ for four different chirping. We will give
detailed explanation in the next section about this symmetry.

##  IV the Symmetry on momentum spectrum

In both of high and low frequency fields, we have observed various types of
symmetries on the momentum spectrum which depend on CEP translation and
spatial scale. Three different symmetries, named exact mirror, approximated
coincidence or/and mirror, and approximated self symmetry, are found,
identified and analyzed by the field model as well as the Eqs. (2) - (5) of
Wigner components. They correspond to the following cases: the exact mirror
symmetry is for $\varphi\leftrightarrow\pi+\varphi$, which is associated the
phases between the first and third quadrant, or the second and fourth one; the
approximated coincidence and/or mirror symmetry are for
$\varphi\leftrightarrow\pi-\varphi$, which is associated the phases between
the first and second quadrant at large or/and small spatial scale; they occur
also for $\varphi\leftrightarrow 2\pi-\varphi$, which is associated the phases
between the first and fourth quadrant, or the second and third one at both of
large and small spatial scales, respectively; Finally the approximated self
symmetry occurs for the special cases of phase $\varphi=4\pi/8$ and
$\varphi=12\pi/8$, where its shifted phase is
$12\pi/8=\pi+4\pi/8=2\pi-4\pi/8$, therefore, it is the combination of exact
mirror and approximated coincidence symmetries.

Figure 8: (color online). The original (left) and exact mirror image (right)
momentum spectrum for $\varphi$ and $\pi+\varphi$ at different spatial scales
in high frequency field. Other field parameters are $\epsilon=0.5$,
$\omega=0.7m$, $\tau=45m^{-1}$ and $b=0.5\omega/\tau=0.0078m^{2}$.

In order to make a full understanding of these symmetries, we give a detailed
analysis in Appendix A for the field model and involving physical quantities
for different phase relations.

####  IV.0.1 Exact mirror symmetry

When $\varphi$ becomes $\pi+\varphi$, the momentum spectrum of them is just
the exactly mirror image with each other so that we call it as the exact
mirror symmetry. In order to see it intuitively, we have shown the momentum
spectrum as the original and exact mirror one for $\varphi$ and $\pi+\varphi$
at different spatial scales in both of high and low- frequency fields, see
Fig. 8 and Fig. 9, respectively, when the fixed chirp is chosen the largest.

Figure 9: (color online). The original and exact mirror image momentum
spectrum for $\varphi$ (left) and $\pi+\varphi$ (right) at different spatial
scales in low frequency field. Other field parameters are $\epsilon=0.5$,
$\omega=0.1m$, $\tau=25m^{-1}$ and $b=1.5\omega/\tau=0.006m^{2}$.

From two figures, we find that for each row of momentum spectrum, the spectrum
on the right is the exact mirror image one of which on the left and vise
versa. We also find that the exact mirror symmetry is strictly valid for all
spatial scales as well as all chirp values. When CEP $\varphi$ shifts towards
to $\pi+\varphi$, the electric field changes sign and propagates in opposite
direction, see Eq. (12). The created particles also move to opposite direction
with opposite signed momentum in the field region and present such exact
mirror image momentum spectrum as shown in Fig. 8 and Fig. 9. We have checked
and ascertained the signs of all physical quantities and Wigner components
which correspond to exact mirror image spectrum. The results are summarized in
Table 3, see the Case i@ in Appendix A. We find that, by a mirror
transformation of momentum, all these physical quantities and Wigner
components still satisfy equations of motion in Eqs. (2) - (5). In this sense,
we have proved the validity of exact mirror symmetry on momentum spectrum,
which can explain or/and understand our numerical results in the previous
section.

By the way, the periodical changes of the peak values and the peak
exchangeable on the momentum spectrum which we mentioned in Table 1 and Table
2 can be also understood by the exact mirror symmetry on the momentum
spectrum.

####  IV.0.2 Approximated coincidence or/and mirror symmetry

For the following two cases, $\varphi\leftrightarrow\pi-\varphi$ and
$\varphi\leftrightarrow 2\pi-\varphi$, we have observed the approximated
coincidence or/and approximated mirror symmetry on the momentum spectrum. It
is worthy to point out that in these two cases, the symmetry is not strictly
valid as in the first case. On the other hand, the types of approximated
symmetries are different for large and small spatial scales. These different
types of approximated symmetries are obtained in Appendix A for two different
correlated translations of CEP by the different approximate treating of the
electric field at large and small spatial scales, respectively.

($1$) The approximated symmetry between $\varphi$ and $\pi-\varphi$.

When the addition of two correlated CEPs is $\pi$, i.e.,
$\varphi\leftrightarrow\pi-\varphi$: At large spatial scale, we observe the
approximated coincidence symmetry on the momentum spectrum for three set of
correlated CEPs as $\pi/8\leftrightarrow 7\pi/8$, $2\pi/8\leftrightarrow
6\pi/8$, $3\pi/8\leftrightarrow 5\pi/8$, in both high and low frequency
fields, see solid black lines in Figs. 2 \- 7. At small spatial scales,
however, we observe the approximated mirror symmetry on the momentum spectrum
for the three set of correlated CEPs that we just mentioned above, see dashed
blue and dashed-dotted red lines in Figs. 2 \- 7.

($2$) The approximated symmetry between $\varphi$ and $2\pi-\varphi$.

When the addition of two correlated CEPs is $2\pi$, i.e.,
$\varphi\leftrightarrow 2\pi-\varphi$: At large (small) spatial scale, we
observe the approximated mirror (coincidence) symmetry on the momentum
spectrum in all chirping cases in both of high and low frequency fields. One
can see the solid black (dashed blue and dashed-dotted red) lines for four set
of correlated CEPs as $\pi/8\leftrightarrow 15\pi/8$, $3\pi/8\leftrightarrow
13\pi/8$, $5\pi/8\leftrightarrow 11\pi/8$, $7\pi/8\leftrightarrow 9\pi/8$ in
Fig. 8, and also see three set of correlated CEPs as $2\pi/8\leftrightarrow
14\pi/8$, $4\pi/8\leftrightarrow 12\pi/8$, $6\pi/8\leftrightarrow 10\pi/8$ in
Fig. 9.

Figure 10: (color online). The reduced momentum spectrum for various chirp
parameters at different small spatial scales in the high frequency field when
$\varphi=4\pi/8$. The chirp values are $b=0.05\omega/\tau=0.00078m^{2}$,
$b=0.1\omega/\tau=0.0016m^{2}$, $b=0.2\omega/\tau\approx 0.0031m^{2}$, and
$b=0.5\omega/\tau\approx 0.0078m^{2}$. Other field parameters are the same as
in Fig. 2.

At small spatial scale, another interesting symmetry occurs for individual CEP
in both high and low frequency fields. We find that the individual momentum
spectrum is approximated self symmetric with respect to $p=0$ for phase of
either $4\pi/8$ or $12\pi/8$. This symmetry is caused by the combination of
two different symmetries, the exact mirror $\varphi\leftrightarrow\pi+\varphi$
and approximated coincidence $\varphi\leftrightarrow 2\pi-\varphi$ symmetry.
In particular, this symmetry is more obvious in high frequency field, see Fig.
10 and Fig. 11.

It is noted that these two types of approximated symmetries are more obvious
in the high frequency field by comparing with that in the low frequency field.
Meanwhile, they pronounce for without/small chirping cases and broken by
strong oscillation on the momentum spectrum for large chirping cases.

Figure 11: (color online). The reduced momentum spectrum for various chirp
parameters at different small spatial scales in the low frequency field when
$\varphi=4\pi/8$. The chirp values are $b=0$, $b=0.5\omega/\tau=0.002m^{2}$,
$b=1.0\omega/\tau=0.004m^{2}$ and $b=1.5\omega/\tau=0.006m^{2}$. Other field
parameters are the same as in Fig. 4.

No matter what kind of approximated symmetry, we can explain them
theoretically by an approximation of the external field and the corresponding
pseudodifferential operator. At large spatial scale, the electric field can be
seen as only time dependent field approximately and the pseudodifferential
operator can be rewritten as in Eq. (15). At small spatial scale, the
pseudodifferential operator is mainly determined by the first term of Eq. (6)
approximately because of the electric field exists in a very small space. Then
we check the signs of all physical quantities according to the symmetry type
which we observed on the momentum spectrum both at large and small spatial
scales. The results are summarized in the Table 3. Finally we verify the
validity of these different types of symmetries by substituting all physical
quantities to the equations of motion in Eqs. (2) - (5), see the Appendix A
for more details.

##  V The reduced particle number density dependence on CEP

To better explain that the sensitivity of momentum spectrum on CEP is
significantly affected by symmetrical chirp and spatial scales, we examine the
ratio of reduced particle number density dependence on CEP at fixed $p=0$ in
both high and low frequency fields.

Here the ratio of reduced particle number density is defined via dividing the
individual reduced particle number density
$\bar{n}_{k}\left(p=0,t,\varphi_{k}\right)$ with the mean value of reduced
particle number density $\bar{n}_{mean}\left(p=0,t,\varphi\right)$, and
written as:

$\bar{n}_{ratio}\left(p=0,t,\varphi\right)=\frac{\bar{n}_{k}\left(p=0,t,\varphi_{k}\right)}{\bar{n}_{mean}\left(p=0,t,\varphi\right)},$
(11)

where
$\bar{n}_{mean}\left(p=0,t,\varphi\right)=\frac{\sum\limits_{k=0}^{15}{\bar{n}_{k}\left(p=0,t,\varphi_{k}\right)}}{16}$.

In high frequency field, at different spatial scales, the ratio of reduced
particle number density depending on CEP for various chirp when $p=0$ are
shown in Fig. 12.

Figure 12: (color online). The ratio of reduced particle number density
dependence on CEP for fixed momentum value $p=0$ at different spatial scales
for various chirp values in high frequency field. The chirp values are
$b=0.2\omega/\tau\approx 0.0031m^{2}$, $b=0.5\omega/\tau\approx 0.0078m^{2}$,
respectively. Other field parameters are the same as in Fig. 2.

We can see that the ratio for the largest chirping is much more greater than
that for small chirping. It demonstrates that momentum spectrum is more
sensitive to CEP in the largest symmetrical chirping case. In the point of
view of spatial scale, the sensitivity of momentum spectrum on CEP is also
suppressed by finite spatial inhomogeneity, and it reduces at small spatial
scales, see dotted red and triangle purple lines in Fig. 12. However, the
sensitivity is stronger even at small spatial scales for largest chirping by
comparing with that for small chirping case in Fig. 12(a). These results are
all consistent with the results of momentum spectrum which we obtained in Fig.
2 and Fig. 3.

In low frequency field, we also plot in Fig. 13, that the ratio of the reduced
particle number density depend on CEP when $p=0$ at different spatial scales
in cases of the smallest and largest chirping.

Figure 13: (color online). The ratio of reduced particle number density
dependence on CEP for fixed momentum $p=0$ at different spatial scales for
various chirp values in low frequency field. The chirp values are
$b=0.5\omega/\tau=0.002m^{2}$, $b=1.5\omega/\tau=0.006m^{2}$. Other field
parameters are the same as in Fig. 4.

We can observe great fluctuation on the value of ratio in both cases. It is
clear that the momentum spectrum is very sensitive to CEP in low frequency
field even with the very small chirping, and this sensitivity is stronger for
the largest chirping. Moreover, the sensitivity of momentum spectrum for CEP
is also suppressed by finite spatial scales, and reduces at small spatial
scales.

Moreover, by the way, we have checked that the behavior of the reduced
particle number density dependence on CEP which found in Fig. 12 and Fig. 13
is similar under other momentum value.

We can conclude that momentum spectrum is more sensitive to CEP in low
frequency field by comparing with the high frequency field. In both of high
and low- frequency fields, the sensitivity is most significant for largest
chirping, meanwhile also suppressed by small spatial scales. These results are
all consistent with that of the momentum spectrum in the previous section.

##  VI CEP effect on the reduced particle number

In this section, we study the CEP effect on the reduced particle number in
both high and low- frequency inhomogeneous fields with symmetrical frequency
chirping.

###  VI.1 High frequency field

When $b=0.2\omega/\tau\approx 0.0031m^{2}$, the reduced particle number
dependence on the CEP is very weak. Here we show the reduced particle number
dependence on CEP at different spatial scales only when
$b=0.5\omega/\tau\approx 0.0078m^{2}$, see Fig. 14.

Figure 14: (color online). The reduced particle number dependence on CEP at
different spatial scales in high frequency field. Field parameters are the
same as in Fig. 3.

We can see that at very small spatial scale $\lambda\leq 2.5m^{-1}$, the
reduced particle number decreases at first and then increases gradually,
minimum reduced particle number occurs for $\varphi=4\pi/8$. However, when
$\lambda\textgreater 2.5m^{-1}$, it is enhanced significantly at the beginning
until CEP reaches $\varphi=4\pi/8$, then it is decreased gradually for
$5\pi/8\leqslant\varphi\leqslant\pi$. The maximum reduced particle number
always occurs for $\varphi=4\pi/8$ except largest spatial scale
$\lambda=1000m^{-1}$. It is just opposite to the changing patterns of cases of
$\lambda\leq 2.5m^{-1}$. We have also noted that the largest reduced particle
number can be obtained for the interval $6m^{-1}\leq\lambda\leq 10m^{-1}$.
However, the maximal enhancement of reduced particle number is most pronounced
for $\lambda=20m^{-1}$ when $0\leqslant\varphi\leqslant 4\pi/8$.

Furthermore, we find that at all spatial scales the reduced particle number is
exactly symmetric about $\varphi=4\pi/8$ in the interval
$0\leqslant\varphi\leqslant\pi$. We would like to point out that this symmetry
is induced by the approximated coincidence or/and mirror symmetry between
$\varphi$ and $\pi-\varphi$ on the momentum spectrum. We have checked and
found that it is also symmetric about $\varphi=\pi$ in the interval
$0\leqslant\varphi\leqslant 2\pi$. This symmetry can be understood by the
exact mirror symmetry between $\varphi$ and $\pi+\varphi$ on the momentum
spectrum.

Figure 15: The optimal spatial scale of reduced particle number for various
CEP with different chirp values: $b=0.05\omega/\tau\approx 0.00078m^{2}$,
$b=0.1\omega/\tau\approx 0.0016m^{2}$, $b=0.2\omega/\tau\approx 0.0031m^{2}$
and $b=0.5\omega/\tau\approx 0.0078m^{2}$. Other field parameters are the same
as in Fig. 3.

In Fig. 15, the optimal spatial scales of reduced particle number for various
CEP with different chirping are plotted. We find that optimal spatial scale is
fixed for increasing CEP when $0\leq b\leq 0.1\omega/\tau\approx 0.0016m^{2}$,
see quadrangle black, dotted red and triangle blue lines in Fig. 15. However,
it starts to sensitive for increasing CEP when chirp reaches
$b=0.2\omega/\tau\approx 0.0031m^{2}$. It shows again that, for large chirp
$b=0.5\omega/\tau\approx 0.0078m^{2}$, the optimal spatial scales are in the
interval $6m^{-1}\leq\lambda_{opt}\leq 10m^{-1}$. These results indicate that
the reduced particle number dependence on CEP is also determined by the
symmetrical chirp value. Therefore, the combined effect of CEP and chirp is
very crucial for pair production process.

###  VI.2 Low frequency field

In Fig. 16 and Fig. 17, we also show the reduced particle number dependence on
CEP for various chirping at small and large spatial scales, respectively. In
both figures, the reduced particle number changes periodically in the interval
$0\leqslant\varphi\leqslant\pi$. In particular, the reduced particle number
decreases at the beginning and takes the minimum when $\varphi=4\pi/8$, then
it increases again when $\varphi\geq 5\pi/8$. Obviously, it is just opposite
to the CEP dependence pattern of reduced particle in high frequency field, see
Fig. 14.

Figure 16: (color online). The reduced particle number dependence on CEP for
various chirp values at small spatial scales in low frequency field . The
chirp values are $b=0$, $b=0.5\omega/\tau=0.002m^{2}$,
$b=1.0\omega/\tau=0.004m^{2}$ and $b=1.5\omega/\tau=0.006m^{2}$. Other field
parameters are the same as in Fig. 4.

Furthermore, we can observe that the curves in the case of without chirping
would be flat with increasing chirp at all spatial scales. It indicates that
the reduced particle number is most sensitive to the increase of CEP in the
case of without chirping, and this sensitivity decreases with increasing
chirp. Because the reduced particle number is enhanced significantly with
increasing chirp no matter whatever the CEP, therefore this sensitivity is
depressed by larger chirping.

Figure 17: (color online). The reduced particle number dependence on CEP for
various chirp values at large spatial scales in low frequency field . The
chirp values are $b=0$, $b=0.5\omega/\tau=0.002m^{2}$,
$b=1.0\omega/\tau=0.004m^{2}$ and $b=1.5\omega/\tau=0.006m^{2}$. Other field
parameters are the same as in Fig. 4.

Now let us turn our attention to the enhancement of the reduced particle
number in the interval $4\pi/8\leqslant\varphi\leqslant\pi$. We found that the
enhancement with CEP is also different for large and small spatial scales. At
small spatial scale, the reduced particle number is enhanced by $1$ to $1.5$
orders of magnitude with CEP for $b=0$ while it is only enhanced about $1.3$
times for the largest chirping, see Fig. 16(a) and (d). At larger spatial
scales, the reduced particle number is enhanced $2$ times for $b=0$ and this
enhancement is decreased to $1.4$ times for the largest chirping, see Fig.
17(a) and (d). It is clear that the reduced particle number enhancement with
increasing CEP is most obvious at small spatial scales without chirping. We
can also conclude that the reduced particle number enhancement is more
sensitive to CEP in low frequency field by comparing with the result of high
frequency field.

Similarly, we find that the reduced particle number is symmetric about
$\varphi=4\pi/8$ in the phase interval $0\leqslant\varphi\leqslant\pi$ except
the largest spatial scale $\lambda=500m^{-1}$. This symmetry is caused by
approximated symmetry on the momentum spectrum. Because three set of
correlated CEPs as $\pi/8\leftrightarrow 7\pi/8$, $2\pi/8\leftrightarrow
6\pi/8$, $3\pi/8\leftrightarrow 5\pi/8$ satisfy the phase relation
$\varphi\leftrightarrow\pi-\varphi$, and cause approximated symmetry on the
momentum spectrum. In this way, the reduced particle is also symmetric about
$\varphi=4\pi/8$ correspondingly. However, this type of approximated symmetry
on the momentum spectrum is not as strict as the exact mirror symmetry,
therefore, it allows that this symmetry is also not strict on the reduced
particle number consequently. As a result, we find the symmetry of reduced
particle number with respect to $\varphi=4\pi/8$ does not hold at the largest
spatial scale $\lambda=500m^{-1}$. Moreover, we examine and find that the
reduced particle number is also exactly symmetric about $\varphi=\pi$ in the
interval $0\leqslant\varphi\leqslant 2\pi$ at all spatial scales. This
symmetry is caused by the exact mirror symmetry on the momentum spectrum.

These results indicate that we may realize enhancement of pair production if
we find out optimal values of CEP and chirping, which is crucial to optimize
the momentum spectrum and the reduced particle number. Therefore, we have also
checked out optimal values of CEP and chirping for the maximum reduced
particle number in both high and low- frequency fields and discussed
qualitatively. We will present and discuss our results in the next section.

##  VII Discussions

In this section we would focus our discussions on the two features, i.e., two
peaks structure on the momentum spectrum and the optimal values of CEP and
chirping in detail.

###  VII.1 Two peaks structure on momentum spectrum

In both of high and low frequency fields, at small spatial scales, we have
observed two momentum peaks on the momentum spectrum. In the high frequency
field, it is caused by the ponderomotive force that have been discussed
extensively in Refs. Kohlfurst:2017hbd ; Ababekri:2020 ; Mohamedsedik:2021 .
Therefore, we just discuss the two momentum peaks which we observed in the low
frequency field when $\varphi=4\pi/8$ here. The reduced momentum spectrum for
various chirp at small spatial scales are shown in Fig. 18.

When $b=0$, we obtain two momentum peaks at small spatial scales as shown in
Fig. 18(a). The reason for these two peaks can be attributed to the changing
of electric field shape. The oscillating mode of the electric field changes
with CEP and turns into two opposite peaks form for $\varphi=4\pi/8$, see
dashed blue line in Fig. 1(a). The particles are created from two opposite
signed field peaks and leave field region in two opposite directions, form the
two opposite positioned peaks (left and right) on the momentum spectrum
consequently. However, these peaks are vanished and replaced by strong
oscillations for large chirp, see Fig. 18(c) and (d). This oscillations could
be understood as the interference effect of created particles.

Figure 18: (color online). The reduced momentum spectrum for various chirp
values at small spatial scales in the low frequency field when
$\varphi=4\pi/8$. The chirp values are $b=0$, $b=0.5\omega/\tau=0.002m^{2}$,
$b=1.0\omega/\tau=0.004m^{2}$ and $b=1.5\omega/\tau=0.006m^{2}$. Other field
parameters are the same as in Fig. 4.

More importantly, when $b=0$, the two momentum peaks are far away more from
the center with increasing spatial scales. It is just opposite to that in the
high frequency field where the two momentum peaks are far away more from the
center with decreasing spatial scales, one can see Fig. 1 (a) and (b) in Ref.
Mohamedsedik:2021 . Because in the low frequency field, more particles are
created by electric field through tunneling when spatial scale is increased,
and momentum spectrum also broadens correspondingly. However, it starts to
shrink with increasing spatial scales when chirp is applied, see Fig. 18(b) -
(d), which is similar to the phenomena of high frequency field we have
mentioned above. It is probably because that low frequency field frequency
contains more and more highfrequency components with increasing chirp, which
are higher than its own original central frequency. In this sense, we have
observed some similar features that found on momentum spectrum in high
frequency field.

###  VII.2 Optimal values of CEP and chirping

In Fig. 19, we also show optimal values of CEP and chirping for maximum
reduced particle number in both of high and low- frequency electric fields. It
is worthy to note that the reduced particle number is symmetric about
$\varphi=4\pi/8$ in both of high and low- frequency fields as we mentioned in
the previous section, and we have $\bar{N}_{\varphi=0}=\bar{N}_{\varphi=\pi}$
and $\bar{N}_{\varphi=3\pi/8}=\bar{N}_{\varphi=5\pi/8}$. In the legend of Fig.
19 we have given the CEP in first quadrant for simplicity. We can see that the
maximum reduced particle number is always occurring for the largest chirping
at any spatial scale, however, it is different for CEP at different spatial
scales in both of high and low- frequency fields.

Figure 19: (color online). The maximum reduced particle number dependence on
CEP for high(a) and low(b) frequency electric fields at different spatial
scales. The field parameters are $\omega=0.7m$, $\tau=45m^{-1}$,
$\epsilon=0.5$, $b=0.5\omega/\tau=0.0078m^{2}$ for (a) and $\omega=0.1m$,
$\tau=25m^{-1}$, $\epsilon=0.5$, $b=1.5\omega/\tau=0.006m^{2}$ for (b),
respectively.

In high frequency field, when $\lambda\leqslant 2.5{m}^{-1}$, the reduced
particle number decreases first and increases gradually with increasing CEP
which we have mentioned in Fig. 14. Therefore the maximum reduced particle
number occurs for $\varphi=0$ or $\pi$. When
$4{m}^{-1}\leqslant\lambda\leqslant 30{m}^{-1}$, the reduced particle number
increases until $\varphi=4\pi/8$ with CEP, the maximum obtains for
$\varphi=4\pi/8$ correspondingly. At very large spatial scale, e.g.,
$\lambda=1000{m}^{-1}$, the maximum reduced particle number occurs for
$\varphi=3\pi/8$ or $5\pi/8$. Nevertheless, it is worthy to emphasize that
although the maximum reduced particle number obtaines for various CEP,
however, in high frequency field, at fixed spatial scale, it is a little bit
larger than the other particle number. For example, when $\lambda=4{m}^{-1}$,
maximum particle number occurs for $\varphi=4\pi/8$ with
$\bar{N}_{{max}{(\varphi=4\pi/8)}}=1.4787$, and the other non maximum particle
numbers that corresponding to other CEP values are
$\bar{N}_{\varphi=3\pi/8}=\bar{N}_{\varphi=5\pi/8}=1.4732$,
$\bar{N}_{\varphi=2\pi/8}=\bar{N}_{\varphi=6\pi/8}=1.46$. We can conclude
again that the reduced particle number has less dependence on CEP in high
frequency field. This result can also be seen in the Fig. 4 of Ref.
Mohamedsedik:2021 .

In the low frequency field, the maximum reduced particle number is always
occurring for $\varphi=0$ or $\pi$ except the case of $\lambda=1.6{m}^{-1}$
where it is for $\varphi=4\pi/8$. It can also be seen in Fig. 16(d). In fact,
we can understand it by the frequency spectrum of electric field with
symmetrical frequency chirp $b=1.5\omega/\tau=0.006m^{2}$ for different two
CEPs of $\varphi=0$ and $\varphi=4\pi/8$ as shown in Fig. 20.

Figure 20: Frequency spectrum of the electric field with the symmetrical chirp
value $b=1.5\omega/\tau=0.006m^{2}$ for different two CEP values in the low
frequency field. The other field parameters are $\epsilon=0.5$, $\omega=0.1m$,
$\tau=25m^{-1}$.

We can see that when $\omega\leq 0.35m$, the frequency amplitude starts from
$0.55$ and increases to $1.7$ for $\varphi=0$ while it starts from $0$ and
increases to $1.65$ for $\varphi=4\pi/8$. It is clear that for fixed frequency
range there are more large amplitude frequency components when $\varphi=0$
than the case of $\varphi=4\pi/8$ one. In this way, the dynamical assisted
role of symmetrical frequency chirp in the low frequency field which we
mentioned in Ref. Mohamedsedik:2021  is more significant in comparison with
the case of $\varphi=4\pi/8$, and it can produce more particles in the field
region. Consequently, we can observe that the maximum particle number in the
case of $\varphi=0$ are higher than that in the case of $\varphi=4\pi/8$.

##  VIII Summary

We have studied CEP effect on the pair production in spatial inhomogeneity
electric field with frequency chirping by using the DHW formalism. We have
exhibited the different impacts of the field parameters, such as CEP,
frequency chirping and spatial scale, on the momentum spectrum and the reduced
particle number. The main results are summarized as follows.

In high frequency field, with increasing CEP, the periodical changes are found
about the interference effect on the momentum spectrum, the momentum peaks and
the reduced particle number. The interference effect can be understood as the
consequence of combined effects of CEP and chirping. The enhancement of the
reduced particle number are most significant in case of largest chirping.
Moreover, it is found that the reduced particle number is exactly symmetric
about $\varphi=4\pi/8$ and also about $\varphi=\pi$ in phase intervals
$0\leqslant\varphi\leqslant\pi$ and $0\leqslant\varphi\leqslant 2\pi$,
respectively, at fixed spatial scales. We also find the different CEP
dependence of the optimal spatial scale in different case of chirping, i.e.,
in the case of without/small chirping, it is almost independent of CEP,
however, in the case of large chirping, it depends on CEP.

In low frequency field, CEP has crucial effect on the shrink or broaden of the
momentum spectrum. When chirping is fixed, we observed the periodical
strong/weak variation of the interference effect with increasing CEP. There
are two opposite positioned momentum peaks (left and right) at
$\varphi=4\pi/8$ and it is discussed qualitatively. The reduced particle
number change is also periodical with increaing CEP. The enhancement of the
reduced particle number in the interval $4\pi/8\leqslant\varphi\leqslant\pi$
is evident and its enhancement degree with CEP is different at large and small
spatial scales. At small spatial scale, the reduced particle number is
enhanced over one order in the small/without chirping case with CEP while it
is only enhanced about fewer times for the large chirping. At larger spatial
scales, the reduced particle number is enhanced fewer times in the
small/without chirping case with CEP but it is decreasing a little evenly for
the large chirping. These results indicate that the reduced particle number is
most sensitive to the CEP in the case of without chirping, and this
sensitivity decreases with chirping. At the small spatial scale, the
sensitivity of particle number to CEP is more suppressed by the larger
chirping in comparison with that at the large spatial scale.

More importantly and interestingly, in both of high and low frequency field,
we observed several different types of symmetries on the two-correlated
momentum spectrum such as exactly mirror symmetry, approximated coincidence
or/and mirror symmetry for the different translations of two-correlated CEPs.
And a specific approximated self symmetry on the individual momentum spectrum
is also found at phase of either $4\pi/8$ or $12\pi/8$, where it is just the
combination of exact mirror and approximated coincidence symmetries. It is
noted that the symmetry is different for large and small spatial scales. We
examined these symmetries by field model and the equations of motion in DHW,
and the validity of the theoretical analysis conforms with our numerical
results.

We also studied the reduced particle number density dependence on CEP at fixed
momentum. It is found that particle number density is more sensitive to CEP in
the largest symmetrical chirping case and this sensitivity is also suppressed
by the small spatial scales in both high and low frequency fields. It is also
noted that the particle number density is more sensitive to CEP in low
frequency field in comparison with the high frequency field. Moreover, the
optimal values of CEP and chirping are also investigated in both frequency
fields. We found that the maximum reduced particle number is always occurring
for the largest chirping at any spatial scale, however, it is different for
CEP at different spatial scales.

These results presented in the paper suggest that, beside the chirping, the
CEP plays also an important role on the pair production in spatial
inhomogeneous electric field, therefore, the exhibited periodicity and the
abundant symmetry of the momentum spectrum and the reduced particle number are
the combinational effects of field parameters. The results and conclusions
have potential implication by an extension to the more realistic field model
such as electromagnetic field with multidimensional spatial scales and also
the practical application for the created positron spectrum through
controlling the field phase.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(NSFC) under Grant No. 11875007, 11935008. The computation was carried out at
the HSCC of the Beijing Normal University.

##  Appendix A Symmetry of momentum spectrum

In this appendix, we give a detailed description for various type of symmetry
which appears in the obtained momentum spectrum shown in the Sec. III and
revealed Sec. IV.

####  A.0.1 Exact mirror symmetry

The exact mirror symmetry holds for two phases as
$\varphi\leftrightarrow\pi+\varphi$. It can be explained by the following
consideration. First we note that when $\varphi$ becomes $\pi+\varphi$, the
original electric field form in Eq. (1) changes sign as

$\displaystyle
E\left(x,t;\pi+\varphi\right)=\epsilon\,E_{cr}\exp\left(-\frac{x^{2}}{2\lambda^{2}}\right)\exp\left(-\frac{t^{2}}{2\tau^{2}}\right)\cos[b|t|t+\omega
t+(\pi+\varphi)]$ (12)
$\displaystyle=-\epsilon\,E_{cr}\exp\left(-\frac{x^{2}}{2\lambda^{2}}\right)\exp\left(-\frac{t^{2}}{2\tau^{2}}\right)\cos(b|t|t+\omega
t+\varphi)$ $\displaystyle=-E\left(x,t;\varphi\right).$

Thus it leads to the momentum $p_{x}$ changes sign while $t$ does not, which
means the $x$ should change the sign correspondingly. Moreover, because
$\Omega(p_{x})$ is an even function of $p_{x}$ so that from Eq. (7), we have
$\mathbbm{s}^{v}\left(x,p_{x},t\right)$ even and
$\mathbbm{v}_{1}^{v}\left(x,p_{x},t\right)$ odd, and finally it concludes that
the particle number density is an even function about $p_{x}$

$n\left(x,p_{x},t\right)_{E\left(\varphi\right)}=n\left(-x,-p_{x},t\right)_{E\left(\varphi+\pi\right)},$
(13)

since that

$\frac{m\mathbbm{s}^{v}\left(x,p_{x},t\right)+p_{x}\mathbbm{v}_{1}^{v}\left(x,p_{x},t\right)}{\Omega\left(p_{x}\right)}=\frac{m\mathbbm{s}^{v}\left(-x,-p_{x},t\right)+p_{x}\mathbbm{v}_{1}^{v}\left(-x,-p_{x},t\right)}{\Omega\left(-p_{x}\right)}.$
(14)

By remembering in mind that
$D_{t}\left(\partial_{t},E_{x},\partial_{p_{x}}\right)=D_{t}\left(\partial_{t},-E_{x},-\partial_{p_{x}}\right)$,
$\mathbbm{p}^{v}\left(x,p_{x},t\right)=-\mathbbm{p}^{v}\left(-x,-p_{x},t\right)$
and
$\mathbbm{v}_{0}^{v}\left(x,p_{x},t\right)=\mathbbm{v}_{0}^{v}\left(-x,-p_{x},t\right)$,
we can conclude that when two phases is correlated as
$\varphi\leftrightarrow\pi+\varphi$, the Eqs. (2) - (5) of all the Wigner
components are invariant under the exchange of $p_{x}$ and $-p_{x}$,
therefore, we call it as the mirror symmetry. The odd/even signs, i.e., $-/+$,
of all involved physical quantities under this exact mirror symmetry are given
in Table 3.

####  A.0.2 Approximated symmetry

For another two cases, $\varphi\leftrightarrow\pi-\varphi$ and
$\varphi\leftrightarrow 2\pi-\varphi$, we observed the approximated
coincidence or/and mirror symmetry in both of high and low frequency fields.
It is worthy to note that, interestingly, the symmetry type is different at
large and small spatial scales.

First we would like to discuss the pseudodifferential operator in Eq. (6) and
obtain the approximated expressions for the electric field at the large and
small spatial scales, respectively. It is well known that when the spatial
scale is very large, we treat the electric field approximately as only time
dependent field by neglecting the spatial inhomogeneity. Thus the
pseudodifferential operator in Eq. (6) can be rewritten as

$D_{t}\approx\partial_{t}+e\int_{-1/2}^{1/2}d\xi\,\,\,E_{x}\left(t\right)\partial_{p_{x}}.$
(15)

On the contrary, when the spatial scale is very small, it means that the
electric field exists only in a very small space, so that the expression of
pseudodifferential operator is mainly determined by the first term of Eq. (6).
In this way, we can write it approximately as $D_{t}\approx\partial_{t}$.

In the following, we will discuss the corresponding symmetry under two
different correlated translations of CEP, i.e.,
$\varphi\leftrightarrow\pi-\varphi$ and $\varphi\leftrightarrow 2\pi-\varphi$,
in both of the large and small spatial scales.

($1$) When the addition of two correlated CEPs is $\pi$, i.e.,
$\varphi\leftrightarrow\pi-\varphi$, the electric field is the odd under this
phase exchange transformation as well as the reversed time.

At the large spatial scale, i.e., quasihomogeneous case, the electric field is
approximate space-independent so that the $p_{x}$ can keep the same due to
both reversed change of $t$ and $x$. It leads to the approximated coincidence
symmetry on the momentum spectrum in both of the high and low frequency
fields. In fact in this case, from Eq. (7) we have
$\mathbbm{s}^{v}\left(x,p_{x},t\right)=\mathbbm{s}^{v}\left(-x,p_{x},-t\right)$
and
$\mathbbm{v}_{1}^{v}\left(x,p_{x},t\right)=\mathbbm{v}_{1}^{v}\left(-x,p_{x},-t\right)$.

At small spatial scales, electric field leads to $p_{x}$ changes sign under
the reversed $t$ only so that we have observed approximated mirror symmetry on
the momentum spectrum in both of high and low frequency fields. Similarly in
this situation, from Eq. (7), we have an even
$\mathbbm{s}^{v}\left(x,p_{x},t\right)=\mathbbm{s}^{v}\left(x,-p_{x},-t\right)$
and an odd
$\mathbbm{v}_{1}^{v}\left(x,p_{x},t\right)=-\mathbbm{v}_{1}^{v}\left(x,-p_{x},-t\right)$
expressions.

We can examine further the odd/even signs of all other physical quantities and
Wigner components which corresponding to different case at different spatial
scales by using the approximated expression of pseudodifferential operator as
well as the equations of motion in Eqs. (2) - (5). They are shown in Table 3
by Cases ii@(a) and ii@(b1), respectively.

($2$) When the addition of two correlated CEPs is $2\pi$, i.e.,
$\varphi\leftrightarrow 2\pi-\varphi$, the electric field is the even under
this phase exchange transformation as well as the reversed time. Therefore, at
large (small) spatial scale, we can observe the approximated mirror
(coincidence) symmetry in both of high and low frequency fields due to the
complete similar reasons about the $-/+$ of $p_{x}$ mentioned above. The
detailed description for the invariant equations of DHW is given in Table 3,
see Cases iii@(a) and iii@(b2).

For two specific phases at either $4\pi/8$ or $12\pi/8$, there exists an
approximated individual symmetry by itself, because that it is the
combinational symmetry by both types of exact mirror symmetry in Case i@ due
to $12\pi/8-4\pi/8=\pi$ and also approximated coincidence symmetry in Case
iii@(b2) due to $12\pi/8+4\pi/8=2\pi$, see Case iv@(b3).

Table 3: Three mutual symmetry for three different CEP translations, Case i@, ii@ and iii@, and one individual self symmetry for phase $4\pi/8$ and $12\pi/8$, Case<EMAIL_ADDRESS>The (a) and (b) denote the large and small spatial scale. The (b1), (b2) and (b3) denote three different symmetry types at small spatial scale, respectively. The $-/+$ sign denotes the odd/even property of various physical variables or/and quantities under the phase translation. Case | Physical quantity | Sign | Symmetry type
---|---|---|---
i@ | $(t,p_{x},x,\Omega,E_{x})$ | ($+$, $-$, $-$, $+$, $-$) | exact mirror
$\varphi\leftrightarrow\pi+\varphi$ | $(\partial_{t},\partial_{p_{x}},\partial_{x},D_{t})$ | ($+$, $-$, $-$, $+$)
all spatial scales |  $(\mathbbm{s}$, $\mathbbm{v}_{1}$, $\mathbbm{p}$, $\mathbbm{v}_{0})$ | ($+$, $-$, $-$, $+$)
ii@(a) | $(t,p_{x},x,\Omega,E_{x})$ | ($-$, $+$, $-$, $+$, $-$) | approximated coincidence
$\varphi\leftrightarrow\pi-\varphi$ | $(\partial_{t},\partial_{p_{x}},\partial_{x},D_{t})$ | ( $-$, $+$, $-$, $-$)
large spatial scales |  $(\mathbbm{s}$, $\mathbbm{v}_{1}$, $\mathbbm{p}$, $\mathbbm{v}_{0})$ | ($+$, $+$, $-$, $+$)
ii@(b1) | $(t,p_{x},x,\Omega,E_{x})$ | ( $-$, $-$, $+$, $+$, $-$) | approximated mirror
$\varphi\leftrightarrow\pi-\varphi$ | $(\partial_{t},\partial_{p_{x}},\partial_{x},D_{t})$ | ($-$, $-$, $+$, $-$)
small spatial scales |  $(\mathbbm{s}$, $\mathbbm{v}_{1}$, $\mathbbm{p}$, $\mathbbm{v}_{0})$ | ($+$, $-$, $+$, $+$)
iii@(a) | $(t,p_{x},x,\Omega,E_{x})$ | ($-$, $-$, $+$, $+$, $+$) | approximated mirror
$\varphi\leftrightarrow 2\pi-\varphi$ | $(\partial_{t},\partial_{p_{x}},\partial_{x},D_{t})$ | ($-$, $-$, $+$, $-$)
large spatial scales |  $(\mathbbm{s}$, $\mathbbm{v}_{1}$, $\mathbbm{p}$, $\mathbbm{v}_{0})$ | ($+$, $-$, $+$, $+$)
iii@(b2) | $(t,p_{x},x,\Omega,E_{x})$ | ($-$, $+$, $-$, $+$, $+$) | approximated coincidence
$\varphi\leftrightarrow 2\pi-\varphi$ | $(\partial_{t},\partial_{p_{x}},\partial_{x},D_{t})$ | ($-$, $+$, $-$, $-$)
small spatial scales |  $(\mathbbm{s}$, $\mathbbm{v}_{1}$, $\mathbbm{p}$, $\mathbbm{v}_{0})$ | ($+$, $+$, $-$, +)
iv@(b3) |  |  |
$4\pi/8$ or/and $12\pi/8$ |  |  i@+iii@(b2) | approximated self symmetry
small spatial scales |  |  |

In summary we have found that no matter what kind of exact or/and approximated
symmetry among the two correlated CEPs and also two specific phases of either
$4\pi/8$ or $12\pi/8$, the invariant form of DHW works in this way. And all
the numerical results present in the paper support and conform with these
symmetries.

## References

  * (1) F. Sauter, Uber das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs, Z. Phys. 69, 742 (1931).
  * (2) W. Heisenberg and H. Euler, Consequences of Dirac’s theory of the positron, Z. Phys. 98, 714 (1936).
  * (3) J. S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82, 664 (1951).
  * (4) F. Gelis and N. Tanji, Schwinger mechanism revisited, Prog. Part. Nucl. Phys. 87, 1 (2016).
  * (5) R. Schützhold, H. Gies and G. Dunne, Dynamically assisted Schwinger mechanism, Phys. Rev. Lett. 101, 130404 (2008); G. V. Dunne, H. Gies and R. Schutzhold, Catalysis of Schwinger vacuum pair production, Phys. Rev. D 80, 111301(R) (2009).
  * (6) A. R. Bell and J. G. Kirk, Possibility of prolific pair production with high power lasers, Phys. Rev. Lett. 101, 200403 (2008).
  * (7) A. Di Piazza, E. Lotstedt, A. I. Milstein and C. H. Keitel, Barrier control in tunneling $e^{+}e^{-}$ photoproduction, Phys. Rev. Lett. 103, 170403 (2009).
  * (8) S. S. Bulanov, V. D. Mur, N. B. Narozhny, J. Nees and V. S. Popov, Multiple colliding electromagnetic pulses: a way to lower the threshold of $e^{+}e^{-}$ pair production from vacuum, Phys. Rev. Lett. 104, 220404 (2010).
  * (9) A. Ringwald, Pair production from vacuum at the focus of an X-ray free electron laser, Phys. Lett. B 510, 107 (2001).
  * (10) T. Heinzl and A. Ilderton, Exploring high intensity QED at ELI, Eur. Phys. J. D 55, 359 (2009).
  * (11) M. Marklund and J. Lundin, Quantum vacuum experiments using high intensity lasers, Eur. Phys. J. D 55, 319 (2009).
  * (12) O. J. Pike, F. Mackenroth, E. G. Hill and S. J. Rose, A photon photon collider in a vacuum hohlraum, Nature Photonics. 8, 434 (2014).
  * (13) A. I. Nikishov, Pair production by a constant external fields, Sov. Phys. JETP. 30, 660 (1970).
  * (14) E. Brézin and C. Itzykson, Pair production in vacuum by an alternating field, Phys. Rev. D 2, 1191 (1970).
  * (15) M. S. Marinov and V. S. Popov, Electron-positron pair creation from vacuum induced by variable electric field, Fortsch. Phys. 25, 373 (1977).
  * (16) V. I. Ritus, Quantum effects of the interaction of elementary particles with an intense electromagnetic field, J. Sov. Laser. Res. 6, 497 (1985).
  * (17) A. I. Nikishov, Problems of intense external-field intensity in quantum electrodynamics, J. Sov. Laser. Res. 6, 619 (1985).
  * (18) F. Hebenstreit, R. Alkofer, G. V. Dunne and H. Gies, Momentum signatures for Schwinger pair production in short laser pulses with sub-cycle structure, Phys. Rev. Lett. 102, 150404 (2009).
  * (19) C. K. Dumlu, Schwinger vacuum pair production in chirped laser pulses, Phys. Rev. D 82, 045007 (2010).
  * (20) F. Hebenstreit, R. Alkofer and H. Gies, Particle self-bunching in the Schwinger effect in spacetime-dependent electric fields, Phys. Rev. Lett. 107, 180403 (2011).
  * (21) C. Kohlfürst and R. Alkofer, Ponderomotive effects in multiphoton pair production, Phys. Rev. D 97, 036026 (2018).
  * (22) M. Ababekri, B. S. Xie and J. Zhang, Effects of finite spatial extent on Schwinger pair production, Phys. Rev. D 100, 016003 (2019).
  * (23) M. Ababekri, Sayipjamal Dulat, B. S. Xie and J. Zhang, Chirp effects on pair production in oscillating electric fields with spatial inhomogeneity, Phys. Lett. B 810, 135815 (2020).
  * (24) C. Kohlfürst, Phase-space analysis of the Schwinger effect in inhomogeneous electromagnetic fields, Eur. Phys. J. Plus 133, 191 (2018).
  * (25) C. Kohlfürst, Effect of time dependent inhomogeneous magnetic fields on the particle momentum spectrum in electron-positron pair production, Phys. Rev. D 101, 096003 (2020).
  * (26) E. Akkermans and G. V. Dunne, Ramsey fringes and time-domain multiple-slit interference from vacuum, Phys. Rev. Lett. 108, 030401 (2012).
  * (27) C. K. Dumlu and G. V. Dunne, Stokes phenomenon and Schwinger vacuum pair production in time-dependent laser pulses, Phys. Rev. Lett. 104, 250402 (2010).
  * (28) C. K. Dumlu and G. V. Dunne, Interference effects in Schwinger vacuum pair production for time-dependent laser pulses, Phys. Rev. D 83, 065028 (2011).
  * (29) B. S. Xie, Z. L. Li and S. Tang, Electron-positron pair production in ultrastrong laser fields, Matter Radiat. Extremes 2, 225 (2017).
  * (30) Z. L. Li, D. Lu and B. S. Xie, Effects of electric field polarizations on pair production, Phys. Rev. D 92, 085001 (2015).
  * (31) Z. L. Li, Y. J. Li and B. S. Xie, Momentum vortices on pairs production by two counter-rotating fields, Phys. Rev. D 96, 076010 (2017).
  * (32) O. Olugh, Z. L. Li, B. S. Xie and R. Alkofer, Pair production in differently polarized electric fields with frequency chirps, Phys. Rev. D 99, 036003 (2019).
  * (33) L. Wang, B.b. Wu, and B. S. Xie, Electron-positron pair production in an oscillating Sauter potential, Phys. Rev. A 100, 2 (2019).
  * (34) O. Olugh, Z. L. Li and B. S. Xie, Effects of polarization and high harmonics of two-color fields on dynamically assisted pair production, Phys. Lett. B 802, 135259 (2020).
  * (35) N. Abdukerim, Z. L. Li and B. S. Xie, Effects of laser pulse shape and carrier envelope phase on pair production, Phys. Lett. B 726, 820 (2013).
  * (36) K. Wang, X. H. Hu, S. Dulat, and B. S. Xie, Effect of symmetrical frequency chirp on pair production, Chin. Phy. B 30, 060204 (2021).
  * (37) M. Mohamedsedik, L. J. Li and B. S. Xie, Schwinger pair production in inhomogeneous electric fields with symmetrical frequency chirp, Phys. Rev. D 104, 016009 (2021).
  * (38) L. J. Li, M. Mohamedsedik and B. S. Xie, Enhanced dynamically assisted pair production in spatial inhomogeneous electric fields with the frequency chirping, Phys. Rev. D 104, 036015 (2021).
  * (39) L. N. Hu, O. Amat, L. J. Li, M. Mohamedsedik and B. S. Xie, Pair production in inhomogeneous electric fields with phase modulation, arXiv:2206.04228.
  * (40) A. Fedotov, A. Ilderton, F. Karbstein, B. King, D. Seipt, H. Taya, G. Torgrimsson, Advances in QED with intense background fields, arXiv:2203.00019.
  * (41) F. Mackenroth, A. Di Piazza, and C. H. Keitel, Determining the carrier-envelope phase of intense few-cycle laser pulses, Phys. Rev. Lett. 105, 063903 (2010).
  * (42) C. Banerjee and M. P. Singh, Electron-positron pair creation by counterpropagating laser pulses: role of carrier envelope phase, Eur. phys. J. D 72, 4 (2018).
  * (43) I. A. Aleksandrov, G. Plunien, and V. M. Shabaev, Pulse shape effects on the electron-positron pair production in strong laser fields, Phys. Rev. D 95, 056013 (2017).
  * (44) J. Braß, R. Milbradt, S. Villalba-Chávez, G. G. Paulus, and C. Müller, Two-color phase-of-the-phase spectroscopy applied to nonperturbative electron-positron pair production in strong oscillating electric fields, Phys. Rev. A 101, 043401 (2020).
  * (45) I. Bialynicki-Birula, P. Gornicki and J. Rafelski, Phase-space structure of the Dirac vacuum, Phys. Rev. D 44, 1825 (1991).
  * (46) D. Vasak, M. Gyulassy and H. T. Elze, Quantum transport theory for abelian plasmas, Annals Phys. 173, 462 (1987).
  * (47) C. Kohlfürst, Electron-positron pair production in inhomogeneous electromagnetic fields, Ph.D. thesis, arXiv:1512.06082.
  * (48) F. Hebenstreit, Schwinger effect in inhomogeneous electric fields, Ph.D. thesis, arXiv:1106.5965.