# Effect of symmetrical frequency chirp on pair production

Kun Wang  School of Physics Science and Technology, Xinjiang University,
Urumqi, Xinjiang 830046 China  Xuehua Hu  School of Physics Science and
Technology, Xinjiang University, Urumqi, Xinjiang 830046 China  Sayipjamal
Dulat<EMAIL_ADDRESS>School of Physics Science and Technology, Xinjiang
University, Urumqi, Xinjiang 830046 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  Beijing Radiation Center, Beijing 100875, China

###### Abstract

By using Dirac-Heisenberg-Wigner formalism we study electron-positron pair
production for linear, elliptic, nearly circular and circular polarizations of
electric fields with symmetrical frequency chirp, and we obtain Momentum
spectra and pair yield. The difference of results among polarized fields is
obvious for the small chirp. When the chirp parameter increases, the momentum
spectra tend to exhibit the multiphoton pair generation that is characterized
by the multi-concentric ring structure. The increase of number density is also
remarkable compared to the case of asymmetrical frequency chirp. Note that the
dynamically assisted Schwinger Mechanism plays an important role for the
enhanced pair production in the symmetrical frequency chirp.

##  I Introduction

At the beginning of its establishment, quantum electrodynamics theoretically
predicted that vacuum would decay and produce electron-positron pair in strong
electric fields, which is called as the Sauter-Schwinger effect Sauter:1931zz
; Heisenberg:1935qt ; Schwinger:1951nm . However, the current laser intensity
(about $10^{22}$$\rm{W/cm^{2}}$) is still far less than the laser intensity
corresponding to the critical field intensity (about
$10^{29}$$\rm{W/cm^{2}}$), so it has not been verified by experiments
Piazza:2012nb . However, with the rapid development of laser technology and
the increasing intensity of electric fields, the generation of pair in the
vacuum is expected to be confirmed by experiments soon. For recent research
progress, please refer to Xie:2017MR ; Gelis:2015kya .

At present, it is a consensus that only the Schwinger tunneling mechanism
cannot generate observable electron-positron pair. Therefore, people have
proposed another mechanism, multiphoton pair generation, which can generate
electron-positron pair in a vacuum by absorbing several high-energy photons
Mocken:2010moc ; Akal:2014al . In addition, to overcome the limitation, that
the current laser equipment does not provide enough high-energy photons to
make observable pair, several catalytic mechanisms for the generation of
observable pair under subcritical conditions are proposed. For example, the
dynamically assisted Schwinger mechanism Li:2014pp ; Schutzhold:2008pz ;
Abdukerim:2012ke  can effectively combine the above two pair generation
mechanisms, thus significantly increasing pair production. Another method to
enhance the number density of pair is to use the electric field with frequency
chirp Dumlu:2010vv ; Jiang:2013vb ; Abdukerim:2017cp ; Olugh:2019pd , a scheme
called chirped pulse amplification was proposed by Strickland and Mourou
Strickland  in the early 1980s which generates intense laser pulses without
destroying the amplification medium. Moreover, the technique is also used to
obtain the existing high-power laser. When it is applied to the theoretical
study of the Sauter-Schwinger effect, proper chirp parameters can increase the
number density of particles produced by several orders of magnitude
Jiang:2013vb ; Abdukerim:2017cp ; Olugh:2019pd .

Here, we report our study about the effect of using the electric field with
symmetrical frequency chirp on pair production using frequency chirp signals.
We consider the relationship between the number density and the symmetrical
frequency chirp signal that varies with time, and then compare the result with
the asymmetrical frequency chirp to analyze the difference. In addition, we
conducted a quantitative analysis of the momentum spectrum and gave
qualitative explanations, which also provided new ideas for the experiment.

This paper is organized as follows: In Sec.II , we briefly introduce the
Dirac-Heisenberg-Wigner (DHW) formalism. In Sec.III , we present the electric
field form. In Sec.IV , we present the numerical results for different
polarization parameters and different chirp parameters, and momentum spectra
with different frequency chirp parameters and four different types of
polarization. In the last section, we present our conclusions.

##  II The DHW formalism

In this paper, we use the DHW formalism which is suitable for pair production
Bialynicki:1991pd ; Heben:2010pd ; Hebenstreit:1ar ; Hebenstreit:2br ; Blinne1
; Blinne2 ; Blinne3 . In the following, we briefly review the DHW formalism,
which begins with the gauge-invariant density operator,

$\hat{\mathcal{C}}\left(r,s\right)=\mathcal{U}\left(A,r,s\right)\
\left[\bar{\psi}\left(r-s/2\right),\psi\left(r+s/2\right)\right]\ ,$ (1)

where we used $\hbar=c=1$, $\psi_{\alpha}(x)$ is the electron’s spinor-valued
Dirac field, $r$ is the center-of-mass coordinate, $s$ is the relative
coordinate, and $\mathcal{U}$ is the Wilson line factor,

$\mathcal{U}\left(A,r,s\right)=\exp\left(\mathrm{i}\ e\
s\int_{-1/2}^{1/2}d\xi\ A\left(r+\xi s\right)\right)\ ,$ (2)

which is added for ensuring the density operator is gauge-invariant and only
related to the background gauge field $A$ and elementary charge $e$
Zhuang:1998pd . Note that the background field is treated in mean-field
(Hartree) approximation,
$F^{\mu\nu}\left({x}\right)\approx\langle\hat{F}^{\mu\nu}\left({x}\right)\rangle$
.

The Wigner operator,

$\hat{\mathcal{W}}\left(r,p\right)=\frac{1}{2}\int d^{4}s\
\mathrm{e}^{\mathrm{i}ps}\ \hat{\mathcal{C}}\left(r,s\right)\ ,$ (3)

which involves the electron’s quantum fluctuations. Then, the covariant Wigner
function can be generated by the vacuum expected value of the Wigner operator
obtained above,

$\mathbbm{W}(r,p)=\langle\Phi|\hat{\mathcal{W}}(r,p)|\Phi\rangle\ .$ (4)

For the accurate representation of matter dynamics in $3+1$ dimensions and the
convenience of numerical calculations, the covariant Wigner function can be
converted to a combination of the Dirac gamma matrix. So we can decompose it
into 16 covariant Wigner components,

$\mathbbm{W}=\frac{1}{4}\left(\mathbbm{1}\mathbbm{S}+\textrm{i}\gamma_{5}\mathbbm{P}+\gamma^{\mu}\mathbbm{V}_{\mu}+\gamma^{\mu}\gamma_{5}\mathbbm{A}_{\mu}+\sigma^{\mu\nu}\mathbbm{T}_{\mu\nu}\right)\,,$
(5)

where the sixteen components $\mathbbm{S}$, $\mathbbm{P}$,
$\mathbbm{V}_{\mu}$, $\mathbbm{A}_{\mu}$ and $\mathbbm{T}_{\mu\nu}$ are
scalar, pseudoscalar, vector, axial vector and tensor. Using an equal-time
approach Ochs:1998ap  to further simplify, and thus the individual Wigner
components can be written as,

$\displaystyle\mathbbm{w}(\textbf{x},\textbf{p},t)=\int\frac{dp_{0}}{2\pi}\
\mathbbm{W}(r,p).$ (6)

Similarly, the $16$ components can also be deduced separately, which are too
long to list them, and the specific derivation can be found in Heben:2010pd ;
Kohlf:11phd . Meanwhile, because of the non-local nature of the pseudo-
differential operators, solving their numerical solutions is very challenging
Hebenstreit:2br ; Kohlfurst:2015niu ; Kohlf:11phd ; Ber:2018pb . For the
homogeneous electric field (13), we can choose vacuum initial conditions as
starting values. The non-vanishing values are

${\mathbbm{s}}_{vac}=\frac{-2m}{\sqrt{{\textbf{p}}^{2}+m^{2}}}\,,\quad{\mathbbm{v}}_{i,vac}=\frac{-2{p_{i}}}{\sqrt{\textbf{p}^{2}+m^{2}}}\,.$
(7)

where $m$ is the mass of an electron. Therefore, sixteen Wigner components can
be simplified into ordinary differential equations Blinne2 , and ten of them
are non-vanishing,

${\mathbbm{w}}=({\mathbbm{s}},{\mathbbm{v}}_{i},{\mathbbm{a}}_{i},{\mathbbm{t}}_{i})\,,\quad{\mathbbm{t}}_{i}:={\mathbbm{t}}_{0i}-{\mathbbm{t}}_{i0}\,.$
(8)

And, the one-particle distribution function can be defined as,

$f({\textbf{q}},t)=\frac{1}{2\Omega(\textbf{q},t)}(\varepsilon-\varepsilon_{vac})\
,$ (9)

where $\varepsilon=m{\mathbbm{s}}+p_{i}{\mathbbm{v}}_{i}$ is the phase space
energy density,
$\varepsilon_{vac}=m{\mathbbm{s}}_{vac}+p_{i}{\mathbbm{v}}_{i,vac}$ is
correspondingly the instantaneous vacuum energy density, q is the canonical
momentum and by definition of ${\textbf{p}}(t)={\textbf{q}}-e{\textbf{A}}(t)$,
and
$\Omega(\textbf{q},t)=\sqrt{{\textbf{p}}^{2}(t)+m^{2}}=\sqrt{m^{2}+(\textbf{q}-e\textbf{A}(t))^{2}}$
is the total energy of electrons. For the convenience, we introduce an
auxiliary three-dimensional quantity $\textbf{v}(\textbf{q},t)$ Blinne2 ,

$v_{i}(\textbf{q},t):={\mathbbm{v}}_{i}(\textbf{p}(t),t)-(1-f({\textbf{q}},t)){\mathbbm{v}}_{i,vac}(\textbf{p}(t),t)\,.$
(10)

Then, by solving the distribution function $f(\textbf{q},t)$ and ten ordinary
differential equations, the following equations can be derived,

$\begin{array}[]{l}\displaystyle\dot{f}=\frac{e}{2\Omega}\,\,\textbf{E}\cdot\textbf{v}\
,\\\\[5.69054pt]
\displaystyle\dot{\textbf{v}}=\frac{2}{\Omega^{3}}\left((e\textbf{E}\cdot\textbf{p})\textbf{p}-e\Omega^{2}\textbf{E}\right)(f-1)-\frac{(e\textbf{E}\cdot\textbf{v})\textbf{p}}{\Omega^{2}}-2\textbf{p}\times\mathbbm{a}_{i}-2m\mathbbm{t}_{i}\
,\\\\[5.69054pt]
\displaystyle\dot{\mathbbm{a}_{i}}=-2\textbf{p}\times\textbf{v}\ ,\\\
\displaystyle\dot{\mathbbm{t}_{i}}=\frac{2}{m}[m^{2}\textbf{v}-(\textbf{p}\cdot\textbf{v})\textbf{p}]\
.\end{array}$ (11)

With initial conditions $f(\textbf{q},-\infty)=0$,
$\textbf{v}(\textbf{q},-\infty)=\textbf{a}(\textbf{q},-\infty)=\textbf{t}(\textbf{q},-\infty)=0$,
the density of the number of pair creation can be obtained by calculating the
integral of the distribution function in the momentum space at time
$t\to+\infty$,

$n=\lim_{t\to+\infty}\int\frac{d^{3}q}{(2\pi)^{3}}f(\textbf{q},t).$ (12)

##  III The external field form

In this section, we establish the following electric field form in order to
study symmetrical frequency chirp,

$\textbf{E}(t)\,\,=\,\,\frac{E_{0}}{\sqrt{1+\delta^{2}}}\,\exp\left(-\frac{t^{2}}{2\tau^{2}}\right)\,\left(\cos(bt|t|+\omega
t+\phi)\textbf{i}+\delta\sin(bt|t|+\omega t+\phi)\textbf{j}\right)\ ,$ (13)

where ${E_{0}}/{\sqrt{1+\delta^{2}}}$ is the amplitude of the electric field,
$\tau$ represents the pulse duration, and $\omega$ is the oscillation
frequency at $t=0$. The parameter $\delta$ ($-1\leq\delta\leq 1$) describes
the ellipticity of the electric field, $\delta=0$ corresponds to linearity and
$\delta=1$ to circular polarization. Besides, the carrier phase $\phi$ is
retained (it is known that the generation of pairs is highly affected by the
phase $\phi$ Kohlfurst:2015niu ; Abdukerim:2013pb ). Since the main concern is
the dependence of symmetrical chirp $b$, the phase $\phi$ is set to zero
below. Note that the form of the effective frequency is
$\omega_{\mathrm{eff}}=\omega+b|t|$. The influence of electric field changes
with time under different frequency modulation pulse parameter $b$ is shown in
Fig.1. It is important to note that the electric field (13) described above,
which varies only over time, can be considered as a standing wave composed of
two laser beams with different polarization and propagating in the opposite
direction, namely the dipole approximation, so the effect of the magnetic
field can be ignored. Meanwhile, considering the electric field parameters
(14), the effects of collision and back-reaction can be ignored.

Figure 1: The electric field $E(t)$ varies with time in linear polarization
($\delta=0$). The parameters are chosen as $E_{0}=0.1\sqrt{2}E_{cr}$,
$\omega=0.6m$, and $\tau=10/m$ where $m$ is the electron mass. The blue solid
line represents the electric field without the chirp parameter $b=0$. The red
dashed line stands for the field with the chirp parameter $b=0.005$ $m^{2}$,
the dark solid line shows the electric field with the chirp parameter $b=0.06$
$m^{2}$.

Figure 2: The Fourier transform of the electric field with the asymmetry chirp
and the symmetry chirp. The other parameters are consistent with those in
Fig.1. Two pictures in the upper left, the upper right, in the lower left, and
the lower right use asymmetrical chirp and symmetrical chirp at $b=0.001$,
$b=0.009$, $b=0.03$, and $b=0.06$, respectively.

In terms of experiments, due to the limitations of the instrument, producing a
perfect circular polarization field is much more difficult than an elliptical
polarization field, for instance, the polarization of the experimental laser
field is as high as $0.93$ Pfeiffer:2012np . Therefore, we include the
numerical calculation of a near-circular elliptical polarization. Besides
three of the parameters of the electric field (13) are fixed as

$E_{0}=0.1\sqrt{2}\,E_{cr},\quad\omega=0.6m,\quad\tau=10/m\,,$ (14)

The form of Keldysh adiabatic parameter is $\gamma=m\omega/eE$, and
multiphoton pair effect and Schwinger (tunnel) effect are determined by
$\gamma\gg 1$ and $\gamma\ll 1$, respectively Keldysh:1965pj . Therefore, for
the known equation (14), not only the influence of the polarization parameter
$\delta$ on the Keldysh adiabatic parameter $\gamma$ should be considered, but
also the frequency $\omega$ will change into the effective frequency when the
chirp parameter $b$ is not zero. For the chirp parameter $b$, we research
several situations where the interval is $0\leq b\leq 0.06$ $m^{2}$ and choose
four different values of $\delta$ according to the polarization state. And we
clearly know that the pulse length in this paper is not enough to get a pure
multiphoton signal, and for the chirp parameter $b=0.06$ $m^{2}$, its value is
beyond the scope of ”normal chirp”. However, the current exploratory research
goal is to qualitatively understand the influence of symmetrical frequency
chirp on the number density and momentum spectra with different types of
polarization and compare with known results.

To explain the following numerical results, we use Fourier transform of the
electric field (13), as shown in Fig.2. With the increasing of the chirp
parameter $b$, the frequency spectrum of the electric field with the
asymmetrical and symmetrical chirped pulses gradually show a multi-peak
structure, and the main peak shifted. Specifically speaking, the main peak of
the asymmetrical pulsed electric field gradually moves to the high frequency
as the increase of $b$, while the main peak of the field with symmetrical
frequency chirp moves to the direction of zero frequency. At the same time,
both the symmetrical electric field and the asymmetrical electric field have
the dynamically assisted Sauter-Schwinger mechanism Li:2014pp ;
Schutzhold:2008pz ; Abdukerim:2012ke . Specifically speaking, the asymmetrical
electric field can be regarded as a low-frequency strong field at first, and
then a high-frequency weak field; while, the symmetrical electric field is a
high-frequency weak field at first, then a low-frequency strong field, and
finally a high-frequency weak field. Both of the above combinations accord
with the basic condition of dynamically assisted Sauter-Schwinger mechanism
Li:2014pp ; Schutzhold:2008pz ; Abdukerim:2012ke . What is more, for the
symmetrical chirp, the mechanism is more obvious and intuitive. Besides, we
will quantitatively explain the formation of the peaks in the momentum
spectrum by considering the Fourier transform of the electric field in the
second subsection of section IV.

##  IV Numerical results

In this section, we show the main results of particle number density under
different symmetrical chirp parameters and different polarizations and the
momentum spectra in the linearly polarized field.

###  IV.1 Pair number density

In this subsection, we study the number density of the created electron-
positron pairs in different polarizations or different chirp parameters.

Figure 3: The number density of the pair production varies with the
polarization parameter $\delta$, for the different symmetrical chirp parameter
$b$. The other parameters are the same as in Fig.1.

In Fig.3, we show the change of the number density with the polarization
parameter $\delta$. The expected symmetry can be seen when mirroring
$\delta\to-\delta$. More specifically, we observe the following from Fig.3.
First, when $b$ is small ($b~{}\textless~{}0.01$ $m^{2}$), the curves
corresponding to different $b$ values are similar and the relative difference
of the number density for different polarizations is large; when $b$ is large
($b~{}\textgreater~{}0.01$ $m^{2}$), the similarity disappeared and the
relative difference becomes smaller. And, with the increasing the chirp
parameter $b$, the peak value of the positron-electron number density also
increases significantly. Especially, when $b$ increases from $0.02$ $m^{2}$ to
$0.03$ $m^{2}$, the number density has been expanded around 15.96 times.
Meanwhile, the difference between the number density of the symmetrical
electric field and the asymmetrical electric field is proportional to b. The
corresponding numbers are provided in Table 1.

Figure 4: The number density of the pair production varies with the chirp
parameter $b$ for the different polarization parameters $\delta=0$(LP),
$\delta=0.5$(EP), and $\delta=1$(CP), respectively.“Sym” and “Asym” represent
the field with symmetrical chirp and with asymmetrical chirp, respectively.
The other parameters are the same as in Fig.1.

In Fig.4, firstly, the number density has been further improved compared with
the asymmetrical chirp, and the changing trend is consistent. Secondly, when
$b\leq 0.018$ $m^{2}$, the number density of linear polarization is
significantly higher than that of elliptical polarization and that of circular
polarization. But when $b\geq 0.018$$m^{2}$, the curves of these three
polarizations have the same changing trend, and the difference between them is
almost indistinguishable (when the chirp parameter $b$ is the same, the
relative error does not exceed $0.1$). The phenomenon has been discussed in
the reference He:2012ed  that the number density produced by linear
polarization is occasionally lower than circular polarization and elliptical
polarization as the increase of the chirp parameter $b$.

Table 1: Numerical results for the number densities (in units of $\lambda_{c}^{-3}=m^{3}$) for circular polarization and some selected chirp parameters in Fig.4 (in units of $m^{2}$). | b ($m^{2}$) | Number Density(Sym) | Number Density(Asym)
---|---|---|---
| $0$ | $7.24\times 10^{-8}$ | $7.24\times 10^{-8}$
| $0.001$ | $7.45\times 10^{-8}$ | $7.28\times 10^{-8}$
| $0.005$ | $1.14\times 10^{-7}$ | $9.53\times 10^{-8}$
| $0.01$ | $2.95\times 10^{-7}$ | $1.89\times 10^{-7}$
| $0.02$ | $2.38\times 10^{-6}$ | $1.43\times 10^{-6}$
| $0.03$ | $3.79\times 10^{-5}$ | $1.61\times 10^{-5}$
| $0.04$ | $3.15\times 10^{-4}$ | $1.40\times 10^{-4}$
| $0.05$ | $9.52\times 10^{-4}$ | $4.42\times 10^{-4}$
| $0.06$ | $1.53\times 10^{-3}$ | $8.58\times 10^{-4}$

The most notable phenomenon in Fig.4 is that the difference between the number
density for symmetrical chirp and that for asymmetrical chirp becomes bigger
when the chirp parameter $b$ increases. The reason is that the increase in
frequency chirp not only causes the increasing in the Keldysh adiabatic
parameter $\gamma$, but also makes multiphoton pair production gradually
dominate. According to Fig.2, we can make a reasonable explanation, when the
electric field with symmetrical chirp has a small chirp parameter
($b~{}\textless~{}0.01$$m^{2}$), the dynamically assisted Sauter-Schwinger
mechanism Li:2014pp ; Schutzhold:2008pz ; Abdukerim:2012ke  is not obvious,
and the pair generation is dominated by the field strength. So it can be
explained in Fig.4 that when the chirp parameter $b$ is small
($b~{}\textless~{}0.01$$m^{2}$), the number density in this paper is a little
different from the number density with asymmetrical chirp. On the contrary,
when the symmetrical chirped electric field has a large chirp parameter
($b~{}\textgreater~{}0.01$$m^{2}$), the effective frequency of the electric
field is large and the generation of vacuum electron-positron pair should be
dominated by multiphoton pair generation process, and the frequency of the
symmetrical chirped electric field becomes higher in the parts of
$t~{}\textgreater~{}0$ and $t~{}\textless~{}0$, which means the dynamically
assisted Sauter-Schwinger mechanism Li:2014pp ; Schutzhold:2008pz ;
Abdukerim:2012ke  becomes intense.

We observe the step-like distortion in Fig.4 which maybe due to the
dynamically assisted Sauter-Schwinger mechanism Li:2014pp ; Schutzhold:2008pz
; Abdukerim:2012ke . In the vicinity of certain specific chirp values, the
effect will be enhanced, which maybe needs further exploration in the future.

###  IV.2 Momentum spectra

In this subsection, we present our results about the momentum spectra in the
linearly polarized field ($\delta=0$). The elliptically polarized field
($\delta=0.5$), circularly polarized field ($\delta=1.0$), and near-circularly
polarized field ($\delta=0.9$) are shown in the appendix. In the case of
linear polarization $(\delta=0)$, the electric field is oriented only along
the $x$-axis, therefore the momentum spectra have rotational symmetry around
the $q_{x}$-axis, as plotted in Fig.5. In the case of no chirp ($b=0$), our
result is consistent with the results of the references Olugh:2019pd ;
Li:2017pd . For the non-zero ($b\neq 0$) chirp parameter, the main result is
that except the symmetry of the momentum spectra, some strong interference
effects also appear, which eventually lead to the momentum spectra of
$e^{+}e^{-}$ pair tend to be multiple concentric ring structure.

Figure 5: Momentum spectra of $e^{+}e^{-}$ pair production for linearly
polarized field ($\delta=0$) in the $(q_{x},q_{y})$-plane and with $q_{z}=0$.
The other parameters are the same as in Fig.1 . Upper row: the small chirp
parameters $b$=0, 0.002$m^{2}$, 0.005$m^{2}$ and 0.009$m^{2}$. Bottom row: the
large chirp parameters $b$=0.02$m^{2}$, 0.03$m^{2}$, 0.04$m^{2}$ and
0.06$m^{2}$. Figure 6: Momentum spectra of pair production for linearly
polarized field ($\delta=0$) at $q_{y}=q_{z}=0$, cf. Fig.5.

More specifically, firstly, for $b=0$, the main peak region of the momentum
spectra is evenly distributed at the center, and the momentum spectra are
symmetrically distributed in $q_{x}$, $q_{y}$ planes. Secondly, by comparing
the momentum spectrum for non-chirp with the momentum spectra for the chirp
parameters $b=0.002$$m^{2}$, $b=0.005$$m^{2}$, and $b=0.009$$m^{2}$, we find
that the shape of momentum spectra changes slightly with the small chirp
parameters ($b<0.01$$m^{2}$), but the peak value of the momentum spectra
improves significantly (when the chirp parameter increases from $b=0$ to
$b=0.009$$m^{2}$, the center value of the momentum spectra is enlarged by
$3.458$ times). Thirdly, when the chirp parameter changes from $b=0.0$ to
$b=0.002$$m^{2}$, the four small peaks around the central region can’t be
observed, and the maximum value of the central region is more than doubled.
Fourthly, when the chirp parameter increases to $b=0.005$$m^{2}$, the overall
shape of the momentum spectra is an elliptical structure, and the distribution
range (red and green regions) becomes larger. Finally, when the chirp
parameter increases to $b=0.009$$m^{2}$, the momentum spectra is still mainly
distributed in the center, and further expanded.

For larger frequency chirp ($b\geq 0.01$$m^{2}$), some more complex structures
are found in the momentum spectra: the entire momentum spectra have changed
significantly, and they are mainly distributed on both sides of the symmetry
center, as shown in the lower of Fig.5. Interestingly, when $b=0.02$$m^{2}$,
the momentum spectrum is divided into two identical main peak regions and two
identical sub-main peak regions which are symmetrically distributed on both
sides of the center. When b increases to $0.03$$m^{2}$, the main peak region
(red) of the momentum spectra on both sides move to the center and merge, and
the sub-peak regions (green) of the momentum spectra also moves to the center,
refer to Fig.6. In the above case, since the chirp parameter $b$ of the
electric field easily changes the distribution of the turning points on the
complex time plane, it is found that the phenomenon is caused by the
interference between multiple pairs of the main turning points Gong:2010pd .

When $b=0.04$, the main peak (red) region of the momentum spectrum is ring-
shaped and evenly distributed around the center of symmetry. When
$b=0.06$$m^{2}$, the main peak region of the momentum spectrum expands away
from the center of symmetry, and the torus-like region becomes narrower. As
the chirp parameter increases, a ring structure affected by interference
effects appears. Because the quantum interference effect between the
amplitudes of the electric field is related to the cycle number of the
electric field during the $e^{+}e^{-}$ pair generation Li2 . In addition,
$e^{+}e^{-}$ pair generation in the vacuum is a typical non-Markovian process,
which means that the evolution of the number density of $e^{+}e^{-}$ pair over
time depends on the complete earlier history. Therefore, small changes of
various parameters contained in the strong background field usually
significantly change the relative phases of the amplitudes Abdukerim:2017cp .
When $b=0.06m^{2}$, the momentum spectrum shows a regular diffraction pattern.

The result of the symmetrical chirp field is the same as expected: the overall
momentum spectrum shape is symmetrical about $q_{x}$ and $q_{y}$. And the
momentum spectrum has been uniformly distributed around the center of symmetry
during the evolution process, which is different from the electric field with
asymmetrical chirp Olugh:2019pd . In the momentum spectrum about $q_{x}$ 6,
the peak of the momentum spectrum is located at $q_{x}=0$ for non-chirp. With
the chirp parameter $b$ increasing, the peak of the momentum spectra will
slightly move to the positive $q_{x}$ direction, and the overall momentum
spectrum maintains symmetry. For $b=0.02m^{2}$, we can not only see four
significant maximums but also other less obvious maximums. Lastly, the maximum
value of the momentum spectra is proportional to the chirp parameter $b$.

Next is our quantitative analysis. From the conservation of energy generated
by multiphoton pairs, we get the general relationship,

$\frac{(n_{1}\omega_{1}+n_{2}\omega_{2}+...)}{2}=\sqrt{{q}^{2}+m_{*}^{2}}~{},$
(15)

where $m_{*}=m\sqrt{1+\frac{e^{2}}{m^{2}}\frac{E_{0}^{2}}{2\omega^{2}}}$ is
called effective mass Ko2014  and $n$ is the number of photons. The momentum
peaks $q_{0}$, $q_{1}$, $q_{2}$, $q_{3}$ with $b=0.06$ in Fig.7 and the
corresponding frequencies in Fig.2 can be calculated by the above equation.
Therefore, the created pairs with momentum $q_{0}$ corresponds to 1 photon
absorption processes with frequency $\omega=1.97$, and $q_{1}$, $q_{2}$,
$q_{3}$ corresponds to two-photon absorption processes (the photon
combinations are $\omega=0.77$ and $\omega=1.28$; $\omega=0.77$ and
$\omega=1.65$; $\omega=0.77$ and $\omega=1.97$ respectively).

Combining the results in the appendix, one obtains some detailed information
about the momentum spectrum. It can be found that the momentum spectra are
very sensitive to the frequency chirp parameter $b$, which includes the
deformation of the ring structure, the appearance of interference effects, and
the significant increasing of the single-particle distribution function. For
example, in all the cases considered, it is more common that when the
frequency chirp parameter increases, the peak value of the momentum spectra
will increase strongly. It is easy to understand when the frequency chirp
parameter b increases, the effective frequency of the strong field
($\omega_{\mathrm{eff}}=\omega+b|t|$) will also increases which means the
probability of the multiphoton pair generation process will increase. In other
words, if a strong field has a constant frequency ($\omega=0.6m$) and a large
frequency chirp, it will contain higher frequency components, so the
probability of $e^{+}e^{-}$ pair generation will increase. Also, the momentum
spectra verify the existence of the dynamically assisted Sauter-Schwinger
mechanism Li:2014pp ; Schutzhold:2008pz ; Abdukerim:2012ke  mentioned above.
During the duration of the symmetrically chirped pulse, the “early-time”, and
“late-time” of the field is also similar to an almost pure multiphoton signal,
so $e^{+}e^{-}$ pair generation under large chirp parameters is dominated by
the multiphoton pair production mechanism. And in different polarizations,
with the chirp parameter $b$ increasing, the momentum spectrum shows more and
more ring structures, which is created by multiphoton.

##  V Summary

Within the DHW formalism, we studied pair production in the four different
types of polarized electric field with symmetrical frequency chirp and
compared it with the electric field with asymmetrical frequency chirp.
Besides, the main results of the number density and spectrum of $e^{+}e^{-}$
pair, which are generated in the arbitrary polarized electric fields with the
symmetrical frequency chirp, are summarized as follows.

(1) Both the difference between the field with the symmetrical frequency chirp
and with the asymmetrical frequency chirp similarly have an effect of the
dynamically assisted Sauter-Schwinger mechanism Li:2014pp ; Schutzhold:2008pz
; Abdukerim:2012ke , but the composition of the former is a high-frequency
weak field at first, then a low-frequency strong field, and finally a high-
frequency weak field. And its high-frequency components are more than the
asymmetrical pulse chirp electric field. Therefore, for different
polarizations, with the increase of $b$, the difference between the number
density of the symmetrical chirp and the asymmetrical chirp also increases.
The specific numerical values are given in Table 1. In addition, with the
increase of the chirp parameter, the number density of linear polarization is
occasionally lower than that of circular polarization and elliptical
polarization when the other parameters are the same.

(2) For the linearly polarized electric field, with the increase of the chirp
parameter $b$, the momentum spectra of $e^{+}e^{-}$ pair production exhibits
peak expansion and splitting and strong interference effects. There is no
doubt that the most complex change in the momentum spectra occurs in the
elliptical polarization. For elliptical polarization, near-circular elliptical
polarization, and circular polarization, it is found that the main peak region
of the momentum spectra will move along the direction of $q_{y}$ with the
increase of the chirp parameter $b$. We think the reason for that is the
electric field form in this paper. Specifically speaking, for the polarization
parameter $\delta=0$ (linear polarization), there is no influence of the
electric field in y-axis $E_{y}$; when $\delta=0.5$ (elliptical polarization),
the influence of $E_{y}$ appears. $E_{y}$ can not only increase the particle
number density but also be equivalent to an accelerating electric field. When
$\delta=1.0$ (circular polarization), the influence of $E_{y}$ is more
significant, and the main peak region also has a more obvious oscillation.

However, the most important discovery is that the electric field with
symmetrical frequency chirp can clearly reflect the existence of the
dynamically assisted Sauter-Schwinger mechanism Li:2014pp ; Schutzhold:2008pz
; Abdukerim:2012ke . When the chirp parameter $b$ increases, the momentum
spectra of arbitrary polarization will eventually tend to a concentric ring
structure, which is caused by the multiphoton process. Because large frequency
chirp can provide a lot of higher frequency components, the “early-time” and
“late-time” of the duration of the symmetrically chirped pulse is similar to
an almost pure multiphoton signal.

In the study, the external laser pulse is limited to a very high electric
field intensity and last very short. For a possible explanation for increasing
the number of pairs production in terms of multiphoton pair production, a
longer pulse study will be necessary. Considering the dramatic increase of the
number density and the associated improved experimental observation potential,
it is certainly feasible to use smaller electric field values and longer pulse
times for research.

###### Acknowledgements.

We are grateful to A. Sawut for the reading and language suggestion on the
manuscript and to Dr. O. Olugh for helpful discussions. This work was
supported by the National Natural Science Foundation of China under the Grant
Nos. 11875007, 11935008 and 11965020.

##  Appendix A Appendix: Momentum spectra

###  A.1 Elliptically polarized field
$\mathbf{\delta}$$\mathbf{=}$$\mathbf{0.5}$

Figure 7: Momentum spectra of $e^{+}e^{-}$ pair production for elliptically
polarized field ($\delta=0.5$) in the $(q_{x},q_{y})$-plane and with
$q_{z}=0$. The other parameters are the same as in Fig.1 . Upper row: the
small chirp parameters $b$=0, 0.002$m^{2}$, 0.005$m^{2}$ and 0.009$m^{2}$.
Bottom row: the large chirp parameters $b$=0.02$m^{2}$, 0.03$m^{2}$,
0.04$m^{2}$ and 0.06$m^{2}$.

For the elliptical polarization ($\delta=0.5$), the momentum spectra are shown
in the Fig.7. On the whole, the momentum spectrum has reflection symmetry.
When the chirp parameter $b$ is small (such as the first row of Fig.7), the
overall change is relatively mild. More specifically, the overall graph
expands away from the center of symmetry, and ring structures gradually
appear. When the chirp parameter $b$ is large (such as the lower row of
pictures in Fig.7), the momentum spectra will be quite complicated reordering,
which is similar to the linear polarization situation. More specifically, a
single extreme value splits into several maxima at first, and then a ring
structure gradually appears. Especially for $b=0.06$$m^{2}$, a ring structure
similar to linear polarization appears, and the overall range is expanded.
Meanwhile, its peak value has been enhanced by four orders of magnitude, from
$9.65\times 10^{-6}$ (when $b=0$) to $8.1\times 10^{-2}$ (when $b=0.06m^{2}$).

###  A.2 Circularly polarized field
$\mathbf{\delta}$$\mathbf{=}$$\mathbf{1.0}$

Figure 8: Momentum spectra of $e^{+}e^{-}$ pair production for circularly
polarized field ($\delta=1$) in the $(q_{x},q_{y})$-plane and with $q_{z}=0$.
The other parameters are the same as in Fig.1 . Upper row: the small chirp
parameters $b$=0, 0.002$m^{2}$, 0.005$m^{2}$ and 0.009$m^{2}$. Bottom row: the
large chirp parameters $b$=0.02$m^{2}$, 0.03$m^{2}$, 0.04$m^{2}$ and
0.06$m^{2}$.

For the circular polarization ($\delta=1$), the momentum distribution of
$e^{+}e^{-}$ pair is shown in the Fig.8. When the chirp parameter $b=0$, the
momentum spectrum shows a ring structure centered at the origin, the weak
interference effect, and the oscillation between the hole and the outer ring.
These phenomena can be also found in references and explained by the effective
scattering potential Dumlu:2010pd  in the semi-classical analysis Olugh:2019pd
; Gong:2010pd . The radius of the ring in momentum spectra can determine the
total number of photons used to produce $e^{+}e^{-}$ pair, by considering the
effective mass formula of $e^{+}e^{-}$ pair generation according to the energy
conservation Vasak:1987ap .

Note that the peak value of the momentum spectra increases significantly as
the frequency chirp $b$ increases, and the main peak region (red) of the
momentum spectrum has the oscillation phenomenon. The main peak region (red)
of the momentum spectrum is evenly distributed around the center of symmetry
for $b=0$. As the chirp parameter $b$ increases, the main peak region moves
toward the negative $p_{y}$ direction firstly and then toward the positive
$p_{y}$ direction, and finally, tend to uniform when the chirp parameter
$b=0.04$$m^{2}$. The reason for the oscillation phenomenon is the electric
field we used, in which $E_{x}$ is symmetrical about the time axis, and the
$E_{y}$ axis is symmetrical about the origin. More specifically, when
$\delta=0$ (linear polarization), there is no influence of $E_{y}$. And for
$\delta=0.5$ (elliptical polarization), the influence of $E_{y}$ appears.
Meanwhile, $E_{y}$ can not only increases the pair generation, but the
remaining $E_{y}$ can also be regarded as an accelerating electric field,
which can accelerate the generated $e^{+}e^{-}$ pair. Consequently, for
$\delta=1.0$ (circular polarization), the influence of $E_{y}$ is more
significant, and with the increasing of the chirp parameter $b$, the central
value also fluctuates more significantly.

###  A.3 Near-circularly polarized field
$\mathbf{\delta}$$\mathbf{=}$$\mathbf{0.9}$

Figure 9: Momentum spectra of $e^{+}e^{-}$ pair production for near-circularly
polarized field ($\delta=0.9$) in the $(q_{x},q_{y})$-plane and with
$q_{z}=0$. The other parameters are the same as in Fig.1 . Upper row: the
small chirp parameters $b$=0, 0.002$m^{2}$, 0.005$m^{2}$ and 0.009$m^{2}$.
Bottom row: the large chirp parameters $b$=0.02$m^{2}$, 0.03$m^{2}$,
0.04$m^{2}$ and 0.06$m^{2}$.

The characteristic shape of the momentum spectrum may be helpful for the
experimental identification of the vacuum $e^{+}e^{-}$ pair generation under a
strong field. Therefore, the momentum spectrum of the near-circular elliptical
polarization ($\delta=0.9$) is calculated, in Fig.9. These results are similar
to the results of helium ionization in strong-field Pfeiffer:2012pl  and the
pair production in an electric field with different polarizations Li2 . Under
a relatively small chirp parameter ($b\leq 0.01$$m^{2}$), we can also see the
violent effect of chirp. On the other hand, for a large chirp parameter
($b\leq 0.01$$m^{2}$), similar to the evolution of circular polarization and
elliptical polarization discussed earlier: the momentum spectrum loses the
symmetry in the direction of $q_{y}$ and finally tends to the structure of
concentric rings.

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Data taxonomy, visualization taxonomy, and task taxonomy for genomic visualizations. The data taxonomy describes how different genomic data types can be encoded as feature sets. A genomic visualization contains one or multiple coordinate systems applying a specific layout, partition, abstraction and arrangement (in case of multiple axes) of sequence coordinates. Feature sets are encoded as tracks and placed on the coordinate systems. Multiple tracks can either be aligned by stacking or overlaying. A visualization can consist of one or multiple views, each containing a set of aligned tracks. Multiple views can show data on one or multiple scales and foci. The task taxonomy explains how different search and query tasks operate on genomic visualizations.

Genomic data visualization is essential for interpretation and hypothesis generation as well as a valuable aid in communicating discoveries. Visual tools bridge the gap between algorithmic approaches and the cognitive skills of investigators. Addressing this need has become crucial in genomics, as biomedical research is increasingly data-driven and many studies lack well-defined hypotheses. A key challenge in data-driven research is to discover unexpected patterns and to formulate hypotheses in an unbiased manner in vast amounts of genomic and other associated data. Over the past two decades, this has driven the development of numerous data visualization techniques and tools for visualizing genomic data. Based on a comprehensive literature survey, we propose taxonomies for data, visualization, and tasks involved in genomic data visualization. Furthermore, we provide a comprehensive review of published genomic visualization tools in the context of the proposed taxonomies.

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Sabrina Nusrat is a Postdoctoral Research Fellow at Harvard Medical School.
She received her PhD in Computer Science from the University of Arizona in 2017.
Her research interests are in the areas of information visualization and visual analytics.

Theresa Anisja Harbig is a Research Associate in the Gehlenborg Lab at Harvard Medical School.
She received her Master's degree in Bioinformatics from the University of Tübingen in 2018. Her main research interest is the visualization of large scale biological data, especially genomic and metagenomic data.

Nils Gehlenborg is an Assistant Professor in the Department of Biomedical Informatics at Harvard Medical School and the Director of the Master in Biomedical Informatics program. The goal of Nils' research is to improve human health by developing visual interfaces and computational techniques that enable scientists, clinicians, and patients to efficiently interact with biomedical data.