# Pair Production in time-dependent Electric field at Finite times

Deepak111Corresponding author.
E-mail address: dsah129<EMAIL_ADDRESS>(Deepak). Manoranjan P.
Singh  Theory and Simulations Lab, Raja Ramanna Centre for Advanced
Technology, Indore-452013, India  Homi Bhabha National Institute, Training
School Complex, Anushakti Nagar, Mumbai 400094, India

###### Abstract

We investigate the finite-time behavior of pair production from the vacuum by
a time-dependent Sauter pulsed electric field using the spinor quantum
electrodynamics (QED). In the adiabatic basis, the one-particle distribution
function in momentum space is determined by utilizing the exact analytical
solution of the Dirac equation. By examining the temporal behavior of the one-
particle distribution function and the momentum spectrum of created pairs in
the sub-critical field limit $(E_{0}=0.2E_{c})$, we observe oscillatory
patterns in the longitudinal momentum spectrum(LMS) of particles at finite
times. These oscillations arise due to quantum interference effects resulting
from the dynamical tunneling. Furthermore, we derive an approximate and
simplified analytical expression for the distribution function at finite
times, which allows us to explain the origin and behavior of these
oscillations. Additionally, we discuss the role of the vacuum polarization
function and its counter term to the oscillations in LMS vacuum excitation. We
also analyse the transverse momentum spectrum (TMS).

###### pacs:

##  I Introduction

The concept of pair production in an electromagnetic field has its roots in
the mid-1920s after the invention of quantum mechanics, with the formulation
of the relativistic wave equation for electrons by Paul Dirac in 1928
Dirac:1928hu . The Dirac Sea model was proposed to explain the enigma of
negative energy solutions. F. Sauter’s work in 1931 Sauter:1931zz
demonstrated that strong electric fields can lead to pair creation through
tunneling with exponential suppression. This paved the way for quantum field
theory, recognizing the vacuum as a polarizable medium influenced by constant
fluctuations. In 1935, W. Heisenberg and H. Euler further explored the
peculiarities of the Dirac equation, revealing non-linear modifications in
Maxwell’s equations due to the interaction of electromagnetic fields with the
electron vacuum loopHeisenberg:1936nmg . J. Schwinger’s groundbreaking work in
1951 precisely calculated the imaginary part of the one-loop effective
Lagrangian in the presence of a static electric field Schwinger:1951nm . As a
result of his seminal work, the phenomenon of vacuum pair creation by electric
fields has since become widely known as the Schwinger effect, and it is also
famously referred to as the Sauter-Schwinger effect in recognition of F.
Sauter’s prior work on solving the Dirac equation in the presence of an
electric field. Schwinger’s pioneering calculation opened up new avenues of
research in quantum field theory and has profoundly impacted our understanding
of particle physics in the presence of strong fields. This extraordinary
property of quantum vacuum producing spontaneous particle-antiparticle pairs
has far-reaching implications for understanding the generation of particle-
antiparticle pairs in the presence of a strong electric field Greiner1985 ;
particle creation in the expanding universeParker:1968mv ; black hole
evaporation as a result of Hawking radiation Parikh:1999mf ; and Unruh
radiation, in which particle production is seen by an accelerating
observerUnruh:1976db . The study of generating electron-positron pairs through
a spatially constant electrical background field was extended to the electric
field with various time dependences. In the 1970s, researchers explored the
occurrence of pair production from the vacuum in the presence of an
oscillating time-dependent electric field Marinov:1977gq Popov:1971iga . Their
investigations revealed different qualitative behaviors for this process under
various interaction regimes. The interaction regimes can be distinguished by
the value of the dimensionless Keldysh parameter
$\gamma=\frac{m\omega}{|e|E_{0}}$ Keldysh:1965ojf , with field amplitude
$E_{0}$, field frequency $\omega$, electron charge $e$ and mass $m$. When
$\gamma>>1$, the process probability shows a perturbative power-law scaling
with field intensity. Instead, for $\gamma<<1$, it exhibits a manifestly non-
perturbative exponential dependence on $\frac{1}{E_{0}}$, similar to the case
of a constant electric field Schwinger:1951nm . Particle production in a
spatially homogeneous single pulse of an electric field has been explored
Nikishov:1969tt , Nikishov:2001ps , and methods for tackling particle creation
in an arbitrary time-dependent electric field have been developed
Troup:1972xkz ,Fradkin:1979ng . Because a single pulse of an electric field is
an idealized form of an electric field created by two colliding laser beams,
particle generation in an alternating electric field has been explored for a
more realistic scenario osti_4400849 . However, it was found that the mass of
the created particle, via the Schwinger mechanism, exponentially suppresses
the pair generation rate, necessitating a very strong electric field to
observe this phenomenon. Therefore, this event has not yet been the subject of
any experimental observations. This makes it unclear how well theoretical
prediction captures the physics of pair production. However, the study of pair
production in strong electric fields subject has attracted sustained interest
from theoreticians in recent years due to the extraordinary progress in
development of the ultra-intense lasers technique and the strong field QED
experimental studies that are planned at upcoming high-intensity laser
facilities, such as the European X-Ray Free-Electron Laser TESLA:2000tvm , the
Extreme-Light Infrastructure Heinzl:2009bmy ; Dunne:2008kc , the Exawatt
Center for Extreme Light Studies xcels . The electric field strength is fast
getting closer to the critical value. Additionally, it is suggested that the
Schwinger mechanism can be indirectly tested in the condensed matter system of
a single monolayer of graphene, where the electrons are roughly characterized
by the massless pseudo-relativistic Dirac equation Allor:2007ei ;
Klimchitskaya:2013fpa ; Fillion-Gourdeau:2015dga .
Particle production can be viewed as evolving a quantum system from an initial
equilibrium configuration to a new final equilibrium configuration via an
intermediate non-equilibrium evolution caused by a strong field background. In
the standard approach, the asymptotic analysis of particle states in the
remote past ( or in-states /before the external field is switched on ) and
future (or out-states/long after the interaction with the external field has
finished) is well understood using the quantum field operator expression in
terms of creation and annihilation operators, connected to one-particle states
in both the present and the future. Then, using the relationship between the
two sets of operators in the past and future, we can compute an equation for
the process’s S-matrix and the number of particles produced throughout the
process. A variety of methodologies have been devised to investigate pair
production in external fields. These techniques encompass the proper-time
method Schwinger:1951nm ; DeWitt:1975ys , the canonical methodMottola:1984ar ,
Green’s function methods Gavrilov:1996pz , semiclassical tunneling
Parikh:1999mf , the Schrödinger-Functional approachHallin:1994ad , functional
techniques Fried:2001ur ; Avan:2002dn , mean-field treatmentSmolyansky:1997ji
, and worldline instantons techniques Kim:2000un ; Affleck:1981bma . In the
literature Alkofer:2001ik Roberts:2002py , pair production in an intense laser
field is studied using analytical ( or numerical) calculations and theoretical
predictions about the pair-production rates as time averages over an infinite
period of time, and these studies only focus on understanding the pair
formation at the asymptotic time final equilibrium state. This idea, however,
is not completely good enough for understanding pair production, at least
theoretically, for several reasons. Firstly, it is rarely mentioned how pair
production processes from vacuum change over time. This raises the question:
Can particles manifest instantly during pair production? And lastly, can these
times at which the formation of physical pairs occur be accessed in
experiments? Motivated by these ideas, we provide a way of understanding the
dynamics of pair production at all times. We look for the evolution of the
quantum system at some initial time $t_{0}$ in the vacuum state, but now what
will be the properties of the quantum system at finite time $t$? What happens
to the system properties at all finite times? For that, the temporal evolution
of particle distribution function in momentum space is good enough observable
to give information about the asymptotic states of the quantum field but also
give a complete description of the process when the matter field and the
strong field background are still interacting. Information about the
properties of the vacuum state before the production of real pairs unveils a
previously undiscovered dimension in quantum non-equilibrium physics, thus
exerting influence across various research domains concerning the emergence of
quasi-particles. These specific time-dependent vacuum states poses questions
about the physical interpretation of time-dependent observables related to
pair production studies and the discussion in the literature is still open
Dabrowski:2016tsx ,Ilderton:2021zej . Secondly, can we provide the physical
meaning of the possible definition of the number of particles in terms of the
number of particles measured well after the finite time $t$ at which the
external background has been switched off? As studied by various authors
Gavrilov:1996pz ,DiPiazza:2004lsj ,Mahajan:2008ya  who derived the adiabatic
number of pairs created after time $T$ where $T$ is larger than the electric
field pulse duration, which is a good approximation for measuring the real
pairs. The second motivation for our work focuses on the possibility of the
formation of real pairs at the finite time when the electric field nearly
vanishes, which may be experimentally accessible in the future. In the present
work, we consider the production of electron-positron pairs from the vacuum in
a time-varying, spatially uniform pulsed electric field is given by
$E\left(t\right)=E_{0}\textrm{sech}^{2}\left(\frac{t}{\tau}\right)$, with
height of $E_{0}$ and width of $\tau$. Such background field has received
extensive attention in the literature Gavrilov:1996pz ,Nikishov:2001ps , with
a focus on the asymptotic behavior of the probability of pair production. This
naturally raises questions about the instantaneous appearance of particles in
pair production and their behavior at intermediate times when using a
dynamical formalism. To this end, we study the evolution of the one-particle
distribution function, $f(\bm{p},t)$ in momentum space, which is rigorously
derived from QED by canonical quantization of the Dirac field and subsequent
Bogoliubov transformation to a quasi-particle representation. In the case of a
time-dependent Sauter-pulse electric field, the exact solution of the one-
particle Dirac equation is possible, and using it, we analytically compute the
particle distribution function,$f(\bm{p},t)$ in terms of Gauss-Hypergeomertic
function. It is well known that pair production from vacuum undergoes three
different stages of evolution: quasielectron-positron plasma (QEPP), the
transient stage, and the final residual electron-positron plasma (REPP) stage.
By temporal evolution of $f(\bm{p},t)$, we find that the occurrence of three
stages is influenced by longitudinal and transverse momentum values
qualitatively and quantitatively. Next, We analyze the created particles’
Longitudinal Momentum Spectrum (LMS) at finite times. In the tunneling regime
($\gamma<1$), we observe oscillatory structure in the LMS at time $t>2\tau$,
and this oscillation pattern continuously changes from $t>2\tau$ up to
$t<6\tau$ for $\gamma<1$. This oscillation behavior at finite time clearly
illustrates the quantum interference effects associated with particle
production, as explained in terms of the vacuum polarisation function,
$u(\bm{p},t)$ and its counter-part depolarisation function, $v(\bm{p},t)$
using dynamical tunneling in the momentum-space representation. We emphasize
that the oscillations seen in the LMS are not artifacts but rather possess
significant physical relevance. In the multi-photon regime, we see that LMS at
finite time near $t=3\tau$ shows the multi-modal structure. Also, we
investigated the impact of transverse momentum value on LMS and revealed that
oscillation present at the finite time disappears for a large value of
$p_{\perp}.$ Further, we find out that the Transverse Momentum Spectrum(TMS)
does not show this quantum interference effect; only some deformation occurs
during the transient stage.

This work is organized as follows: In Sec.II detailed theoretical formulations
of our problem are given this largely follows the derivation from Grib1988 ;
Grib1972 . In Sec. III, we present expressions for the particle momentum
distribution function and pair correlation function obtained within the
adiabatic basis. Results are discussed in Sec. IV. The article is concluded in
Sec. V.
Throughout the paper, we use natural units and set $\hslash=c=m=|e|=1$, the
electric charge $e<0$ and express all variables in terms of the electron mass
unit.

##  II Theoretical Framework

In this section, we will briefly discuss the formalism required for our
present work based on the original literature Grib1988 ; Grib1972 . This
formalism serves as the theoretical framework that underlies our study and
enables us to analyze and interpret the phenomenon of electron-positron
creation under a strong electric field. To construct one-particle solutions of
the Dirac equation in such a field, we start by writing the Dirac equation for
a particle in an electromagnetic field, which takes the following form:

$\bigl{(}\textrm{i}\gamma^{\mu}\partial_{\mu}-e\gamma^{\mu}A_{\mu}-m\bigr{)}\Psi(x)=0.$
(1)

where the metric convention is taken as where, $A_{\mu}$ is the 4-vector
electromagnetic potential, $m$ is the mass of electron, $e$ is the charge of
the electron $\Psi(x)$ is the four-component spinor and
$\gamma^{\mu}$-matrices(or Dirac matrices) we chose the Weyl or chiral basis
Ryder:1985wq

$\gamma^{0}=\begin{pmatrix}\mathbb{I}&0\\\
0&-\mathbb{I}\end{pmatrix},\gamma^{i}=\begin{pmatrix}0&-\sigma^{i}\\\
\sigma^{i}&0\end{pmatrix},i=1,2,3,$ (2)

where $\mathbb{I}$ is the identity matrix and $\sigma^{i}$ are the Pauli
matrices The Dirac matrices satisfy the anticommutation relations:

$\\{\gamma^{\mu},\gamma^{\nu}\\}=2g^{\mu\nu}$ (3)

with the metric tensor, $g^{\mu\nu}=diag(+---).$
Four coupled differential equations result from the Dirac equation for the
spinor, and it is typically challenging to find precise analytical solutions
in the presence of external fields. Feynmann and Gell-Mann were able to
circumvent this difficulty by taking into account a two-component form of the
Dirac equation Feynman:1958ty .
Accordingly, we turn this equation into a second-order differential equation
by assuming the existence of a bispinor $\chi(x)$ such that

$\Psi(x)=\bigl{(}\textrm{i}\gamma^{\nu}\partial_{\nu}-e\gamma^{\nu}A_{\nu}+m\bigr{)}\chi(x).$
(4)

and inserting Eq. (4) into (1), it follows that $\chi(x)$ satisfies the
quadratic Dirac equation

$[(i\partial_{\mu}-eA_{\mu})^{2}-\frac{e}{2}\sigma^{\mu\nu}\mathcal{F}_{\mu\nu}-m^{2}]\chi(x)=0$
(5)

where $\chi(x)$ is a four component spinor. Here, we consider the case where
the electromagnetic field tensor is $\mathcal{F}^{\mu 0}=(0,0,0,E(t))$ with
$E(t)$ linearly polarized time-dependent quasi-classical spatially uniform
electric field along the $z$-axis which is characterized by the four-vector
potential $A^{\mu}(x)=(0,{\bm{A}}(t))\equiv(0,0,0,A(t))$, with an arbitrary
$A(t)$ such that $E(t)=-\frac{dA(t)}{dt}.$
Then the second-order Dirac equation becomes

$(\Box+e^{2}A^{2}(t)+2iA(t)\partial_{3}-ie\partial_{t}A(t)\gamma^{0}\gamma^{3}+m^{2})\chi(x)=0.$
(6)

Here,$\Box=\partial_{\mu}\partial^{\mu}$ is D’Alembert operator. It is worth
noting that this equation has twice as many solutions as the Dirac equation.
As a result, in order to have a one-to-one correspondence between both sets of
solutions, we must reduce the number of solutions of Eq. (6). The method used
here reduces the problem to solving a differential equation for a single
scalar function. To illustrate this, we first point out that the problem is
spatial homogeneity, which implies that there exist solutions of the form

$\chi(x)=\textrm{e}^{\textrm{i}\bm{p}\cdot{\bm{x}}}\chi_{\bm{p}}(t),$ (7)

where, $\chi_{\bm{p}}(t)$ is independent of the position ${\bm{x}}$ and we
label it by momentum of a particle ${\bm{p}}$.
Now, Eq. (6) becomes

$\Bigl{(}\partial_{t}^{2}+\textrm{i}eE(t)\gamma^{0}\gamma^{3}+\omega^{2}(\bm{p},t)\Bigr{)}\chi_{\bm{p}}(t)=0.$
(8)

with

$\omega^{2}(\bm{p},t)=\epsilon_{\perp}^{2}(\bm{p})+(p_{\|}-eA(t))^{2},\hskip
22.1124pt\epsilon_{\perp}^{2}({\bm{p}})={\bm{p}}_{\perp}^{2}+m^{2},\hskip
22.1124ptP(t)=p_{\parallel}-eA(t)$ (9)

Here, ${\bm{p}}_{\perp}$ and $p_{\|}$ are the momentum components
perpendicular and parallel to the external field direction. Now we expand the
function $\chi_{\bm{p}}(t)$ in the basis of eigenvectors of
$\gamma^{0}\gamma^{3}.$ The matrix representation of $\gamma^{0}\gamma^{3}$ is

$\gamma^{0}\gamma^{3}=\begin{pmatrix}\mathbb{I}&0\\\
0&-\mathbb{I}\end{pmatrix}\begin{pmatrix}0&\sigma^{3}\\\
-\sigma^{3}&0\end{pmatrix}=\begin{pmatrix}0&\sigma^{3}\\\
\sigma^{3}&0\end{pmatrix},$ (10)

making it easy to recognize the eigenvector. They are given by

$R_{1}=\begin{pmatrix}1\\\ 0\\\ 0\\\ 0\end{pmatrix},\quad
R_{2}=\begin{pmatrix}0\\\ 0\\\ 0\\\ 1\end{pmatrix},\quad
R_{3}=\begin{pmatrix}0\\\ 1\\\ 0\\\ 0\end{pmatrix},\quad
R_{4}=\begin{pmatrix}0\\\ 0\\\ 1\\\ 0\end{pmatrix},$ (11)

Two doubly degenerate eigenvalues exist for $\gamma^{0}\gamma^{3}$ with
$R_{1},R_{2}$ have eigenvalue $`1^{\prime}$ and $R_{3},R_{4}$ have eigenvalue
$`-1^{\prime}.$ But it turns out that picking one of them is sufficient
Grib1988 . Now, we shall seek the solutions of Eq. (8) in the form

$\chi_{\bm{p}}(t)\equiv\chi_{\bm{p}r}(t)=\psi_{{\bm{p}}}(t)R_{r},$ (12)

where $\gamma^{0}\gamma^{3}R_{r}=R_{r},$ satisfying
$R_{r}^{\dagger}R_{r^{\prime}}=2\delta_{rr^{\prime}},r,r^{\prime}=1,2.$
Solving a differential equation for a scalar function $\psi_{\bm{p}}(t)$
simplifies the problem,

$\Bigl{(}\partial_{t}^{2}+\textrm{i}eE(t)+\omega^{2}(\bm{p},t)\Bigr{)}\psi_{\bm{p}}(t)=0.$
(13)

Now, let’s examine the resulting solutions. It follows from (13) that, in
vanishing electric field, $E(t)\rightarrow 0$ ( $t\rightarrow-\infty$), the
term $\omega(\bm{p},t)$ becomes $\omega(\bm{p})$ and , the scalar function
$\psi(\bm{p},t)$ satisfies the asymptotic equation,

$\Bigl{(}\partial_{t}^{2}+\omega^{2}(\bm{p})\Bigr{)}\psi_{\bm{p}}(t)=0,$ (14)

There are two linearly independent solutions to this harmonic oscillator
equation, which correspond to energy $\omega({\bm{p}})$ and
$-\omega({\bm{p}})$. In what follows, we will label these solutions with
superscripts $\lambda=+$ and $\lambda=-$ respectively. Solutions of this
equation are clearly given by plane waves and thus

$\psi_{\bm{p}}^{(\lambda)}(t)\underset{t\rightarrow-\infty}{\sim}\textrm{e}^{-\textrm{i}\lambda\omega({\bm{p}})t}.$
(15)

We will interpret these solutions as describing an electron ($\lambda=+$) and
its anti-particle, i.e., a positron ($\lambda=-$). in the electric field.
Finally, the corresponding solutions of (6) have the form,

$\chi_{r}^{(\lambda)}(x)=\textrm{e}^{\textrm{i}{\bm{p}}\cdot{\bm{x}}}\psi_{\bm{p}}^{(\lambda)}(t)R_{r},$
(16)

those of the Dirac equation, however, are derived

$\displaystyle\Psi_{r}^{(\lambda)}(x)=\Bigl{[}\textrm{i}\gamma^{0}\partial_{t}$
$\displaystyle-{\bm{p}}\cdot{\bm{\gamma}}+eA(t)\gamma^{3}+m\Bigr{]}\textrm{e}^{\textrm{i}{\bm{p}}\cdot{\bm{x}}}\psi_{\bm{p}}^{(\lambda)}(t)R_{r}.$
(17)

At the asymptotic time, equation (17) describes an electron or positron
(depending on the parameter $\lambda=\pm$) with momentum ${\bm{p}}$, and these
particles are in a superposition of spin-up and down states.
The newly constructed eigenstates of the Dirac equation representing an
electron/positron in a time-dependent electric field provide a complete and
orthonormal relation

$\sum_{\lambda=\pm}\sum_{r=\pm}\int\frac{d^{3}{\bm{p}}}{(2\pi)^{3}}\,\Psi_{{\bm{p}}r}^{(\lambda)}(x)[\Psi_{{\bm{p}}r}^{(\lambda)}(x^{\prime})]^{\dagger}=\delta({\bm{x}}-{\bm{x}}^{\prime}).$
(18)

$\Psi_{\bm{p}r}^{(\lambda)}(x)$ provide a full set of orthonormal solutions of
the Dirac equation in a time-dependent electric field Eq. (1) Grib1988 . We
can now build the Dirac fermion field operator in the second quantization
using these one-particle solutions. Now, the Dirac fermion field operator
$\hat{\Psi}(x)$ is given by

$\hat{\Psi}(x)=\sum_{r}\int\frac{d^{3}{\bm{p}}}{(2\pi)^{3}}\Bigl{(}\Psi_{{\bm{p}}r}^{(+)}(x)\hat{b}_{{\bm{p}}r}+\Psi_{-{\bm{p}}r}^{(-)}(x)\hat{d}_{{\bm{p}}r}^{\dagger}\Bigr{)},$
(19)

where $\Psi^{(\lambda)}_{\bm{p}}(x)$ are the one particle solutions of the
Dirac equation (17), whereas $\hat{b}_{{\bm{p}}r}$ and $\hat{d}_{{\bm{p}}r}$
are the annihilation operators of electron and positron with momentum $\bm{p}$
and spin $r$ in the vacuum state at $t\rightarrow-\infty$ through the
conditions that $\hat{b}_{{\bm{p}}r}|{0_{in}}\rangle=0$ and
$\hat{d}_{-{\bm{p}}r}|{0_{in}}\rangle=0$. Moreover, they satisfy the standard
fermionic anti-commutation relations,

$\\{\hat{b}_{{\bm{p}}r},\hat{b}_{{\bm{p}}^{\prime}r^{\prime}}^{\dagger}\\}=\\{\hat{d}_{{\bm{p}}r},\hat{d}_{{\bm{p}}^{\prime}r^{\prime}}^{\dagger}\\}=\delta({\bm{p}}-{\bm{p}}^{\prime})\delta_{rr^{\prime}},$
(20)

with the remaining anti-commutators being zero. Now the Hamiltonian can be
calculated from the energy-momentum tensor which yields,

$\hat{H}(t)=\textrm{i}\int\hat{\Psi}^{\dagger}(x)\dot{\hat{\Psi}}(x)d^{3}\bm{x}$
(21)

After a lengthy calculation, we determine the diagonal and off-diagonal parts
of the Hamiltonian as

$\displaystyle\hat{H}_{diag}(t)=\textrm{i}\sum_{r}\int\frac{d^{3}{\bm{p}}}{(2\pi)^{3}}\Bigl{[}\varepsilon^{(++)}_{\bm{p}}(t)\hat{b}_{{\bm{p}}r}^{\dagger}\hat{b}_{{\bm{p}}r}+\varepsilon^{(--)}_{\bm{p}}(t)\hat{d}_{-{\bm{p}}r}\hat{d}_{-{\bm{p}}r}^{\dagger}\Bigr{]},$
(22)
$\displaystyle\hat{H}_{offdiag}(t)=\textrm{i}\sum_{r}\int\frac{d^{3}{\bm{p}}}{(2\pi)^{3}}\Bigl{[}\varepsilon^{(+-)}_{\bm{p}}(t)\hat{b}_{{\bm{p}}r}^{\dagger}\hat{d}_{-{\bm{p}}r}^{\dagger}+\varepsilon^{(-+)}_{\bm{p}}(t)\hat{d}_{-{\bm{p}}r}\hat{b}_{{\bm{p}}r}\Bigr{]},$
(23)

where the factors $\varepsilon_{\bm{p}}^{(\lambda\lambda^{\prime})}(t)$ are
expressed as

$\varepsilon_{\bm{p}}^{(\lambda\lambda^{\prime})}(t)=\left\\{\begin{array}[]{ll}\omega^{2}({\bm{p}},t)\Bigl{(}\dot{\psi}_{\bm{p}}^{(\lambda)}(t)[\psi_{\bm{p}}^{(\lambda)}(t)]^{*}-\psi_{\bm{p}}^{(\lambda)}(t)[\dot{\psi}_{\bm{p}}^{(\lambda)}(t)]^{*}\Bigr{)}+\displaystyle\textrm{i}P(t)\Bigl{(}|\dot{\psi}_{\bm{p}}^{(\lambda)}(t)|^{2}+\omega^{2}({\bm{p}},t)|\psi_{\bm{p}}^{(\lambda)}(t)|^{2}\Bigr{)}&\quad{\mbox{if}}\,\,\lambda=\lambda^{\prime},\\\
\displaystyle\omega^{2}({\bm{p}},t)\Bigl{(}\dot{\psi}_{\bm{p}}^{(\lambda)}(t)\psi_{-\bm{p}}^{(\lambda^{\prime})}(t)-\psi_{\bm{p}}^{(\lambda)}(t)\dot{\psi}_{-\bm{p}}^{(\lambda)}(t)\Bigr{)}+\displaystyle\textrm{i}P(t)\Bigl{(}\dot{\psi}_{\bm{p}}^{(\lambda)}(t)\dot{\psi}_{-\bm{p}}^{(\lambda^{\prime})}(t)+\omega^{2}({\bm{p}},t)\psi_{\bm{p}}^{(\lambda)}(t)\psi_{-\bm{p}}^{(\lambda^{\prime})}(t)\Bigr{)}&\quad{\mbox{if}}\,\,\lambda\neq\lambda^{\prime},\end{array}\right.$
(24)

We shall stress here that with the above Hamiltonian, having non-vanishing
off-diagonal elements, the positive and negative energy modes mix and thus no
clear interpretation in terms of particles and antiparticles is possible.

In order to calculate the spectrum, we have to diagonalize the Hamiltonian;
this is achieved by a basis transformation, where we introduce new time-
dependent operators $\hat{B}_{{\bm{p}}r}(t)$ and $\hat{D}_{{\bm{p}}r}(t)$. The
relation between the operators $\hat{b}_{{\bm{p}}r},\hat{d}_{{\bm{p}}r}$and
$\hat{B}_{{\bm{p}}r}(t),\hat{D}_{{\bm{p}}r}(t)$ is given by a Bogoliubov
transformation

$\displaystyle\hat{B}_{{\bm{p}}r}(t)$
$\displaystyle=\alpha_{\bm{p}}(t)\hat{b}_{{\bm{p}}r}+\beta_{\bm{p}}(t)\hat{d}^{\dagger}_{-{\bm{p}}r},$
(25) $\displaystyle\hat{D}_{{\bm{p}}r}(t)$
$\displaystyle=\alpha_{-{\bm{p}}}(t)\hat{d}_{{\bm{p}}r}-\beta_{-{\bm{p}}}(t)\hat{b}_{-{\bm{p}}r}^{\dagger}.$
(26)

It introduces a new set of creation and annihilation operators for
quasiparticles at time $t$, such that the Hamiltonian is diagonalized in that
”quasi-particle representation” (new basis). Hence, the instantaneous vacuum
state is defined as $\hat{B}_{{\bm{p}}r}(t)|{0_{t}}\rangle=0$ and
$\hat{D}_{{\bm{p}}r}(t)|{0_{t}}\rangle=0$. Note that this transformation
preserves the anti-commutation relations of the creation and annihilation
operators provided that, at every time $t$, unknown functions
$\alpha_{\bm{p}}(t)$ and $\beta_{\bm{p}}(t)$ satisfy the condition,

$|\alpha_{\bm{p}}(t)|^{2}+|\beta_{\bm{p}}(t)|^{2}=1.$ (27)

We can also express $\hat{\Psi}(x)$ in terms of new operators as the
Bogoliubov transformation provides us with the desired change of basis.

$\hat{\Psi}(x)=\sum_{r}\int\frac{d^{3}{\bm{p}}}{(2\pi)^{3}}\Bigl{[}\Phi_{{\bm{p}}r}^{(+)}(x)\hat{B}_{{\bm{p}}r}(t)+\Phi_{-{\bm{p}}r}^{(-)}(x)\hat{D}_{{\bm{p}}r}^{\dagger}(t)\Bigr{]},$
(28)

with the bispinors $\Phi_{{\bm{p}}r}^{(\lambda)}(x)$ such that

$\displaystyle\Phi_{{\bm{p}}r}^{(+)}(x)$
$\displaystyle=\alpha_{\bm{p}}^{*}(t)\Psi_{{\bm{p}}r}^{(+)}(x)+\beta_{\bm{p}}^{*}(t)\Psi_{{\bm{p}}r}^{(-)}(x),$
(29) $\displaystyle\Phi_{{\bm{p}}r}^{(-)}(x)$
$\displaystyle=\alpha_{\bm{p}}(t)\Psi_{{\bm{p}}r}^{(-)}(x)-\beta_{\bm{p}}(t)\Psi_{{\bm{p}}r}^{(+)}(x).$
(30)

It follows from here that $\Phi_{{\bm{p}}r}^{(\lambda)}(x)$ should have the
same spinor form as $\Psi_{{\bm{p}}r}^{(\lambda)}(x)$. Thus, we propose that

$\displaystyle\Phi_{{\bm{p}}r}^{(\lambda)}(x)$
$\displaystyle=\Bigl{[}\textrm{i}\gamma^{0}\partial_{t}-{\bm{p}}\cdot{\bm{\gamma}}+eA(t)\gamma^{3}+m\Bigr{]}\textrm{e}^{\textrm{i}{\bm{p}}\cdot{\bm{x}}}\phi_{\bm{p}}^{(\lambda)}(t)R_{r},$
(31)

where, $\phi_{\bm{p}}^{(\lambda)}(t)$ are unknown functions. The function
$\phi_{\bm{p}}^{(\lambda)}(t)$ are the mode functions in the quasi-particle
representation, which are chosen according to the ansatz,

$\phi_{\bm{p}}^{(\lambda)}(t)=\frac{e^{-\textrm{i}\lambda\Theta_{\bm{p}}(t)}}{\sqrt{2\omega(\bm{p},t)\bigl{(}\omega(\bm{p},t)-\lambda
P_{\|}(t)\bigr{)}}}$ (32)

The functions $\phi_{\bm{p}}^{(\lambda)}(t)$ are chosen such that they
coincide with the mode functions $\psi_{\bm{p}}^{\lambda}(t)$ in the case of a
vanishing vector potential.

Now, combining Eqs. (17), (29), (30), and (31), we obtain that

$\displaystyle\psi_{\bm{p}}^{(+)}(t)=\alpha_{\bm{p}}(t)$
$\displaystyle\textrm{e}^{-\textrm{i}\Theta_{\bm{p}}(t)}\phi_{\bm{p}}^{(+)}(t)-\beta_{\bm{p}}^{*}(t)\textrm{e}^{\textrm{i}\Theta_{\bm{p}}(t)}\phi_{\bm{p}}^{(-)}(t),$
(33) $\displaystyle\psi_{\bm{p}}^{(-)}(t)=\beta_{\bm{p}}(t)$
$\displaystyle\textrm{e}^{-\textrm{i}\Theta_{\bm{p}}(t)}\phi_{\bm{p}}^{(+)}(t)+\alpha_{\bm{p}}^{*}(t)\textrm{e}^{\textrm{i}\Theta_{\bm{p}}(t)}\phi_{\bm{p}}^{(-)}(t).$
(34)

with the accumulated dynamical
phase,$\Theta_{\bm{p}}(t_{0},t)=\int^{t}_{t_{0}}dt^{\prime}\omega_{\bm{p}}(t^{\prime}).$
and the coefficients $\alpha_{\bm{p}}(t)$ and $\beta_{\bm{p}}(t)$ are given by

$\displaystyle\alpha_{\bm{p}}(t)=i\phi_{\bm{p}}^{(-)}(t)\epsilon_{\perp}(t)\textrm{e}^{\textrm{i}\Theta_{\bm{p}}(t)}(\partial_{t}-i\omega(\bm{p},t))\psi_{\bm{p}}^{(+)}(t)$
(35)
$\displaystyle\beta_{\bm{p}}(t)=-i\phi_{\bm{p}}^{(+)}(t)\epsilon_{\perp}(t)\textrm{e}^{-\textrm{i}\Theta_{\bm{p}}(t)}(\partial_{t}+i\omega(\bm{p},t))\psi_{\bm{p}}^{(+)}(t)$
(36)

From the above equations, if we know $\psi_{\bm{p}}(t)$ from a solution of
differential Eq. (13) for a specific electric field, we are able to find out
the Bogolyubov transformation coefficients and correspondingly momentum
distribution function of the created particle. Let us introduce the occupation
number of electrons in the given eigenmode $\bm{p}r$ of the fermionic field
using the time-dependent creation and annihilation operators,

$\displaystyle f_{r}(\bm{p},t)$
$\displaystyle=\langle{0_{in}}|\hat{B}_{{\bm{p}}r}^{\dagger}(t)\hat{B}_{{\bm{p}}r}(t)|{0_{in}}\rangle$
(37)

Similarly, one can also define the occupation number of the positron,

$\displaystyle\bar{f}_{r}(-\bm{p},t)$
$\displaystyle=\sum_{r}\langle{0_{in}}|\hat{D}_{-{\bm{p}}r}^{\dagger}(t)\hat{D}_{-{\bm{p}}r}(t)|{0_{in}}\rangle$
(38)

Because of the invariance of charge conjugation,

$\displaystyle f_{r}(\bm{p},t)$ $\displaystyle=\bar{f}_{r}(-\bm{p},t)$ (39)

Under the quasi-particle model, $f_{r}(\bm{p},t)$ and $\bar{f}_{r}(-\bm{p},t)$
will act as one-particle distribution functions. The temporal evolution of
$f_{r}(\bm{p},t)$ satisfies

$\displaystyle f(\bm{p},t)$
$\displaystyle=\sum_{r}|\beta_{\bm{p}}(t)|^{2}=2|\beta_{\bm{p}}(t)|^{2}$ (40)
$\displaystyle f(\bm{p},t)$
$\displaystyle=\lim_{t\to\infty}2|\beta_{\bm{p}}(t)|^{2}$ (41)

Here, we emphasized that the $f(\bm{p},t)$ is interpreted as the distribution
function of real particles at asymptotic times only where the electric field
vanished.

##  III Pair Production in Sauter-pulse electric field

A spatially uniform external background is a common approximation of the
electromagnetic field of two counter-propagating laser pulses generating a
standing wave Nakajima:2004tn ; Blaschke:2008wf . In general, the pair-
production process takes place close to the electric-field maximum (or near to
critical field limit) where the magnetic-field vanishes. Despite the fact that
laser fields typically include many optical cycles, in this case, we examine a
relatively simple model of the external field made up of the Sauter profile,
which can be thought of as an extremely short laser pulses.

$E\left(t\right)=E_{0}\textrm{sech}^{2}\left(\frac{t}{\tau}\right),$ (42)

where, $\tau$ is the duration of pulse and $E_{0}$ is field strength. This
electric field,$E(t)$ exponentially goes to zero for $|t|>>\tau.$ In the limit
of $\tau\rightarrow\infty$ the electric field becomes homogeneous in time. We
can choose a gauge in which $A_{0}=0$ and the vector potential associated with
the electric field is $A(t)=(0,0,A_{3}=-\int dtE(t)).$ After the integration,
we find the Sauter-type gauge potential,

$A\left(t\right)=-E_{0}\tau\left(1+\textrm{tanh}\left(\frac{t}{\tau}\right)\right).$
(43)

Figure 1 illustrates the corresponding time dependence of (42) and (43). Its
peak height is attained at $t=0$, its half height is reached at $t=\pm
0.81\tau$, and at $t=\pm\tau$, the pulse amplitude has already dropped to
$41\%$ of its peak height, followed by a further dramatic reduction well below
$10\%$ at $t=\pm 2\tau$.
Now, in the presence of external electric field Eq. (42) equation of motion
for mode function Eq. (13) reads

$\Bigl{(}\partial_{t}^{2}+\textrm{i}E_{0}\textrm{sech}^{2}\left(\frac{t}{\tau}\right)+\omega^{2}(\bm{p},t)\Bigr{)}\psi_{\bm{p}}(t)=0.$
(44)

Figure 1: Time evolution of $E(t)$ (left) and $A(t)$ (right).

where we have skipped the irrelevant index $\lambda$ of the mode function
$\psi_{\bm{p}}(t).$ This equation can be solved by converting it into a
hypergeometric differential equation abramowitz , by the change of the time
variable as $y=\frac{1}{2}\left(1+tanh(\frac{t}{\tau})\right)$ . The new
variable $y$ transforms the equation as

$\left(\frac{4}{\tau^{2}}y\left(1-y\right)\partial_{y}y\left(1-y\right)\partial_{y}+\omega^{2}(\bm{p},y)+4\textrm{i}E_{0}y\left(1-y\right)\right)\psi_{\bm{p}}(t)=0.$
(45)

Further by using the following ansatz

$\psi_{\bm{p}}(y)=y^{k}\left(1-y\right)^{l}\eta_{\bm{p}}(y)$ (46)

with, $k=\frac{-\textrm{i}\tau\omega_{-}}{2},$
$l=\frac{\textrm{i}\tau\omega_{+}}{2}$ and
$\omega_{\pm}^{2}=m^{2}+\bm{p}_{\perp}^{2}+\left(p_{\parallel}\pm
eE_{0}\tau\right)^{2}$ in order to combining Eqs. (45) and (46), we have
$\eta_{\bm{p}}(y)$ satisfying the following hyper-geometric differential
equation

$\left(y\left(1-y\right)\partial_{y}^{2}+\left(c-\left(a+b+1\right)y\right)\partial_{y}-ab\right)\eta_{\bm{p}}(y)=0.$
(47)

Here,

$\displaystyle a$
$\displaystyle=-\textrm{i}E_{0}\tau^{2}-\frac{\textrm{i}\tau\omega_{-}}{2}+\frac{\textrm{i}\tau\omega_{+}}{2}=i\zeta_{1}$
$\displaystyle b$
$\displaystyle=1+\textrm{i}E_{0}\tau^{2}-\frac{\textrm{i}\tau\omega_{-}}{2}+\frac{\textrm{i}\tau\omega_{+}}{2}=1+i\zeta_{2},$
(48) $\displaystyle c$
$\displaystyle=1-\textrm{i}\tau\omega_{-}=1+i\zeta_{3},$

The two linearly independent solutions of Eq.(47) are
$\eta_{\bm{p}}^{(\pm)}(y)$ in the neighborhood of the singular point $y=0$

$\eta_{\bm{p}}^{(+)}(t)=N^{(+)}{}_{2}\mathcal{F}_{1}\left(a,b,c;y\right),$
(49)
$\eta_{\bm{p}}^{(-)}(t)=N^{(-)}{}_{2}\mathcal{F}_{1}\left(1-a,1-b,2-c;y\right).$
(50)

with ${}_{2}\mathcal{F}_{1}\left(a,b,c;y\right)$ denoting the Gauss hyper-
geometric function and $N^{(\pm)}$ being the normalization constants fixed by
initial conditions.
To get the mode functions $\psi_{\bm{p}}^{(\pm)}(y)$ we have to resubstitute
$\eta_{\bm{p}}^{(\pm)}(t)$ in Eq.(46)

$\displaystyle\psi^{(+)}_{\bm{p}}(y)=N^{(+)}(\bm{p})y^{k}\left(1-y\right)^{l}{}_{2}\mathcal{F}_{1}\left(a,b,c;y\right),$
(51)
$\displaystyle\psi^{(-)}_{\bm{p}}(y)=N^{(-)}(\bm{p})y^{-k}\left(1-y\right)^{-l}{}_{2}\mathcal{F}_{1}\left(1-a,1-b,2-c;y\right)$
(52)

where $N^{(\pm)}(\bm{p})$ as normalization constants. These constants are
chosen such that

$\psi^{(\pm)}_{\bm{p}}(y\rightarrow 0)=\phi^{(\pm)}_{\bm{p}}(y\rightarrow 0)$

using the initial condition, we get

$N^{(\pm)}=\frac{e^{\mp\textrm{i}\tilde{\Theta}(\bm{p},y_{0},0)}}{\sqrt{2\omega_{-}\left(\omega_{-}\mp
P\left(0\right)\right)}}$ (53)

with accumulated phase,
$\tilde{\Theta}_{\bm{p}}(y_{0},0)=\frac{2}{\tau}\int^{0}_{y_{0}}dy^{\prime}\frac{\omega_{\bm{p}}(y^{\prime})}{y^{\prime}\left(1-y^{\prime}\right)}.$
We can now proceed with the calculation of the One-particle distribution
function using Eq. (41)

$\displaystyle f(\bm{p},t)=2|\beta(\bm{p},t)|^{2}.$ (54)

To continue, we will convert all functions in $\beta(\bm{p},t)$ given by
Eq.(36) to the new time variable $y$.This transformation yields

$\displaystyle|\beta(\bm{p},y)|^{2}=\frac{\epsilon_{\perp}^{2}(\textbf{p})}{2\omega(\bm{p},y)\bigl{(}\omega(\bm{p},y)-P(p_{\parallel},y)\bigr{)}}\left|\Bigl{(}\frac{2}{\tau}y(1-y)\partial_{y}+\textrm{i}\omega(\bm{p},y)\Bigr{)}\psi^{(+)}(\bm{p},y)\right|^{2}$
(55)

By utilizing the equation above and carrying out a detailed calculation, we
obtain an analytical expression for the one-particle distribution function in
terms of the transformed time variable $y:$

$\displaystyle f(\bm{p},y)$
$\displaystyle=|N^{+}(\bm{p})|^{2}\Big{(}1+\frac{(p_{\parallel}-eA(y))}{\omega(\bm{p},y)}\Big{)}\Biggl{(}\frac{4}{\tau^{2}}y^{2}(1-y)^{2}|\frac{ab}{c}f_{1}|^{2}+(\omega(\bm{p},y)-(1-y)\omega_{-}-y\omega_{+})^{2}|f_{2}|^{2}$
$\displaystyle+\frac{4}{\tau}y(1-y)(\omega(\bm{p},t)-(1-y)\omega_{-}-y\omega_{+})\Re(\frac{ab}{c}f_{1}\bar{f_{2}})\Biggr{)},$
(56)

where, $f_{1}={}_{2}\mathcal{F}_{1}\left(1+a,1+b,1+c;y\right)$,
$f_{2}={}_{2}\mathcal{F}_{1}\left(a,b,c;y\right).$

##  IV Result

Figure 2: Time evolution of $\mathbf{f}(\bm{p},t)$ with $E_{0}=0.2E_{c}$ and
$\tau=10[m^{-1}]$ and all the units are taken in electron mass unit.

Quantum vacuum becomes unstable under the action of the external electric
field. As a consequence, virtual particle-antiparticle pairs are created in an
off-mass-shell configuration. These virtual charged particles are accelerated
by the electric field to enough energy to become real particles in an on-shell
mass configuration. During the action of the external force, pair annihilation
processes occur simultaneously with the pair creation, giving rise to a
dynamical quasiparticle plasma. This results in different states of in and
out-vacuums. The most complete description of vacuum pair creation from in-
state to out-state is given by the one-particle distribution function. The
time evolution of the one-particle distribution function shows that virtual
particle-antiparticle (electron-positron plasma (EPP) ) excited from vacuum
passes through three different stages: the quasiparticle electron-positron
plasma (QEPP) stage, the transient stage, and the final residual electron-
positron plasma(REPP) in the out-state as pointed by authors of
Blaschke:2011is , see figure 2. The initial stage QEPP and the final REPP
stage are separated by the fast oscillation of EPP by the transient stage. The
transient stage is considered to begin at time $t_{in}$ where oscillating of
$f(t)$ attains the REPP level for the first time. The time $t_{out}$ where the
transient stage ends is taken as the time when the average level of the
oscillating $f(t)$ hits the REPP; after that, the REPP stage begins. At the
REPP stage, quasi-particles become independent, and real particle-
antiparticles are observed with a lesser value off than that at the electric
field maximum at $t=0.$
Each of the three stages contributes to the various physical effects, such as
vacuum polarization effects Smolyansky:2010as , the emission of annihilation
photons originating from the focal point of colliding laser beams
Blaschke:2005hs ; Blaschke:2010vs ; Blaschke:2011af ; Blaschke:2011is ,the
birefringence effect DiPiazza:2006pr , and various other secondary effects. In
order to estimate the contributions of the various stages in measuring
observable effects, a detailed analysis of each period of the EPP’s evolution
is quite helpful.
The dependence of the temporal evolution of the quasi-particle distribution
functions $f(\bm{p},t)$ on the momentum, $\bm{p}(p_{\parallel},p_{\perp})$ is
shown in Fig. 3. As depicted in the left panel of Figure 3, the distribution
function $f(p_{\parallel},t)$ demonstrates a consistent increase over time
within the region of QEPP. Notably, it showcases a higher value for negative
longitudinal momentum value than positive ones.On closer inspection, it
becomes evident that the distribution function reaches its peak precisely when
the longitudinal quasi-momentum of particles, $P(t)$, becomes zero. This
particular peak occurrence is influenced by the specific $p_{\parallel}$
value. After $t=0$, where the electric field reaches its maximum value,
$f(p_{\parallel},t)$ decreases. $f(p_{\parallel},t)$ after decreasing to a
certain value shows rapid oscillations and thus transitions from QEPP to a
transient region. One observes a gradual narrowing and a disappearance of the
fluctuations in the transient domain for higher $p_{\parallel}-$ value. This
is because the role of the vacuum polarization effect decreases with
increasing the longitudinal momentum $p_{\parallel}$, and The transient region
appears later for a higher $p_{\parallel}-$ value.

Figure 3:  The relationship between momentum and the transient domain’s time
of occurrence. Left panel: for longitudinal momentum Right panel: for
transverse momentum with $E_{0}=0.2E_{c}$ and $\tau=10[m^{-1}]$ and all the
units are taken in electron mass unit.

We confirm this by quantifying the period of the transient stage, $t_{in}$ and
$t_{out}$ as shown in Table 1. Also, we can say that transient regions have a
time period $\approx 10[m^{-1}]$, and the transient stage starts nearly after
$t\approx\tau$ see Table 1.

$p_{\parallel}[m]$ |  $t_{in}[m^{-1}]$ |  $f(p_{\parallel},t_{in})$ |  $t_{out}[m^{-1}]$ |  $f(p_{\parallel},t_{out})$
---|---|---|---|---
0.0 |  12.57 |  $7.203\times 10^{-7}$ |  24.038 |  $3.309\times 10^{-7}$
0.25 |  13.37 |  $5.576\times 10^{-7}$ |  24.803 |  $3.112\times 10^{-7}$
0.50 |  13.78 |  $2.242\times 10^{-7}$ |  24.320 |  $1.599\times 10^{-7}$
0.75 |  14.53 |  $7.769\times 10^{-8}$ |  25.799 |  $7.033\times 10^{-8}$
1.00 |  16.76 |  $1.748\times 10^{-8}$ |  26.866 |  $1.082\times 10^{-8}$
Table 1:  Table 1 :Transient region time labelled by longitudinal momentum
$p_{\parallel}.$

After that, it reaches the REPP stage, where the distribution
function,$f_{out}$, becomes constant.$f_{out}$ being largest for
$p_{\parallel}=0$ and the same for positive and negative $p_{\parallel}-$
value.
Next, we also point out the influence of transverse momentum on temporal
stages, as seen from the right panel of Fig. 3. We can observe an interesting
behavior for the higher transverse momentum value, $f(p_{\perp})$ slowly
reaches to REPP stage with the residual value $f(p_{\perp},t>t_{out})$ which
is lowest in comparison to small transverse momentum value $p_{\perp}.$ These
stages behaviors are mainly decided by the double quasi-energy
$2\omega(p_{\parallel},p_{\perp},t)$, as we increase the transverse momentum
corresponding transverse energy $\epsilon_{\perp}$also increases that come in
the dynamical energy gap $2\omega(\bm{p},t)$ and due to which to reach on-
shell condition (REPP stage) takes a longer time.

###  IV.1 Longitudinal momentum

In this section, we discuss about longitudinal momentum spectrum (LMS) of the
created particle for $E_{0}=0.2E_{c}$ and $\tau=10[m^{-1}]$.

Figure 4: Time evolution of quasi-particle distribution function in the
longitudinal momentum space $f(p_{\parallel},t)$ at different times. The
transverse momentum is considered to be zero, and all the units are taken in
the electron mass unit.The field parameters are $E_{0}=0.2E_{c}$ and
$\tau=10[m^{-1}].$

Figure 4, we show how particle creation proceeds. At early times,
$t=-10[m^{-1}]$ in the QEPP region when the electric field is increasing, the
created particle shows a smooth Gaussian-like structure of momentum spectrum
with a peak around $p_{\parallel}\approx 2.$ The electric field propels the
newly created particles towards the negative $z-$direction. The movement of
the distribution function’s peak from $p_{\parallel}=+2$
$(\leavevmode\nobreak\ +eE_{0}\tau)$ to $-2$ $(\leavevmode\nobreak\
-eE_{0}\tau)$ can be understood through the concept of quasi-momentum
$P(t)=(p_{\parallel}-eA(t)),$, as depicted in the time evolution shown in
Figure 4(a) to (c). This phenomenon implies that the momentum distribution of
the generated particles is expected to be distributed over a range of $\Delta
p_{\parallel}=2$, a spread determined by the electric field’s magnitude. At
time $t=0[m^{-1}]$ where the electric field is maximum ( $E(t)=E_{0}$),
magnitude of longitudinal momentum distribution function
$f(p_{\parallel}=0,t=0)\approx 5\times 10^{-3}$ that follows the tendency
$E^{2}(t)/8$ Schmidt:1998zh . For $t>0$, when there’s a decrease in the
strength of the field, an interesting phenomenon occurs in the momentum
distribution function $f(p_{\parallel})$. It rapidly drops to $99.2\%$ of its
maximum value, driven by its reliance on the strength of the electric field at
$t=10\,[m]]^{-1}$, as illustrated in figure 4(c). The $f(p_{\parallel})$ peak
corresponds to newly generated particles. However, within a narrow range of
$-1<p_{\parallel}<1$, during what we term the transient stage, the typically
smooth Gaussian shape of the distribution is disrupted. This disruption
coincides with a weakening of the accelerating field’s effect, which can be
visualized in figure 4(d). Near $t\approx 2\tau$, when the electric field’s
magnitude dwindles to roughly $93\%$ of its maximum, a secondary peak emerges
within the Longitudinal Momentum Spectrum (LMS). This peak manifests around a
longitudinal momentum value of zero, accompanied by observable oscillations
within a confined range of $p_{\parallel}$, as demonstrated in figure 4(e). In
proximity to $p_{\parallel}=0$, this secondary peak experiences growth over
time as the electric field weakens, while concurrently, the dominant peak
begins to diminish. Progressing to $t=3\tau$, the previously secondary peak
has risen to dominance, showcasing oscillatory tendencies aligned with the
onset of the transient stage and the initial stages of the REPP phase. As
depicted in figure 4(f), this oscillatory behavior occurs within a narrow
range of longitudinal momentum when the electric field’s intensity subsides to
approximately one-hundredth of the maximum value. Intriguingly, this
oscillation exhibits an asymmetry, with its amplitude being more pronounced
for negative longitudinal momentum compared to its positive longitudinal
momentum. Within figure 4(f), the Gaussian bump centered around
$p_{\parallel}\approx-2$ arises from particles generated during the initial
phases of the process. Conversely, the dominant peak, characterized by the
initiation of oscillation, comprises particles formed at later instances,
which have encountered relatively less acceleration since their inception.
This oscillation in the LMS finds its origin in the quantum interference
effect, a phenomenon stemming from the outcomes of dynamical tunneling, as
elucidated in reference Keldysh2016 . Around $t\approx 4\tau$, a minor peak at
$p_{\parallel}=-2$ value is nearly diminished. Only the dominant peak at
$p_{\parallel}=0$ persists, with a faint onset of oscillatory behavior
superimposed on a Gaussian-like structure. Eventually, as depicted in figures
4(g) to (h), the oscillation gradually fades away by $t=50\,\text{m}^{-1}$.
Dynamical tunneling can be grasped as a process evolving over time involving
inter-band dynamics in particle momentum representation. Within momentum
space, various channels offer possibilities for particle tunneling. As time
advances, distinct scenarios emerge: (i) At the time $t$, a particle can
tunnel directly, adopting a momentum value denoted as
$p_{\parallel}^{\prime}$, (ii) In the early stages, a particle exhibiting
lower momentum ($p_{\parallel}<p_{\parallel}^{\prime}$) can tunnel, followed
by acceleration leading to momentum $p_{\parallel}^{\prime},$ (iii)
Conversely, a particle initially possessing higher momentum
($p_{\parallel}>p_{\parallel}^{\prime}$) can tunnel at time $t_{1}$ and
subsequently decelerate to reach the specific momentum
$p_{\parallel}^{\prime}$. Finally, when the individual probability amplitudes
of this process are added together, they result in quantum interference at
time $t$. At the asymptotic time, those processes do not share the same phase
information and become random due to averaging over particle paths that do not
show quantum interference effects, which leads to a smooth distribution
function. Several research articles, including the work by Dumlu et al.
Dumlu:2011rr , have pointed out that LMS doesn’t exhibit an oscillatory
structure for the Sauter pulsed field as time approaches infinity. Instead, it
displays a single-peaked Gaussian-like structure. This observation is
consistent with the outcome illustrated in figure 4(i) at
$t=100\,\text{[m]}^{-1}$.
Certainly, we can establish certain time scales linked to the quantum
signature manifesting as oscillations within the LMS of generated particles
during the REPP stage. Drawing from the emergence of a secondary peak, we
define three distinct time scales: $(i)$ $t_{cp}$ (Central Peak Formation):
This time scale is characterized by the emergence of the secondary peak and
the beginning of its development, $(ii)$ $t_{sep}$ (Peak Separation): At this
time, the central peak becomes dominant or the time after the two peaks become
distinctly separated, $(iii)$ $t_{dis}$ (Disappearance of Oscillation): This
indicates the time when the oscillations within the central peak fade away or
the time after the primary peak (or the left-side peak) ceases to exist. By
identifying and quantifying these time scales, we can better characterize the
intricate quantum dynamics reflected in the LMS during the REPP stage.

$E_{0}[E_{c}]$ |  $t_{cp}[m^{-1}]$ |  $t_{sep}[m^{-1}]$ |  $t_{dis.}[m^{-1}]$
---|---|---|---
0.1 |  47 |  57 |  70
0.2 |  22 |  30 |  50
0.3 |  15 |  20 |  35
0.4 |  9 |  15 |  28
0.5 |  6 |  12 |  23
Table 2:  Table 2: time labelled by different electric field strength $E_{0}.$

Furthermore, it’s important to highlight that these time scales are influenced
by the electric field strength $E_{0}$, as indicated in Table 2. A remarkable
pattern becomes apparent upon examining the table—a consistent behavior of
these three-time scales emerges with an increase in $E_{0}$. Additionally,
there is a clear trend where these events tend to occur earlier in time as
$E_{0}$ increases.

Moreover, our observations extend to the spread of $\Delta p$ within the LMS,
which exhibits changes depending on the value of $E_{0}$. This relationship is
intuitively valid, as higher values of $E_{0}$ correspond to greater kinetic
momentum.
During the acceleration of the quasi-particle, we see the first Gaussian peak
shift to the negative z-direction, and deformation in the tail gives the
second peak in the spectrum around $t=2.3\tau$ that is clearly visible with
interference visibility of $0.5$. There are two distinct peaks here: one shows
the smooth Gaussian-like structure, and the other shows the interference with
effect as an oscillation at time $t=2.65\tau$ with visibility (or degree of
coherence) of $0.314$ The Gaussian-like onset oscillation peak is then
dominated by $t=3\tau$, reducing visibility to $0.15$. This oscillation was
observed in the Gaussian-like structure, which fades at $t=3.34\tau$ tau and
has a visibility of $0.08$. From these, we can say that these oscillations are
seen in LMS at the Compton time scale during the formation of electrons or
positrons. The LMS figure shows that coherence abruptly disappears at
$t=4.15\tau$, where visibility is $0.02$.
We still do not talk about the Keldysh parameter,$\gamma$, on which we
categorize the pair production process, whether it is governed by multi-photon
or tunneling mechanisms. We also test whether this oscillation structure
behavior explicitly depends on $\gamma$ or not. We choose the $\gamma$ value
using two different combinations of $E_{0}$ and $\tau$ to see its behavior at
a finite time. To show that the same value of Keldysh parameter,$\gamma$,
gives a different qualitative picture of LMS at a finite time, here we take
$\gamma=1$ corresponding to two different configurations of parameter
$(E_{0},\tau)$ related to Sauter pulse electric field. As we know that
$\gamma=1$ gives an intermediate regime of pair production via tunneling and
multi-photon processes, which are known as the nonperturbative
Avetissian:2002ucr . Fig. 5 shows LMS for Sauter pulse electric field with
$E_{0}=0.2$ and $\tau=5[m^{-1}].$ From fig. 5 (a) to 5(c) we can see that due
to the electric field smooth uni-modal Gaussian structure accelerated towards
negative z-direction and after the pulse duration $t=6[m^{-1}]$ we observed
some disturbance near the $p_{\parallel}=1.$ At $t=11.5[m^{-1}]$ interference
effect is observed due to which smooth Gaussian structure becomes a multi-
modal structure as shown in figure 5(d) and 5(e). That interference effect
nearly disappear at $t\approx 4\tau.$ Comparing figures 6(d) and 6(e),
distinct structural differences become evident. These disparities contribute
to the presence of two qualitatively distinct behaviors at finite times, both
under the condition of $\gamma=1$.

It’s important to emphasize that this behavior during finite times is
primarily influenced by the width of the pulse $\tau$. Depending on the
classification of short and long pulses, we can outline two distinct
behaviors: (i) Long Pulse Behavior ($\tau>8$): In this regime, two Gaussian-
pulse profiles are observable. One of these profiles forms at the origin,
potentially with the onset of oscillations. (ii) Short Pulse Behavior
($\tau<8$): In contrast, shorter pulses lead to a unique outcome. Initially, a
smooth unimodal structure experiences a splitting near the REPP region. This
leads to the formation of a multi-modal Gaussian structure, which eventually
converges into a single-peaked Gaussian profile as time progresses towards the
asymptotic state.

Figure 5: Time evolution of quasi-particle distribution function in the
longitudinal momentum space $f(p_{\parallel},t)$. The transverse momentum is
considered to be zero and all the units are taken in the electron mass
unit.The field parameters are $E_{0}=0.2E_{c}$ and $\tau=5[m^{-1}].$

Figure 6: Time evolution of quasi-particle distribution function in the
longitudinal momentum space $f(p_{\parallel},t)$. The transverse momentum is
considered to be zero, and all the units are taken in the electron mass
unit.The field parameters are $E_{0}=0.1E_{c}$ and $\tau=10[m^{-1}].$

###  IV.2 Approximate analytical expression for $f(\bm{p},t)$ at finite time

One-particle distribution function as a function of the transformed time
variable $y$ is given by

$\displaystyle f(\bm{p},t)$
$\displaystyle=|N^{+}(\bm{p})|^{2}\Big{(}1+\frac{(p_{\parallel}-eA(t))}{\omega(\bm{p},t)}\Big{)}\Biggl{(}\frac{4}{\tau^{2}}y^{2}(1-y)^{2}|\frac{ab}{c}f_{1}|^{2}+(\omega(\bm{p},t)-(1-y)\omega_{-}-y\omega_{+})^{2}|f_{2}|^{2}$
$\displaystyle+\frac{4}{\tau}y(1-y)(\omega(\bm{p},t)-(1-y)\omega_{-}-y\omega_{+})\Re(\frac{ab}{c}f_{1}\bar{f_{2}})\Biggr{)}$
(57)

where, $f_{1}={}_{2}\mathcal{F}_{1}\left(1+a,1+b,1+c;y\right)$,
$f_{2}={}_{2}\mathcal{F}_{1}\left(a,b,c;y\right)$
To investigate the behavior of the function $f(\bm{p},t)$ as time approaches
infinity, we employ a variety of approximations based on collections of
formulas associated with Gamma and Gauss-hypergeometric functions. These
approximations enable us to deduce simplified analytical expressions for
$f(\textbf{p},t)$ in the late-time regime. First of all, we start with
approximating the Gauss-hypergeometric function as $y$ approaches 1. It is
crucial to ensure a smooth convergence towards the limit of
${}_{2}\mathcal{F}_{1}\left(a,b,c;y\rightarrow 1\right)$, which requires a
thorough grasp of the limit itself. This task is complicated by the intricate
nature of the variables $a$, $b$, and $c$ in this specific context, making it
essential to exercise caution when dealing with this limit. Therefore, it is
beneficial to perform a transformation in the argument by substituting $y$
with $(1-y).$ This transition can be accomplished by employing the following
mathematical identity

$\displaystyle{}_{2}\mathcal{F}_{1}\left(a,b,c;z\right)$
$\displaystyle=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c)\Gamma(c-a-b)}{}_{2}\mathcal{F}_{1}\left(a,b,1-c+a+b;1-z\right)$
$\displaystyle+\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}(1-z)^{c-a-b}{}_{2}\mathcal{F}_{1}\left(c-a,c-b,1+c-a-b;1-z\right)$
(58)

In general Gauss-Hypergeometric function,

$\displaystyle{}_{2}\mathcal{F}_{1}\left(a,b;c;z\right)$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}}\frac{z^{n}}{n!}$
(59)

where $()_{n}$ denotes the Pochhammer symbol.

$\displaystyle{}_{2}\mathcal{F}_{1}\left(a,b;c;z\right)$
$\displaystyle=1+\frac{ab}{c}z+\frac{a(a+1)b(b+1)}{c(c+1)}\frac{z^{2}}{2!}+\frac{a(a+1)(a+2)b(b+1)(b+2)}{c(c+1)(c+2)}\frac{z^{3}}{3!}+...$
(60)

The series continues with additional terms involving higher powers of $z.$
Each term in the series involves the parameters $a,b,$ and $c$, as well as the
variable $z$ raised to a specific power. Using the above relation we
approximate the Gauss-Hypergeometric function $f_{1}$ and $f_{2}$ that present
in the relation of the particle distribution function and compute simple
analytical expression near to $y\rightarrow 1.$

$\displaystyle\frac{2}{\tau}y(1-y)\frac{ab}{c}f_{1}$
$\displaystyle=\frac{2}{\tau}y(1-y)\frac{ab}{c}{}_{2}\mathcal{F}_{1}\left(1+a,1+b;1+c;y\right)$
$\displaystyle=\frac{2}{\tau}y(1-y)\frac{ab}{c}\Bigg{[}\frac{\Gamma(1+c)\Gamma(c-a-b-1)}{\Gamma(c-a)\Gamma(c-b)}{}_{2}\mathcal{F}_{1}\left(1+a,1+b,2+a+b-c;1-z\right)$
$\displaystyle+\frac{\Gamma(1+c)\Gamma(1+a+b-c)}{\Gamma(1+a)\Gamma(1+b)}(1-y)^{(c-a-b-1)}{}_{2}\mathcal{F}_{1}\left(c-a,c-b,c-a-b;1-z\right)\Bigg{]}$
(61)

Now, taking limit $y\rightarrow 1,$ and approximate the
${}_{2}\mathcal{F}_{1}\left(1+a,1+b,2+a+b-c;1-z\right)$ and
${}_{2}\mathcal{F}_{1}\left(c-a,c-b,c-a-b;1-z\right)$ using the series
relation of Gauss-hypergeometric function (60) and consider only zeroth order
term.
We have,

$\displaystyle\frac{2}{\tau}y(1-y)\frac{ab}{c}f_{1}$
$\displaystyle=\frac{2}{\tau}y(1-y)ab\frac{\Gamma(c)\Gamma(c-a-b-1)}{\Gamma(c-a)\Gamma(c-b)}+\frac{2}{\tau}y(1-y)^{c-a-b}(a+b-c)\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}$
$\displaystyle=\frac{2}{\tau}y(1-y)ab\Gamma_{1}+\frac{2}{\tau}y(1-y)^{c-a-b}(a+b-c)\Gamma_{2}$
(62)

where, $\Gamma_{1}=\frac{\Gamma(c)\Gamma(c-a-b-1)}{\Gamma(c-a)\Gamma(c-b)}$
and $\Gamma_{2}=\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}$
Similarly,

$\displaystyle(\omega(\bm{p},y)-(1-y)\omega_{-}-y\omega_{+})f_{2}$
$\displaystyle=(\omega(\bm{p},y)-(1-y)\omega_{-}-y\omega_{+})\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}{}_{2}\mathcal{F}_{1}\left(a,b,1-c+a+b;1-y\right)$
$\displaystyle+(1-y)^{c-a-b}\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}{}_{2}\mathcal{F}_{1}\left(c-a,c-b,1+c-a-b;1-y\right)$
$\displaystyle=(\omega(\bm{p},y)-(1-y)\omega_{-}-y\omega_{+})\Bigg{[}(c-a-b-1)\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}$
$\displaystyle+(1-y)^{c-a-b}\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}\Bigg{]}$
$\displaystyle=(\omega(\bm{p},y)-(1-y)\omega_{-}-y\omega_{+})\Bigg{(}(c-a-b-1)\Gamma_{1}+(1-y)^{c-a-b}\Gamma_{2}\Bigg{)}$
(63)

Since we will come across the gamma function $\Gamma(z)$ in the subsequent
content, we introduce it at this point. The Gamma function generally
satisfies,

$\displaystyle\Gamma(1+z)=z\Gamma(z),$ (64)
$\displaystyle\Gamma(1-z)\Gamma(z)=\frac{\pi}{\sin(\pi z)}$ (65)

from which we can derive the following useful relations,

$\displaystyle|\Gamma(\textrm{i}z)|^{2}$ $\displaystyle=\frac{\pi}{z\sinh(\pi
z)}$ $\displaystyle|\Gamma(1+\textrm{i}z)|^{2}$ $\displaystyle=\frac{\pi
z}{\sinh{\pi z}}$ $\displaystyle|\Gamma(\frac{1}{2}+\textrm{i}z)|^{2}$
$\displaystyle=\frac{\pi}{\cosh{\pi z}}$ (66)

Using that we are able to derive the following new set of equations

$\displaystyle|\Gamma_{1}|^{2}$
$\displaystyle=\frac{\delta_{2}\tau\omega_{-}}{\delta_{0}\delta_{1}(1+\delta_{0}^{2})}\Bigl{(}\frac{\sinh{\pi\delta_{1}}\sinh{\pi\delta_{2}}}{\sinh{(\pi\tau\omega_{-})}\sinh{(\pi\delta_{0})}}\Bigr{)}$
$\displaystyle|\Gamma_{2}|^{2}$
$\displaystyle=\frac{\delta_{3}\tau\omega_{-}}{\delta_{0}\delta_{4}}\Bigl{(}\frac{\sinh{\pi\delta_{3}}\sinh{\pi\delta_{4}}}{\sinh{(\pi\delta_{0})}\sinh{(\pi\tau\omega_{-})}}\Bigr{)}$
(67)

To compute $\Gamma_{1}\overline{\Gamma_{2}}$, we employ approximate formulas
for the gamma function, specifically Stirling’s formula for the gamma
function.

$\displaystyle\Gamma(z)\approx z^{z-1/2}e^{-z}\sqrt{2\pi}$ (68)

Then, we derive the set of equations employing Stirling’s formula for the
gamma function, which was used to determine the gamma function in the
computation of the distribution function.

$\displaystyle\Gamma(1+\textrm{i}x)$
$\displaystyle\sim\sqrt{2\pi}e^{(\frac{1}{2}ln(x)-\frac{\pi}{2}x)+\textrm{i}(x(ln(x)-1)+\frac{\pi}{4})}$
$\displaystyle\Gamma(-\textrm{i}x)$
$\displaystyle\sim\sqrt{2\pi}e^{(\frac{\pi}{2}x-\frac{1}{2}ln(x))+\textrm{i}(x(1-ln(x))-\frac{\pi}{4})}$
$\displaystyle\Gamma(\textrm{i}x)$
$\displaystyle\sim\sqrt{2\pi}e^{(-\frac{\pi}{2}x-\frac{1}{2}ln(x))+\textrm{i}(x(ln(x)-1)-\frac{\pi}{4})}$
(69)

Now,

$\displaystyle\Gamma_{1}\overline{\Gamma_{2}}$
$\displaystyle=\left(\frac{\Gamma(c)\Gamma(c-a-b-1)}{\Gamma(c-a)\Gamma(c-b)}\right)\overline{\Bigl{(}\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}}\Bigr{)}$
(70)

Subsequently, following certain algebraic manipulations, we obtain :

$\displaystyle\Gamma_{1}\bar{\Gamma_{2}}$
$\displaystyle=\frac{\tau\omega_{-}}{2\sinh{(\pi\tau\omega_{-})}}\frac{e^{\rho}}{\sqrt{1+\tau^{2}\omega_{+}^{2}}}\exp{\Bigl{[}\pi+\ln{\Bigl{(}\frac{\delta_{1}^{\delta_{1}}\delta_{2}^{\delta_{2}}\delta_{4}^{\delta_{4}}}{\delta_{0}^{2\delta_{0}}\delta_{3}^{\delta_{3}}}}\Bigr{)}+arctan(-\tau\omega_{+})\Bigr{]}}$
(71)

where,
$\rho=ln{\Bigl{(}\sqrt{\frac{(\delta_{2}\delta_{3})}{\delta_{1}\delta_{4}\delta_{0}^{2}}}}\Bigr{)},$
$\delta_{0}=\tau\omega_{+},\delta_{1}=\frac{\tau}{2}((\omega_{-}+\omega_{+})-2E_{0}\tau),\delta_{2}=\frac{\tau}{2}((\omega_{-}+\omega_{+})+2E_{0}\tau),$
$\delta_{3}=\frac{\tau}{2}((\omega_{-}-\omega_{+})+2E_{0}\tau),\delta_{4}=\frac{\tau}{2}((\omega_{+}-\omega_{-})+2E_{0}\tau).$

Now, employing this approximation, equation 57 can be re-expressed as follows:

$\displaystyle f(\bm{p},t)$
$\displaystyle=|N^{+}(\bm{p})|^{2}\Bigg{(}1+\frac{(p_{\parallel}-eA(t))}{\omega(\bm{p},t)}\Bigg{)}\Big{(}|\mathcal{Z}_{1}(\bm{p},t)|^{2}+|\mathcal{Z}_{2}(\bm{p},t)|^{2}+2\Re(\mathcal{Z}_{1}(\bm{p},t)\bar{\mathcal{Z}}_{2}(\bm{p},t))\Big{)}$
(72)

where,

$\displaystyle|\mathcal{Z}_{1}(\bm{p},t)|^{2}$
$\displaystyle=|\Gamma_{1}|^{2}(\frac{4}{\tau}y^{2}(1-y)^{2}(\mu^{2}+\nu^{2})+(1+\tau^{2}\omega_{+}^{2})(\omega(\bm{p},y)-y\omega_{+}-(1-y)\omega_{-})^{2})$
$\displaystyle+\frac{4}{\tau}y(1-y)(\omega(\bm{p},y)-y\omega_{+}-(1-y)\omega_{-})$
(73) $\displaystyle|\mathcal{Z}_{2}(\bm{p},t)|^{2}$
$\displaystyle=|\Gamma_{2}|^{2}((\omega(\bm{p},y)+y\omega_{+})^{2}-2\omega_{-}(\omega(\bm{p},y)+y\omega_{+})(1-y)+\omega_{-}^{2}(1-y)^{2})$
(74)
$\displaystyle\Re(\mathcal{Z}_{1}(\bm{p},t)\bar{\mathcal{Z}}_{2}(\bm{p},t))$
$\displaystyle=\frac{2}{\tau}(y(1-y)\omega-y(1-y)^{2}\omega_{-}+y^{2}(1-y)\omega_{+})\Bigl{[}\sin{(\tau\omega_{+}\ln{(1-y)})}(\mu\Re(\Gamma_{1}\bar{\Gamma_{2}})$
$\displaystyle-\nu\Im(\Gamma_{1}\bar{\Gamma_{2}}))+\cos{(\tau\omega_{+}\ln{(1-y)})}(\nu\Re(\Gamma_{1}\bar{\Gamma_{2}})+\mu\Im(\Gamma_{1}\bar{\Gamma_{2}}))\Bigr{]}$
$\displaystyle+(\omega^{2}+(1-y)^{2}\omega_{-}^{2}-2\omega_{-}(1-y)-y^{2}\omega_{+}^{2})\Bigl{[}\sin{(\tau\omega_{+}\ln{(1-y)})}\biggl{(}\tau\omega_{+}\Re(\Gamma_{1}\bar{\Gamma_{2}})$
$\displaystyle+\Im(\Gamma_{1}\bar{\Gamma_{2}})\biggr{)}+\cos{(\tau\omega_{+}\ln{(1-y)})}\biggl{(}-\Re(\Gamma_{1}\bar{\Gamma_{2}})+\tau\omega_{+}\Im(\Gamma_{1}\bar{\Gamma_{2}})\biggr{)}\Bigr{]}$
(75)

Now, using the equation (71) for approximating the
$\Gamma_{1}\bar{\Gamma_{2}}$

$\displaystyle\Re(\mathcal{Z}_{1}(\bm{p},t)\bar{\mathcal{Z}}_{2}(\bm{p},t))$
$\displaystyle=\frac{e^{\rho}\tau\omega_{-}}{\sqrt{1+\tau^{2}\omega_{+}^{2}\sinh{(\pi\tau\omega_{-})}}}\Bigg{[}\frac{2}{\tau}(y(1-y)\omega-y(1-y)^{2}\omega_{-}+y^{2}(1-y)\omega_{+})$
$\displaystyle\sqrt{\mu^{2}+\nu^{2}}\sin{(\tau\omega_{+}\ln(1-y)+\Xi_{1})}+(\omega^{2}+(1-y)^{2}\omega_{-}^{2}-2\omega_{-}(1-y)-y^{2}\omega_{+}^{2})$
$\displaystyle\sin{(\tau\omega_{+}ln(1-y)-\Xi_{2})}\sqrt{1+\tau\omega_{+}^{2}}\Bigg{]}$
(76)

where,
$\rho=ln{\Bigl{(}\sqrt{\frac{(\delta_{2}\delta_{3})}{\delta_{1}\delta_{4}\delta_{0}^{2}}}}\Bigr{)},$
$\mu=\Bigl{(}E_{0}\tau^{2}-\frac{\tau}{2}(\omega_{-}-\omega_{+})\Bigr{)}\Bigl{(}E_{0}\tau^{2}+\frac{\tau}{2}(\omega_{-}-\omega_{+})\Bigr{)},$
$\nu=-\Bigl{(}E_{0}\tau^{2}-\frac{\tau}{2}(\omega_{-}-\omega_{+})\Bigr{)},$

$\Xi_{1}=\pi+\arctan\Bigl{(}\frac{1-\delta_{0}\delta_{3}}{\delta_{0}-\delta_{3}}\Bigr{)}+\ln{\Bigl{(}\frac{\delta_{1}^{\delta_{1}}\delta_{2}^{\delta_{2}}\delta_{4}^{\delta_{4}}}{\delta_{0}^{2\delta_{0}}\delta_{3}^{\delta_{3}}}}\Bigr{)}$
$\Xi_{2}=\ln{\Bigl{(}\frac{\delta_{0}^{2\delta_{0}}\delta_{3}^{\delta_{3}}}{\delta_{1}^{\delta_{1}}\delta_{2}^{\delta_{2}}\delta_{4}^{\delta_{4}}}}\Bigr{)}-\frac{\pi}{2}$

To further simplify,

$\displaystyle f(\bm{p},t)$
$\displaystyle=|N^{+}(\bm{p})|^{2}\Bigg{(}1+\frac{(p_{\parallel}-eA(t))}{\omega(\bm{p},t)}\Bigg{)}\Bigg{(}|\Gamma_{1}|^{2}(\frac{4}{\tau}y^{2}(1-y)^{2}(\mu^{2}+\nu^{2})+(1+\tau^{2}\omega_{+}^{2})(\omega(p,y)-y\omega_{+}-(1-y)\omega_{-})^{2})$
$\displaystyle+\frac{4}{\tau}y(1-y)(\omega-y\omega_{+}-(1-y)\omega_{-})+|\Gamma_{2}|^{2}((\omega+y\omega_{+})^{2}-2\omega_{-}(\omega+y\omega_{+})(1-y)+\omega_{-}^{2}(1-y)^{2})$
$\displaystyle+\frac{(-e^{\rho})\tau\omega_{-}}{2\sinh{(\pi\tau\omega_{-})}\sqrt{1+\tau^{2}\omega_{+}^{2}}}\Bigl{(}\frac{2}{\tau}\sqrt{\mu^{2}+\nu^{2}}\sin{(\tau\omega_{+}ln(1-y)+\Xi_{1})}(y(1-y)\omega-y(1-y)^{2}\omega_{-}$
$\displaystyle+y^{2}(1-y)\omega_{+})+\sin{(\tau\omega_{+}ln(1-y)-\Xi_{2})}\sqrt{1+\tau\omega_{+}^{2}}(\omega^{2}+(1-y)^{2}\omega_{-}^{2}$
$\displaystyle-2\omega_{-}(1-y)-y^{2}\omega_{+}^{2})\Bigr{)}\Bigg{)}$ (77)

Since we are interested in examining the behavior of the vacuum state at
finite times (or late times), we aim to derive an analytical expression for
the longitudinal momentum distribution function,$f(p_{\parallel},t)$ as a
power series of $\sum_{n}C_{n}(1-y)^{n}$, where value of $y$ changes with $t$
following the equation $y=\frac{1}{2}\left(1+\tanh(\frac{t}{\tau})\right)$. As
time advances, particularly in the context of large time limit such as
$t=10\tau$, it becomes evident that $(1-y)$ approaches zero. Consequently, the
dominant influence originates from the zeroth-order terms, where the
coefficients of $(1-y)^{0}$ precisely depict
$f(p_{\parallel},t\rightarrow\infty)$. By considering the other higher-order
terms, we can investigate the distribution function’s behavior at finite
times.
In this context, we consider the transverse momentum value to be zero and
focus solely on exploration in the longitudinal direction. Now, quasi-
energy,$\omega(p_{\parallel},y)$ can be approximated as follows:

$\displaystyle\omega(p_{\parallel},y)=\sqrt{1+(p_{\parallel}+yE_{0}\tau)^{2}}\Bigg{(}1-(1-y)\frac{E_{0}\tau(p_{\parallel}+yE_{0}\tau)}{1+(p_{\parallel}+yE_{0}\tau)^{2}}\Bigg{)}$
(78)

Subsequently, through algebraic manipulation, we derive:

$\displaystyle|\mathcal{Z}_{1}(\bm{p},t)|^{2}$
$\displaystyle=|\Gamma_{1}|^{2}\Biggl{[}\frac{4}{\tau^{2}}y^{2}(1-y)^{2}(\mu^{2}+\nu^{2})$
$\displaystyle+(1+\tau^{2}\omega_{+}^{2})\Biggl{(}y^{2}\omega_{+}^{2}+2y(1-y)\omega_{+}\omega_{-}-2y\omega_{+}\sqrt{1+(p_{\parallel}+E_{0}\tau
y)}$ $\displaystyle\left(1-(1-y)\frac{E_{0}\tau(p_{\parallel}+E_{0}\tau
y)}{1+(p_{\parallel}+E_{0}\tau y)^{2})}\right)+(1+(p_{\parallel}+E_{0}\tau
y)^{2})(1-2(1-y)\frac{E_{0}\tau(p_{\parallel}+E_{0}\tau
y)}{1+(p_{\parallel}+E_{0}\tau y)^{2}}$
$\displaystyle+(1-y)^{2}\frac{(E_{0}\tau(p_{\parallel}+E_{0}\tau
y))^{2}}{(1+(p_{\parallel}+E_{0}\tau
y)^{2})^{2}})+(1-y)^{2}\omega_{-}^{2}-2\omega_{-}(1-y)\sqrt{(1+(p_{\parallel}+E_{0}\tau
y)^{2})}$ $\displaystyle\left(1-(1-y)\frac{E_{0}\tau(p_{\parallel}+E_{0}\tau
y)}{1+(p_{\parallel}+E_{0}\tau
y)^{2}}\right)\Biggr{)}+\frac{4}{\tau}y(1-y)(\mu\tau\omega_{+}-\nu)\Big{(}\sqrt{1+(p_{\parallel}+E_{0}\tau
y)^{2}}$ $\displaystyle\left(1-(1-y)\frac{E_{0}\tau(p_{\parallel}+E_{0}\tau
y)}{1+(p_{\parallel}+E_{0}\tau
y)^{2}}\right)-(1-y)\omega_{-}-y\omega_{+}\Big{)}\Biggr{]}$ (79)
$\displaystyle|\mathcal{Z}_{2}(\bm{p},t)|^{2}$
$\displaystyle=|\Gamma_{2}|^{2}\Biggl{[}y^{2}\omega_{+}^{2}-2y(1-y)\omega_{+}\omega_{-}+2y\omega_{+}\sqrt{1+(p_{\parallel}+E_{0}\tau
y)^{2}}\left(1-(1-y)\frac{E_{0}\tau(p_{\parallel}+E_{0}\tau
y)}{1+(p_{\parallel}+E_{0}\tau y)^{2}}\right)$
$\displaystyle+(1+(p_{\parallel}+E_{0}\tau
y)^{2})\left(1-2(1-y)\frac{E_{0}\tau(p_{\parallel}+E_{0}\tau
y)}{1+(p_{\parallel}+E_{0}\tau
y)^{2}}+(1-y)^{2}\frac{E_{0}\tau(p_{\parallel}+E_{0}\tau
y)^{2}}{(1+(p_{\parallel}+E_{0}\tau y)^{2})^{2}}\right)$
$\displaystyle+(1-y)^{2}\omega_{-}^{2}-2(1-y)\omega_{-}\sqrt{1+(p_{\parallel}+E_{0}\tau
y)^{2}}\left(1-(1-y)\frac{E_{0}\tau(p_{\parallel}+E_{0}\tau
y)}{1+(p_{\parallel}+E_{0}\tau y)^{2}}\right)\Biggr{]}$ (80)
$\displaystyle\Re(\mathcal{Z}_{1}(\bm{p},t)\bar{\mathcal{Z}}_{2}(\bm{p},t))$
$\displaystyle=\frac{\tau\omega_{-}e^{\rho}}{2\sinh{\pi\tau\omega_{-}}}\Biggl{[}\frac{2}{\tau}\sqrt{\frac{\mu^{2}+\nu^{2}}{1+\tau\omega_{+}^{2}}}\left(y(1-y)\sqrt{1+(p_{\parallel}+E_{0}\tau
y)^{2}}+y^{2}(1-y)\omega_{+}\right)$
$\displaystyle\sin{(\tau\omega_{+}\ln(1-y)+\Xi_{1})}+\sin{(\tau\omega_{+}\ln(1-y)-\Xi_{2})}\biggl{(}(1+(p_{\parallel}+E_{0}\tau
y)^{2})$ $\displaystyle\left(1-2(1-y)\frac{E_{0}\tau(p_{\parallel}+E_{0}\tau
y)}{1+(p_{\parallel}+E_{0}\tau
y)^{2}}\right)-y^{2}\omega_{+}^{2}-2(1-y)\omega_{-}\sqrt{1+(p_{\parallel}+E_{0}\tau
y)^{2}}\biggr{)}\Biggr{]}$ (81)

Now, we can construct an analytical and simplified algebraic expression for
the distribution function during a finite time $(t>2\tau)$. This expression
takes the form of a power series involving $(1-y)^{n}$, where $n$ can be 0, 1,
or 2. We omit higher-order terms from consideration due to the diminishing
significance of the $(1-y)$ term as time progresses.

$\displaystyle f(p_{\parallel},t)$
$\displaystyle=\Bigg{(}\frac{|N^{+}(\bm{p})|^{2}}{2\left[p_{\parallel}+E_{0}\tau(y-(1-y))^{2}\right]}\Bigg{)}\Bigg{[}{\color[rgb]{0,0,1}(1-y)^{2}}|\Gamma_{1}|^{2}\Bigg{(}\frac{4y^{2}(\mu^{2}+\nu^{2})}{\tau^{2}}+\Bigl{(}\omega_{-}+\frac{E_{0}\tau(p_{\parallel}+E_{0}\tau
y)}{\sqrt{1+(p_{\parallel}+E_{0}\tau y)^{2}}}\Bigr{)}^{2}$
$\displaystyle-\frac{4}{\tau^{2}}y\omega_{-}(\mu\tau\omega_{+}-\nu)-\frac{4y}{\tau}\frac{E_{0}\tau(p_{\parallel}+E_{0}\tau
y)}{\sqrt{1+(p_{\parallel}+E_{0}\tau
y)^{2}}}\Bigg{)}+|\Gamma_{1}|^{2}{\color[rgb]{0,0,1}(1-y)}\Bigg{(}2y\omega_{-}\omega_{+}(1+\tau^{2}\omega_{+}^{2})$
$\displaystyle+2y\omega_{+}(1+\tau^{2}\omega_{+}^{2})\frac{E_{0}\tau(p_{\parallel}+E_{0}\tau
y)}{\sqrt{1+(p_{\parallel}+E_{0}\tau
y)^{2}}}-2E_{0}\tau(1+\tau^{2}\omega_{+}^{2})(p_{\parallel}+E_{0}\tau y)$
$\displaystyle-2(1+\tau^{2}\omega_{+}^{2})\omega_{-}\sqrt{1+(p_{\parallel}+E_{0}\tau
y)^{2}}-\frac{4}{\tau}y^{2}\omega_{+}(\mu\tau\omega_{+}-\nu)+\frac{4}{\tau}y(\mu\tau\omega_{+}-\nu)\sqrt{1+(p_{\parallel}+E_{0}\tau
y)^{2}}\Bigg{)}$
$\displaystyle-2|\Gamma_{2}|^{2}{\color[rgb]{0,0,1}(1-y)}\Bigg{(}y\omega_{+}\omega_{-}+y\omega_{+}\frac{E_{0}\tau(p_{\parallel}+E_{0}\tau
y)}{\sqrt{1+(p_{\parallel}+E_{0}\tau
y)^{2}}}+E_{0}\tau(p_{\parallel}+E_{0}\tau
y)+\omega_{-}\sqrt{1+(p_{\parallel}+E_{0}\tau y)^{2}}\Bigg{)}$
$\displaystyle+|\Gamma_{1}|^{2}(1+\tau^{2}\omega_{+}^{2})\Bigl{(}y\omega_{+}-\sqrt{1+(p_{\parallel}+E_{0}\tau
y)^{2}}\Bigr{)}^{2}+|\Gamma_{2}|^{2}\Bigl{(}y\omega_{+}+\sqrt{1+(p_{\parallel}+E_{0}\tau
y)^{2}}\Bigr{)}^{2}$
$\displaystyle-\frac{e^{\rho}\tau\omega_{-}}{2\sinh{(\pi\tau\omega_{-})}\sqrt{1+\tau^{2}\omega_{+}^{2}}}\Biggl{(}\sqrt{1+\tau^{2}\omega_{+}^{2}}\sin{(\tau\omega_{+}ln(1-y)-\Xi_{2})}(1+(p_{\parallel}+E_{0}\tau
y)^{2}-y^{2}\omega_{+}^{2})$
$\displaystyle+{\color[rgb]{0,0,1}(1-y)}\biggl{(}\frac{2}{\tau}\sqrt{\mu^{2}+\nu^{2}}\sin{(\tau\omega_{+}ln(1-y)+\Xi_{1})}(y\sqrt{1+(p_{\parallel}+E_{0}\tau
y)^{2}}+y^{2}\omega_{+})$
$\displaystyle-2\sqrt{1+\tau^{2}\omega_{+}^{2}}\left(E_{0}\tau(p_{\parallel}+E_{0}\tau
y)+\omega_{-}\sqrt{1+(p_{\parallel}+E_{0}\tau
y)^{2}}\right)\sin{(\tau\omega_{+}\ln(1-y)-\Xi_{2})}\Big{)}\biggr{)}\Bigg{]}$
(82)

Figure 7: Time evolution of the components defined in equation ((72)) of the
approximate analytical expression of $f(\bm{p},t)$ black curve
:$\mathcal{Z}_{1}(p_{\parallel},t)$ blue
curve:$\mathcal{Z}_{2}(p_{\parallel},t)$ yellow curve:
$(\mathcal{Z}_{2}(p_{\parallel},t)+\Re(\mathcal{Z}_{1}(p_{\parallel},t)\bar{\mathcal{Z}_{2}}(p_{\parallel},t)))$magenta
curve:interference
term,$\Re(\mathcal{Z}_{1}(p_{\parallel},t)\bar{\mathcal{Z}_{2}}(p_{\parallel},t))$

On carefully examining the approximate expression for the distribution
function, we can say that in the early stages of $e^{+}-e^{-}$ formation,
first term $\mathcal{Z}_{1}(p_{\parallel},t)$ responsible for the primary peak
at $p_{\parallel}\approx-2$ due to the presence of $|\Gamma_{1}|^{2}$ that
dependent on $(E_{0},\tau)$ and as time progresses that primary peak
diminished. Secondary peak at $p_{\parallel}=0$ starts build-up due to term
$\mathcal{Z}_{2}(p_{\parallel},t)$ and
$\Re(\mathcal{Z}_{1}(p_{\parallel},t)\bar{\mathcal{Z}_{2}}(p_{\parallel},t))$
responsible for onset oscillation on that peak. The magnitude of
$\Re(\mathcal{Z}_{1}(p_{\parallel},t)\bar{\mathcal{Z}_{2}}(p_{\parallel},t))$
term oscillation changes due to the presence of $``ln(1-y)^{\prime\prime}$ in
the sinusoidal function. It becomes suppressed and, as a result of the only
secondary peak at $p_{\parallel}=0$, visible also explicitly verified in
figure 7. Also, we point out that
$\Re(\mathcal{Z}_{1}(p_{\parallel},t)\bar{\mathcal{Z}_{2}}(p_{\parallel},t))$
an oscillatory finite function whose magnitude depends on $t$. Its magnitude
determines the nature of $f(p_{\parallel},t)$ dynamics in $p_{\parallel}-$
space at finite times.
We further derive an asymptotic expression for the distribution function in
the limit $t\rightarrow\infty$ $(y\approx 1).$ The term
$\mathcal{Z}_{1}(p_{\parallel},t)$ and
$\Re(\mathcal{Z}_{1}(p_{\parallel},t)\bar{\mathcal{Z}_{2}}(p_{\parallel},t))$
becomes zero and only $\mathcal{Z}_{2}(p_{\parallel},t)$ term exist. By
lengthy calculations, we found an analytical expression for the distribution
function for asymptotic times as follows:

$\displaystyle f(p_{\parallel},y\rightarrow 1)$
$\displaystyle=\frac{2\sinh(\pi\tau(2E_{0}\tau+\omega_{-}-\omega_{+})/2)\sinh(\pi\tau(2E_{0}\tau-\omega_{-}+\omega_{+})/2)}{\sinh(\pi\tau\omega_{-})\sinh(\pi\tau\omega_{+})}$
(83)

###  IV.3 Multi-photon regime

In this section, we explored the LMS of the created particle in the
multiphoton regime. We choose the parameters of the laser pulse in such a way
that the Keldysh parameter,$\gamma>>1.$ Fig. 8 shows LMS for short pulse-
duration $\tau=4[m^{-1}]$ and $E_{0}=0.1E_{c}.$ The Keldysh parameter, in this
case, is close to $2.5$, i.e., $\gamma>>1$ corresponds to n-th order
perturbation theory, with $n$ being the minimum number of photons to be
absorbed in order to overcome the threshold energy for pair creation
$n\omega>2m.$ Here, pair production via multi-photon processes is discussed
for $\gamma=2.5.$ In the early time of the creation of pairs, $t=-4[m^{-1}],$
spectrum has a unimodal Gaussian-like profile peak at
$p_{\parallel}\simeq-0.4$, and as time proceeds, we see shifting of peak
$p_{\parallel}\simeq-eE_{0}\tau$ to $p_{\parallel}\simeq eE_{0}\tau$ due to
the action of Lorentz force and its peak value shows the maximum value for
$t=0$, i.e., $f(p_{\parallel})=1.4\times 10^{-3}.$(see figure 8 (a-c)).
However, that smooth unimodal spectrum now shows slight modulation Fig. 8(d)
when $t=9[m^{-1}]$( which is greater than the effective pulse duration length
($2\tau$)). At finite time, $t=12[m^{-1}]$spectrum has a quad-modal profile,
as seen in figure 8(e). The central peak, which is located at
$p_{\parallel}=0$, is much more prominent than the two unequal peaks at
$p_{\parallel}\simeq\pm 0.4$ and another very small peak at
$p_{\parallel}\simeq 0.7.$

Fig. 8(f-g) shows the merging of those multimodal peaks as a result of fading
of different peaks that occur and the single smooth Gaussian peaks observed at
$t>5\tau$ as shown in figure 8 (h-i). An interesting qualitative contrast
becomes evident when comparing the previous scenario with $\gamma=0.5$ to the
current situation. The presented figure illustrates this distinct behavior.
Notably, a multi-modal pattern emerges at a specific moment, characterized by
the presence of more than two peaks, occurring approximately at $3\tau$.
As time progresses, the multi-modal pattern gradually dissipates, eventually
leading to a singular smooth peak above $t>5\tau$. Intriguingly, a similar
evolution is observable in the case of $\gamma=0.5$, signifying a shared
progression in both scenarios despite representing a non-multiphoton process.

Figure 8: Time evolution of quasi-particle distribution function in the
longitudinal momentum space $f(p_{\parallel},t)$ at different times. The
transverse momentum is considered to be zero, and all the units are taken in
the electron mass unit.The field parameters are $E_{0}=0.1E_{c}$ and
$\tau=4[m^{-1}].$

###  IV.4 Correlation function

In order to describe the creation and annihilation of particles in our
physical system, one needs to define a term that is able to describe such
creation and annihilation processes due to an external field - the time-
dependent pair correlation function and its complex conjugate, respectively

$\displaystyle\mathcal{C}(\bm{p},t)$
$\displaystyle=\langle{0}|\hat{D}_{-{\bm{p}}r}^{\dagger}(t)\hat{B}_{{\bm{p}}r}^{\dagger}(t)|{0}\rangle=2\alpha_{\bm{p}}^{*}(t)\beta_{\bm{p}}(t)$
(84) $\displaystyle\mathcal{C}^{*}(\bm{p},t)$
$\displaystyle=\langle{0}|\hat{D}_{-{\bm{p}}r}(t)\hat{B}_{{\bm{p}}r}(t)|{0}\rangle=2\beta_{\bm{p}}^{*}(t)\alpha_{\bm{p}}(t)$
(85)

As can easily be seen, this function $\mathcal{C}(\bm{p},t)$ consisting of
creation operators for a particle and an anti-particle with the opposite
momentum describes the process of production of $e^{-}e^{+}$ pair. In numerous
research papersSchmidt:1998vi ; Banerjee:2018fbw ; Blaschke2016 ;
Fedotov:2010ue , the particle-antiparticle correlation function undergoes
redefinition by utilizing the slowly varying component of the time-dependent
creation and annihilation operators for quasi-particles and
anti-(quasi)particles.

$\hat{\mathcal{B}}_{{\bm{p}}r}(t)=\hat{B}_{{\bm{p}}r}(t)\textrm{e}^{-\textrm{i}\Theta(\bm{p},t)}$
$\hat{\mathcal{D}}_{-{\bm{p}}r}(t)=\hat{D}_{-{\bm{p}}r}(t)\textrm{e}^{-\textrm{i}\Theta(\bm{p},t)}$

where,$\Theta(\bm{p},t)=\int^{t}dt^{\prime}\omega(\bm{p},t^{\prime})$
So that,

$\displaystyle\mathcal{C}(\bm{p},t)$
$\displaystyle=\langle{0}|\hat{\mathcal{D}}_{-{\bm{p}}r}^{\dagger}(t)\hat{\mathcal{B}}_{{\bm{p}}r}^{\dagger}(t)|{0}\rangle$
$\displaystyle=\textrm{e}^{2\textrm{i}\Theta(\bm{p},t)}\langle{0}|\hat{D}_{-{\bm{p}}r}^{\dagger}(t)\hat{B}_{{\bm{p}}r}^{\dagger}(t)|{0}\rangle$
(86)

Now, using the above relation for the Pair correlation function in the
presence of Sauter-pulse electric field $E(t)$

$\displaystyle\mathcal{C}(\bm{p},t)$
$\displaystyle=|N^{+}(\bm{p})|^{2}\sqrt{1-\frac{(p_{\parallel}-eA(t))^{2}}{\omega^{2}(\bm{p},t)}}\Biggl{(}\frac{4}{\tau^{2}}y^{2}(1-y)^{2}|\frac{ab}{c}|^{2}|f_{1}|^{2}+(\omega^{2}(\textbf{p},y)-(y\omega_{+}+(1-y)\omega_{-})^{2})|f_{2}|^{2}$
$\displaystyle+\frac{2\textrm{i}}{\tau}y(1-y)((1-y)\omega_{-}+y\omega_{+})(\frac{ab}{c}f_{1}\bar{f_{2}}-\frac{\bar{ab}}{\bar{c}}\bar{f_{1}}f_{2})\Biggr{)}$
(87)

As we know, the vacuum polarization effects play a crucial role in the process
of pair production. This effect is defined through the functions
$u(\bm{p},t)=\Re{(\mathcal{C}(\bm{p},t))}$ and
$v(\bm{p},t)=\Im{(\mathcal{C}(\bm{p},t))}$.

$\displaystyle u(\bm{p},t)$
$\displaystyle=|N^{+}(\bm{p})|^{2}\sqrt{1-\frac{(p_{\parallel}-eA(t))^{2}}{\omega^{2}(\bm{p},t)}}\Biggl{(}\frac{4}{\tau^{2}}y^{2}(1-y)^{2}|\frac{ab}{c}|^{2}|f_{1}|^{2}+(\omega^{2}(\textbf{p},y)-(y\omega_{+}+(1-y)\omega_{-})^{2})|f_{2}|^{2}$
$\displaystyle+\frac{2}{\tau}y(1-y)((1-y)\omega_{-}+y\omega_{+})(\Im{(\frac{\bar{ab}}{\bar{c}}\bar{f_{1}}f_{2})}-\Im{(\frac{ab}{c}f_{1}\bar{f_{2}})})\Biggr{)}$
(88) $\displaystyle v(\bm{p},t)$
$\displaystyle=|N^{+}(\bm{p})|^{2}\sqrt{1-\frac{(p_{\parallel}-eA(t))^{2}}{\omega^{2}(\bm{p},t)}}\Biggl{(}\frac{2}{\tau}y(1-y)((1-y)\omega_{-}+y\omega_{+})(\Re{(\frac{ab}{c}f_{1}\bar{f_{2}})}-\Re{(\frac{\bar{ab}}{\bar{c}}\bar{f_{1}}f_{2})})\Biggr{)}$
(89)

The function $u(\bm{p},t)$ depicts vacuum polarization effects and pair
production phenomena. The function $v(\bm{p},t)$ serves as counter terms to
the pair production, effectively representing the pair annihilation in the
vacuum excitation process. By considering these functions, we can gain
valuable insights into the interplay between particle creation and
annihilation in the complex dynamics of the vacuum polarization phenomenon.
Through algebraic manipulation, it becomes feasible to deduce the analytical
expressions for $u(\bm{p},t)$ and $v(\bm{p},t)$ which are as follows:

$\displaystyle u(\bm{p},t)$
$\displaystyle=|N^{+}(\bm{p})|^{2}\sqrt{1-\frac{\bigl{(}p_{\parallel}-eA(t)\bigr{)}^{2}}{\omega(\bm{p},t)}}\Bigl{(}\frac{4}{\tau^{2}}y^{2}\Bigl{(}1-y)^{2}(\mu^{2}+\nu^{2})|\Gamma_{1}|^{2}+\tau\omega_{+}|\Gamma_{2}|^{2}$
$\displaystyle-2(1-y)\sqrt{\frac{\mu^{2}+\nu^{2}}{1+\tau^{2}\omega_{+}^{2}}}\frac{e^{\rho}\tau^{2}\omega_{+}^{2}}{\sinh{(\tau\omega_{+}^{2})}}\sin{(\tau\omega_{+}\ln(1-y)-\Xi_{1})}$
$\displaystyle+(\omega^{2}-((1-y)\omega_{-}+y\omega_{+})^{2})((1+\tau^{2}\omega_{+}^{2})|\Gamma_{1}|^{2}+|\Gamma_{2}|^{2}+2\frac{e^{\rho}\tau^{2}\omega_{+}^{2}}{\sinh{(\tau\omega_{+}^{2})}}\cos{(\tau\omega_{+}\ln(1-y)-\Xi_{1})}$
$\displaystyle+(1-y)\sqrt{\frac{\mu^{2}+\nu^{2}}{1+\tau^{2}\omega_{+}^{2}}}\frac{e^{\rho}\tau^{2}\omega_{+}^{2}}{\sinh{(\pi\tau\omega_{+})}}\cos{(\Xi_{1}-\xi)}\sin{(\tau\omega_{+}\ln(1-y)+\xi)}$
(90) $\displaystyle v(\bm{p},t)$
$\displaystyle=(-1)|N^{+}(\bm{p})|^{2}\sqrt{1-\frac{\bigl{(}p_{\parallel}-eA(t)\bigr{)}^{2}}{\omega(\bm{p},t)}}\Bigg{(}\frac{2y}{\tau}(\omega+(1-y)\omega_{-}+y\omega_{+})\Bigl{(}2(1-y)\sqrt{(1+\tau^{2}\omega_{+}^{2})(\mu^{2}+\nu^{2})}$
$\displaystyle|\Gamma_{1}|^{2}\cos{(\xi-\eta)}-(1-y)\sqrt{\frac{\mu^{2}+\nu^{2}}{1+\tau^{2}\omega_{+}^{2}}}\frac{e^{\rho}\tau^{2}\omega_{+}^{2}}{\sinh{(\pi\tau\omega_{+})}}\cos{(\Xi_{1}-\xi)}\cos{(\tau\omega_{+}\ln(1-y)+\xi)}$
$\displaystyle+\frac{e^{\rho}\tau^{2}\omega_{+}^{2}}{\sinh{(\pi\tau\omega_{+})}}\sin{(\tau\omega_{+}\ln{(1-y)}+(\xi+\eta-\Xi_{1}))}\Bigr{)}\Bigg{)}.$
(91)

with, $\xi=arctan(-1/\delta_{3})$ and $\eta=arctan(\delta_{0})$

Figure 9: Evolution of $u(t,p_{\parallel})$ and $v(t,p_{\parallel})$ for
$p_{\parallel}=$0(blue),0.5(pink),-0.5(brown),1(red), -1(green) with
$p_{\perp}=0,$ $E_{0}=0.2E_{c}$ and $\tau=10[m^{-1}]$ . All the measurements
are taken in electron mass units.

For a better understanding of the phenomenon of particle creation under the
strong electric field, we will also trace the evolution of the vacuum
polarization function, $u(\bm{p},t)$ its counter term $v(\bm{p},t)$. Figure 9
shows the time evolution of $u(p_{\parallel},t)$ and $v(p_{\parallel},t)$ for
different values of $p_{\parallel}$ at zero transverse momentum. Due to pair
annihilation being stronger than pair creation in the early times,
depolarisation function $v(p_{\parallel},t)$ dominates over
$u(p_{\parallel},t)[\ref{fig:uv}(a)-(c)].$ It appears that in the polarization
function, there is a sinusoidal-type structure. On the other hand, the
depolarization function $v(p_{\parallel},t)$ shows an unimodal Gaussian peak
in its temporal evolution. Both $u(p_{\parallel},t)$ and $v(p_{\parallel},t)$
show oscillations with varying amplitudes during the initial transient stage.
These oscillations are particularly pronounced when $p_{\parallel}=0$, as
evident from Figure 9(a) and 9(c). As time progresses, irregular oscillations
are observed in the transient stage. However, the oscillations become regular
and stable as the system enters the REPP stage. Both functions,
$u(p_{\parallel},t)$ and $v(p_{\parallel},t)$, now exhibit regular
oscillations centered around the zero value. Moreover, as the momentum value
$p_{\parallel}$ increases, the amplitudes of these oscillations diminish, as
shown in Figure 9(b) and 9(d). During the REPP stage, one interesting finding
is that $u(p_{\parallel},t)$ and $v(p_{\parallel},t)$ demonstrate balancing
behavior characterized by similar oscillatory patterns. This balance is a
result of the formation of real free electron-positron pairs.

####  IV.4.1 Momentum Spectra of $u(p_{\parallel},t)$ and $v(p_{\parallel},t)$

The longitudinal momentum $(p_{\parallel})$ influences the vacuum polarization
function’s qualitative nature. To gain a comprehensive understanding of its
dependency on $p_{\parallel}$, both $u(p_{\parallel},t)$ and its corresponding
counter term $v(p_{\parallel},t)$ are subjected to analysis using the LMS,
which allows us to study their characteristics more comprehensively. Figure 10
displays the LMS of $u(p_{\parallel},t)$ and $v(p_{\parallel},t).$ In the
initial stages of particle formation, specifically at $t<0$, both
$u(p_{\parallel},t)$ and $v(p_{\parallel},t)$ display asymmetric Gaussian
peaks with roughly similar profiles. During this period, $v(p_{\parallel},t)$
dominates over $u(p_{\parallel},t)$. Upon closer examination, it becomes
evident that the peaks occur at different positions around
$p_{\parallel}\approx 1$, as illustrated in Figure 10(a). At $t=0$, the
initial Gaussian-shaped structure of $u(p_{\parallel},t)$ deformed, becoming a
bi-modal asymmetric Gaussian-like structure. This means that
$u(p_{\parallel})$ displays peaks at specific non-zero values of
$p_{\parallel}\simeq-0.35[m]$ and a dip (or valley) at
$p_{\parallel}\simeq+0.35[m]$, resulting in
$u(p_{\parallel})=u(-p_{\parallel})$ for the maximum value. In contrast, the
spectrum of $v(p_{\parallel},t)$ retains its Gaussian-like unimodal profile,
with the peak located at $p_{\parallel}=0$. This implies that the behavior of
$u(p_{\parallel},t)$ undergoes significant changes, becoming asymmetric, while
$v(p_{\parallel},t)$ remains relatively unchanged in its overall shape during
this time. Due to the force factor $``eE(t)^{\prime\prime}$ and corresponding
longitudinal kinetic quasi-momentum $P(t)=(p_{\parallel}-eA(t))$, the spectrum
moved to the left side of the origin, and its peak now located at
$p_{\parallel}\approx-2$ for $v(p_{\parallel},t)$ whereas small dip in LMS of
$u(p_{\parallel},t)$ with some disruption in the tail ( $-1<p_{\parallel}<1$)
as shown in the figure 10(c-d) near to the transient stage. The behavior of
$v(p_{\parallel},t)$ and $u(p_{\parallel},t)$ at this stage explicitly counter
each other, as they exhibit distinct shifts and variations due to the applied
electric field and longitudinal kinetic quasi-momentum.

Figure 10:  Created particle LMS at finite time Blue curve: polarisation
function( $u(p_{\parallel},t)$ ) and Magenta curve: de-polarisation function(
$v(p_{\parallel},t).$ ).The transverse momentum is considered to be zero and
all the units are taken in the electron mass unit.The field parameters are
$E_{0}=0.2E_{c}$ and $\tau=10[m^{-1}].$

At $t=24$, we observe two distinct structures in the polarization and
depolarization functions as time progresses. The first structure occurs at
$p_{\parallel}=-2$, as a consequence of earlier vacuum excitation responsible
for pair formation. The second structure is located in the range
$-1<p_{\parallel}<1$, exhibiting varying amplitude oscillations, with the
maximum occurring at zero longitudinal momentum throughout the process. At
this stage, $u(p_{\parallel})$ and $v(p_{\parallel})$ compete with each other,
as shown in Figure 10(d) and 10(e). As we approach the beginning of the REPP
stage, at $t=3\tau$, both functions, $u(p_{\parallel},t)$ and
$v(p_{\parallel},t)$, indeed exhibit a single oscillating structure resembling
a sine or cosine function with varying amplitude. The maximum now appears near
$p_{\parallel}=0$, dominating in comparison to the secondary peak (and dip for
$u(p_{\parallel},t)$) that was present on the left side of the origin at
$p_{\parallel}\approx-2$. As time progresses further, this left side structure
slowly vanishes, as shown in Figure 10(e) and 10(f). Eventually, as $A(t)$
reaches a constant value, a balance is achieved between the processes of
particle creation and annihilation, as depicted in Figure 10(g). In the late
REPP stage, where the particle distribution function $f(p_{\parallel},t)$ is
constant, the polarization function $u(p_{\parallel},t)$ is balanced by its
counterpart $v(p_{\parallel},t)$. This results in very regular and rapid
oscillations with varying amplitude within the Gaussian envelope, as
explicitly shown in Figure 10(h-i). During this stage, the system exhibits a
stable oscillatory behavior, indicating a well-established equilibrium between
the processes of particle creation and annihilation. The overall behavior
observed in both $u(p_{\parallel},t)$ and $v(p_{\parallel},t)$ during this
process provides valuable insights into the complex dynamics of particle-
antiparticle pair creation and annihilation under the influence of the
electric field.

During the formation of electron-positron pairs from the quantum vacuum,
various processes can occur. Electrons are prominently produced and
annihilated simultaneously, resulting in electron acceleration and
deceleration throughout the formation of real $e^{-}e^{+}$ independent pairs
from the virtual $e^{-}e^{+}$ pairs. In this situation, polarization functions
$u(\bm{p},t)$ and its counter term $v(\bm{p},t)$ are responsible for the
acceleration and deceleration of electrons, respectively. In that sense, when
an electron Electrons are typically created in the direction of the external
field with positive momentum and are subsequently decelerated and may then, as
soon as $p_{\parallel}<0$, be annihilated again; these processes can be
understood in terms of $u(\bm{p},t)$ and $v(\bm{p},t)$ which represents the
acceleration and declaration of electrons. In quasi-momentum space, there are
many possibilities that particles can find with specific momentum $p_{0}$ with
time $t_{0}.$ Suppose particles at lower momentum level with $p_{0}-\delta
p_{0}$ in earlier time reach to momentum value $p_{0}$ by acceleration process
showed by $u(p_{\parallel},t)$ and it is also possible that particles at
higher momentum level with $p_{0}+\delta p_{0}$ followed by deceleration
process ($p_{0}-\delta p_{0}$) and finally come to momentum value $p_{0}$ see
figure 11.

Figure 11: A Schematic representation of particle occupies different momentum
states in different scenarios

In the quasi-particle longitudinal momentum spectrum at a specific time
$t_{0}$, two possible events can give rise to the quantum interference effect.
This expectation is explicitly confirmed in the middle panel of Figure 4,
where oscillations are observed in a bell-shaped profile around $t\approx
2\tau$. These oscillations in the LMS are observed for a very short duration,
during which coherence is maintained in the LMS of $u(p_{\parallel},t)$ and
$v(p_{\parallel},t)$, as evident from Figure 10(e-f). However, these
oscillations slowly fade away in the late REPP region around $t=4\tau$. The
disappearance of oscillations in the particle LMS can be understood in terms
of the loss of coherence that was maintained in the LMS of
$u(p_{\parallel},t)$ and $v(p_{\parallel},t)$. After the loss of coherence,
the LMS of $u(p_{\parallel},t)$ and $v(p_{\parallel},t)$ become identical at
$t=6\tau$ (see Figure 10). The envelope of the Gaussian with a cosine or sine
variation results in a smooth bell-shaped particle momentum distribution. This
behavior suggests a well-established equilibrium in the system, with
interference effects subsiding and a stable particle-antiparticle distribution
being achieved.

###  IV.5 Dependence on Transverse momentum

Figure 12:  LMS of created particles for different values of the transverse
momentum $p_{\perp}$ ( $p_{\perp}=0$(red), $p_{\perp}=0.4$(blue),
and$p_{\perp}=0.8$ (brown).)

In this section, we extensively investigate the effect of transverse momentum
on the particle momentum distribution function by plotting the time evolution
of the LMS for different fixed values of transverse momentum, $p_{\perp}$.
Figure 12 demonstrates that the distribution function is diminished with
increasing $p_{\perp}$ as if the extra mass has been added to the electrons
and positrons. Higher transverse momentum requires more energy for the
creation of $e^{-}e^{+}$ pairs, resulting in a decrease in the number of
produced particles at that time. Figure 12 (c) highlights an interesting
feature due to higher transverse momentum, where the oscillations are nearly
washed out at $t=3\tau$. This loss of coherence is because of less particle
produced or acceleration in comparison to lower transverse momentum at that
finite time might be responsible for this phenomenon.
Furthermore, as we previously identified some time scales relative to the
occurrence of quantum interference patterns discussed in section IV.1, we now
observe that the quantum signature depends on the transverse momentum, as
indicated in Table 3. Lower $p_{\perp}$ values exhibit oscillation patterns
that become visible earlier in time compared to higher momenta.

$p_{\perp}[m]$ |  $t_{cp}[m^{-1}]$ |  $t_{sep}[m^{-1}]$ |  $t_{dis.}[m^{-1}]$
---|---|---|---
0.00 |  22 |  30 |  50
0.25 |  24 |  32 |  53
0.50 |  31 |  38 |  63
0.75 |  39 |  47 |  71
1.00 |  53 |  60 |  83
Table 3:  Table 3: Time labelled by different transverse mode value
$p_{\perp}.$

###  IV.6 Transverse momentum spectrum

As we discussed in the previous section about the dependence of transverse
momentum,$p_{\perp}$ on the longitudinal momentum distribution function at
finite times. Understanding the Transverse Momentum Spectrum (TMS) of the
created pairs can provide valuable insights into the pair production process
and offer more information about the particles involved. The time evolution of
TMS of created quasi-particle pairs is depicted in figure 13. The shape of the
transverse momentum distribution is changed when the electric field is
switched on. At the initial time, $t=-10[m^{-1}],$ we see that the TMS shows a
Gaussian structure peak at $p_{\perp}=0.$ After $t\approx\tau/2,$ the smooth
Gaussian structure becomes distorted, and we see inconstancy in spectrum
structure either a dip at the origin with an off-axis maximum or a peak at
zero transverse momentum with small peaks up to $t\approx 2\tau$ observed.
During the transient stage and at the beginning of the REPP stage, the
spectrum shows a prominent peak at zero transverse momentum, accompanied by
weakly pronounced peaks at $p_{\perp}\approx\pm 0.67$ and $p_{\perp}\approx\pm
0.84$ as seen in the figure 13(c),(d). The width of the momentum spectra
changes with the evolution time, with the momentum distribution of quasi-
particle pairs having a width of the order of the electron mass. The half-
width of the $p_{\perp}$ distribution is determined by the field strength at
that time, which is explicitly confirmed by Figure 13. The motion of the
longitudinal and transverse directions remains coupled through the quasi-
energy $\omega(p_{\parallel},p_{\perp},t)$ during the depletion of the
accelerating field (i.e., transient and the beginning of the REPP stage),
leading to the observed substructure in the TMS. As the electric field decays
to $98.6\%$ of its maximum value, the width of the $p_{\perp}$ distribution
becomes of the order of $1$, and the substructure in TMS disappears in the
REPP stage ( see figure 13 (g)). At this stage, the TMS shows a smooth Gauss-
like distribution, with the maximum value $f(p_{\perp})$ occurring at
$p_{\perp}=0$ as shown in figure 13(h). The distribution function
$f(p_{\perp})$ can be well understood by assuming that particle creation is
exponentially suppressed with $\exp{(-\frac{m^{2}+p_{\perp}^{2}}{eE_{0}})}$.
Furthermore, in the absence of an electric field, the transverse particle
spectrum exhibits a Gaussian distribution, as shown in Figure 13(i). In this
case, the absence of quantum interference arises because, although all
particles have acquired a phase, the distribution of phase information only
varies in the direction of $p_{\parallel}$. Hence, when summing over the
phases in the direction of $p_{\perp}$, no interference pattern emerges.

Figure 13: Time evolution of quasi-particle distribution function in the
transverse momentum space $f(p_{\perp},t)$ at different times. The
longitudinal momentum is considered to be zero, and all the units are taken in
the electron mass units.The field parameters are $E_{0}=0.2$ and $\tau=10.$

##  V Conclusion

We have conducted a detailed analysis of the electron-positron pair creation
from the vacuum under the influence of a time-dependent Sauter pulse electric
field. By deriving the exact analytic solution for the mode function, we
computed the one-particle distribution function in momentum space for the
Sauter-pulse electric field. Our investigation reveals that the process of
transition from initially virtual particles to real particles occurs in three
distinct stages, which crucially depend on the longitudinal and transverse
modes in momentum space. Moreover, we have quantified the initiation of the
REPP stage, determined by the momentum value, with higher momenta resulting in
narrower oscillations in the transient region. We meticulously examined the
Longitudinal Momentum Spectrum (LMS) and Transverse Momentum Spectrum (TMS) to
understand the momentum-dependent behavior during these stages. In the LMS,
one interesting feature at the beginning of the REPP stage is that the
Longitudinal momentum distribution function exhibits an oscillating structure
as imprints of a quantum signature at the finite time where the electric field
is nearly zero, which can be understood in the Dynamical Tunneling picture.
The two-peaked structure in which the central Gaussian-peak structure has
onset oscillation, and this quantum interference pattern evolves and fades
away. Based on this observation, we identified three distinct time scales
associated with this behavior. Consequently, we concluded that these
oscillations are prominent in the LMS at the Compton time scale during the
formation of electron-positron pairs. Additionally, we investigated the impact
of different electric field strengths ($E_{0}$) on the duration of the
interference pattern’s formation and disappearance. Subsequently, we examined
whether or not the oscillation structure behavior at finite time explicitly
depends on the Keldysh parameter ($\gamma$). Remarkably, we found that for two
different configurations of parameters $(E_{0},\tau)$ under the same
$\gamma=1$, distinct behaviors were observed due to their placement in the
intermediate regime of pair production. In the multiphoton regime, for
$\gamma=2.5$, we explored the time evolution of LMS in different stages of
pair production, and the spectrum shows the splitting of a smooth uni-modal
structure into a multi-modal Gaussian structure near the REPP region, after
which it merges into a single peak Gaussian profile at the asymptotic time
limit.

We also derived approximate analytical expressions for the one-particle
distribution function $f(\bm{p},t)$ at finite times. Utilizing these
expressions, we further unveiled that the LMS structure mainly comprises three
distinct functional behaviors. One function dominantly governs the early
times, while the combination of the second and third functions leads to the
central peak structure with onset oscillation. The third term contributes to
oscillations, owing to the presence of a sinusoidal function, but its
amplitude diminishes with time, resulting in a smooth spectrum profile in late
time. Furthermore, we investigated the role of vacuum polarization,
$u(p_{\parallel},t)$, and its counterpart, $v(p_{\parallel},t)$, by plotting
their time evolution. In the REPP region, both $u(p_{\parallel},t)$ and
$v(p_{\parallel},t)$ exhibit nearly identical oscillations with the same
amplitude, and this oscillation amplitude decays for higher $p_{\parallel}$
values. This observation implies that the qualitative nature of the vacuum
polarization function is significantly influenced by the longitudinal
momentum. To elucidate this further, we plotted the LMS of
$u(p_{\parallel},t)$ and $v(p_{\parallel},t)$, showcasing two distinct
structures at the begining of the REPP stage. One structure was characterized
by a Gaussian-like structure, while the other contained a deformed oscillating
profile within the Gaussian envelope. These oscillations, exhibiting varying
amplitudes, were confined to a small window of longitudinal momentum
($-1<p_{\parallel}<1$).Over time, the regular oscillations of
$u(p_{\parallel},t)$ and $v(p_{\parallel},t)$ became balanced, as explicitly
observed in the plot of the LMS of the created particles for late time, where
the oscillating behavior was absent due to the equilibrium between
$u(p_{\parallel},t)$ and $v(p_{\parallel},t).$ As we recognize the pivotal
roles of $u(p_{\parallel},t)$ and $v(p_{\parallel},t)$ functions in
accelerating (creation of $e^{-}$) and decelerating (annihilation of $e^{-}$)
electrons, respectively, the observed oscillations in the LMS of
$f(p_{\parallel},t)$ at finite times can be attributed to the characteristic
behavior imprints of these functions, resulting from the acceleration and
deceleration of electrons in momentum representation, which manifests as the
observed oscillating structure. Next, we also discuss the influence of
transverse momentum,$p_{\perp}$ on LMS, which diminishes the value of
$f(p_{\parallel})$ and smooths out the oscillation for higher $p_{\perp}$
values. Moreover, we emphasized that these oscillations, seen in the late REPP
stage as transverse momentum increases and also impact the formation time of
pairs. Finally, we study the transverse momentum spectrum(TMS), uncovering
interesting results not previously considered by any other author. In the TMS,
fluctuating substructures change regularly in the transient region up to the
start of the REPP region, after which they disappear. The appearance of these
substructures in the smooth Gauss-like distribution is attributed to the
motion of an electron in the longitudinal direction (i.e., z-direction)
coupled to the transverse direction motion in the transient region, where
electron-positron pair dependence is evident. However, upon reaching the REPP
stage, the coupling between the longitudinal and transverse directions is not
observed, as real independent electron-positron pairs are formed where
electric fields decay out.

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