NeuroHammer: Inducing Bit-Flips in Memristive Crossbar Memories
This work was funded by the Federal Ministry of Education and Research (BMBF, Germany) in the project NEUROTEC II (Project Nos. 16ME0398K and 16ME0399).

Felix Staudigl1,
Hazem Al Indari1,
Daniel Schön3,
Dominik Sisejkovic1,

Farhad Merchant1,
Jan Moritz Joseph1
Vikas Rana2
Stephan Menzel3,
Rainer Leupers1

Institute for Communication Technologies and Embedded Systems, RWTH Aachen University, Germany

Peter Grünberg Institut (PGI-10) Forschungszentrum Juelich GmbH, Juelich, Germany

Peter Grünberg Institut (PGI-7) Forschungszentrum Juelich GmbH, Juelich, Germany

{staudigl, alindari, sisejkovic, merchantf, joseph<EMAIL_ADDRESS>
{st.menzel, schoen<EMAIL_ADDRESS>
Emerging non-volatile memory (NVM) technologies offer unique advantages in energy efficiency, latency, and features such as computing-in-memory. Consequently, emerging NVM technologies are considered an ideal substrate for computation and storage in future-generation neuromorphic platforms. These technologies need to be evaluated for fundamental reliability and security issues. In this paper, we present NeuroHammer, a security threat in ReRAM crossbars caused by thermal crosstalk between memory cells. We demonstrate that bit-flips can be deliberately induced in ReRAM devices in a crossbar by systematically writing adjacent memory cells. A simulation flow is developed to evaluate NeuroHammer and the impact of physical parameters on the effectiveness of the attack. Finally, we discuss the security implications in the context of possible attack scenarios.

ReRAM, memristor, hardware security, thermal crosstalk, reliability, neuromorphic computing

§ INTRODUCTION

Working principle of NeuroHammer.

Emerging non-volatile memory (NVM) technologies offer promising features that give them an advantage over classical RAM technologies. These advantages are density, low leakage power, and computing-in-memory (CIM) capabilities <cit.>. Specifically,  CIM capabilities help alleviate the von Neumann bottleneck by significantly reducing data movements <cit.>.

Reliability and disturbance errors in contemporary solid-state storage technologies, like dynamic random access memory (DRAM), have emerged due to a technology push resulting in high memory densities <cit.>. These disturbance errors can form security vulnerabilities like RowHammer, compromising user information through privilege escalation, as demonstrated in Google's Project Zero <cit.>, or denial of service in clouds <cit.>.

Similar disturbance errors have been reported in phase-change memory, which suffers from thermal crosstalk in technology nodes below 20 <cit.>. Cai et al. <cit.> discuss thermal crosstalk in ReRAM structures and their impact on the reliability of neuromorphic systems. These disturbance errors can cause an unintended malfunction in memory cells. The authors in <cit.> discuss the impact of the filament temperature on the switching kinetics of ReRAM cells. In this paper, we combine the reported thermal crosstalk in dense memristive crossbars with the temperature impact on the switching kinetics to define an attack that intentionally causes malfunctions.

Contributions: We introduce NeuroHammer, a security attack on emerging NVMs, which deliberately causes bit-flips. To the best of our knowledge, this paper provides the first investigation of undesired bit-flips in emerging NVMs in the context of a security attack. The major contributions are as follows. (1) A security threat based on disturbance errors in ReRAM structures through thermal crosstalk observed during the writing process of memory cells. (2) A simulation methodology for the characterization of NeuroHammer using state-of-the-art crossbar models. (3) A discussion on NeuroHammer-induced security threats based on the provided characterization.

§ BACKGROUND

RowHammer: In 2014, Kim et al. <cit.> reported the impact of disturbance errors on DRAMs. The DRAM memory cell consists of a capacitor connected to a transistor. The capacitor's charge decreases with the scaling down of the DRAM process technology. Consequently, the memory cell is more susceptible to disturbance errors based on electromagnetic inference. The RowHammer attack uses this phenomenon to deliberately flip certain bits in DRAM memories by hammering/reading particular rows. The consciously triggered bit-flips violate a fundamental concept of secure and reliable computing systems: memory isolation, which ensures strict separation of application memory to mitigate malicious changes in its internal state.

ReRAM: Redox-based resistive memories (ReRAMs) are an emerging class of non-volatile memories, which store binary information in terms of resistances, i.e., the low resistive state (LRS) and the high resistive state (HRS). ReRAMs typically consist of a simple metal/insulator/metal structure, enabling a high integration density.

Depending on the ionic defect type, one can distinguish between valence change memories (VCM) and electrochemical metallization cells (ECM) <cit.>. VCM cells rely on the motion of oxygen defects in the insulating oxide material. ECM cells are based on the migration of (typically) Ag or Cu cationic defects, which were injected from the chemically active electrode consisting of Ag or Cu.

The most studied ReRAMs are filamentary VCM cells in which oxygen defects form very small conducting filamentary regions. This enables high scalability down to a few nm<cit.> and fast switching speed down to 50 ps <cit.> at moderate switching voltages. This fast switching speed is achieved by local Joule heating, which accelerates the ion migration exponentially <cit.>.

§ MECHANICS OF NEUROHAMMER

Simulation methodology: (a) the thermal coupling in a 5x5 memristive crossbar, (b) low-level simulation setup and its boundary conditions, and (c) circuit-level simulation setup and its internal components.

Von Witzleben et al. <cit.> investigated the impact of the temperature on the switching kinetics of redox-based ReRAM cells, concluding that the switching time depends significantly on the temperature of the filament. The proposed attack uses this phenomenon to deliberately induce bit-flips in ReRAM crossbar memories. In the following, we define a pulse as a rectangular electrical pulse with a fixed amplitude of $V_\text{SET} = 1.05\text{V}$ and a given pulse length. The pulse length is defined by the duration in which the signal is active. The attack is divided into four distinct phases, as illustrated in Fig. <ref>:

§.§.§ Hammering
The NeuroHammer attack hammers a single memory cell to trigger a bit-flip. In Fig. <ref>, the red (blue) cell symbolizes the attacked (target) cell. The red cell should be initially switched to LRS to maximize the resulting current flowing through the cell. The $V/2$ scheme is used to apply constant stress to the blue cell, which equals $V_\text{SET}/2$ of the voltage pulse applied to the red cell.

§.§.§ Temperature increase
The recurring voltage pulses applied on the red cell result in a current through the device, and hence, a temporary temperature increase of the filament. As a result, the temperature of the blue cell increases due to thermal coupling. In addition, the $V/2$ scheme applies a respective voltage pulse ($V_\text{SET}/2$) to the blue cell, which simultaneously leads to a temperature increase in the filament.

§.§.§ Switching kinetics
The increased temperature of the blue cell changes the switching kinetics of the ReRAM device. Consequently, the ReRAM device is more susceptible to pulses, which have either a larger amplitude or a longer pulse length.

§.§.§ Bit-flip
At this point, the blue cell gradually switches its internal state based on the changed switching kinetics and the constantly applied $V/2$ voltage pulses. These pulses would typically not be sufficient to switch the state of the blue cell. However, considering the thermal coupling effects of dense crossbar structures, the target cell is susceptible to pulses with even lower amplitudes. Eventually, the blue cell switches its internal state and a bit-flip occurs.

§ SIMULATION METHODOLOGY

To verify the attack presented in Chapter <ref>, a simulation framework was developed based on the simulator COMSOL Multiphysics <cit.> and the circuit simulator Cadence Virtuoso <cit.>. Initially, the crossbar simulation aims to simulate the thermal crosstalk, based on which we extract thermal crosstalk coefficients, which are called alpha values. These values serve as input for the circuit simulation to enable the framework to simulate thermal coupling in dense ReRAM crossbar structure.

§.§ Crossbar Simulation

Fig. <ref> illustrates the basic structure of the simulation model, which consists of a memristive crossbar array of electrodes on a Si/SiO$_2$ substrate. To calculate the temperature of the adjacent cells which surround the target cell, the static heat transfer equation and the current continuity equation are solved for the temperature \(T\) and the potential \(\phi\), shown in Equation <ref> and <ref>.

\begin{equation}\label{eq:static_heat}
-\nabla\cdot(\kappa\nabla T) = \textbf{j}\cdot\textbf{E}
\end{equation}

\begin{equation}\label{eq:current}
\nabla\cdot\textbf{j} = -\nabla\cdot(\sigma\nabla\phi) = 0
\end{equation}

Here, \(\kappa\) denotes the thermal conductivity, \(\sigma\) the electric conductivity, \(\textbf{j}\) the local current density, and \(\textbf{E}\) the electric field. The electric conductivity and thus also the thermal conductivity (see Wiedemann–Franz law) of the filament is adjusted so that a certain current flows through the device. All other surfaces are thermally and electrically insulated.

To determine the alpha values of a geometry with specific materials for a temperature prediction of the adjacent cells, a variation of the dissipated power in the selected cell is required. Through a voltage sweep of \(V_\text{SET}\), a temperature matrix which contains the temperature of each cell can be extracted from every simulation (see Fig. <ref>). The thermal resistance \(R_\text{th}\) of the selected cell can then be determined as a fit parameter of a linear regression of the respective temperature \(T(P_\text{LRS})\) and dissipated power \(P_\text{LRS}=V_\text{SET}\cdot I\) (Equation <ref>). With further linear regression for every neighboring cell, the alpha values can be determined (Equation <ref>).

\begin{equation}\label{eq:linear_reg_rth}
T(P_\text{LRS}) = T_0 + R_\text{th} \cdot P_\text{LRS}
\end{equation}

\begin{equation}\label{eq:linear_reg_alpha}
T_{ij}(P_\text{LRS}) = T_0 + R_{th} \cdot P_\text{LRS} \cdot \alpha_{ij}
\end{equation}

Here, \(T_{ij}(P_\text{LRS})\) denotes the temperature of the cell \(i,j\) depending on the dissipated power \(P_\text{LRS}\) and \(\alpha_{ij}\), the alpha value of the cell \(i,j\).

§.§ Circuit Simulation

The circuit-level simulation and its internal modules are illustrated in Fig. <ref>. The framework consists of three major parts: memristive crossbar, memory controller, and the crosstalk hub. The platform can be parameterized based on configuration files and the standard graphical user interface (GUI) of the Cadence Virtuoso tool. The remainder of this section describes the modules of the framework in detail.

Memory controller: In general, crossbar structures are interfaced over their rows (word lines) and columns (bit lines). The memory controller is responsible for generating and driving the respective pulse for a certain input line of the crossbar. The stimuli file stores the explicit characteristics, i.e. pulse length, duty cycle, and amplitude of each input pulse, while the init file holds the initial state of every ReRAM cell.

Crosstalk hub: The hub calculates the temperature increase of a ReRAM cell due to thermal coupling with all surrounding cells. The temperature is calculated based on the alpha values extracted from the crossbar simulation (Section <ref>) and the filament temperatures of the adjacent cells:

\begin{equation}
T_\text{in}(\bm{\alpha},\textbf{T}) = \sum_{\substack{
0<i<m \\
\alpha_{ij} T_{ij,\text{out}},
\end{equation}

where \(i,j \in \mathbb{N}\) indicate the row and column of the crossbar excluding the attacked cell itself, \(m,n \in \mathbb{N}\) indicate the number of rows and columns, and \(T_\text{in}\) the additional temperature based on the thermal crosstalk.

Memristive crossbar:
The passive 5x5 memristive crossbar array represents the central part of the simulation framework in which the instantiated memristive devices connect the bit lines to the respective word lines. The JART VCM v1b model employed in this study was developed for filamentary switching VCM cells and fitted to a nanocrossbar Pt/HfO$_2$/TiO$_\text{x}$/Ti device <cit.>. The dissipated power $P_\text{d}$ in the cell increases the local temperature $T$ according to

\begin{equation}
T = R_\text{th,eff} \cdot P_d + T_0,
\end{equation}

where $T_0$ is the ambient temperature and $R_\text{th,eff}$ is the effective thermal resistance (in K/W), describing the heat dissipation to the immediate cell surrounding and the thermal properties of the materials. The complete equation system and the used parameters can be found in <cit.>. It should be noted that the "deterministic" model version is used here.

The original model was adjusted to enable the memristive model to exchange parameters with the simulation framework. Thus, we introduced two interface variables to communicate the temperature values to the crosstalk hub, and receive the additional temperature generated from the adjacent cells.

§ RESULTS

[row sep=
,col sep=   ]
time      v1

10       5530

15       3281

20       2466

30       1564

45      1064

50       943

60       797

75       629

100     472

ylabel=# pulses to trigger a bit-flip,
xlabel=Pulse length in ,
ylabel style=at=(-0.004,0.5),anchor=north,
ytick=0, 1, 10, 100, 1000, 10000,
xtick=10, 20, 30, 40, 50, 60, 70, 80, 90, 100,
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xlabel style =font=,
tick label style=font=,
[draw=cg5,mark=*] table[x=time,y=v1] ;

[row sep=
,col sep=   ]
time     v1     v2     v3

10     1594    	1062    	797

50     6946    	4629    	3474

90     31294    	20849    	15648

ylabel=# pulses to trigger a bit-flip,
xlabel=Electrode spacing,
ylabel style=at=(-0.01,0.5),anchor=north,
legend style=at=(0.5,0.96), font=,
anchor=north,legend cell align=left,
legend image post style=scale=1.4,
enlarge x limits=abs=1.1cm,
ymajorgrids = true,
bar width=.5cm,
ymode = log,
width=9.5cm, height=6cm,
ytick=0, 1, 10, 100, 1000, 10000, 100000,
xtick align=inside,
xticklabel style=align=center,
xticklabels =
ylabel style =font=,
xlabel style =font=,
tick label style=font=,
legend entries=50, 75 , 100,
legend columns=6,
[draw=black,fill=cg2] table[x=time,y=v1] ;
[draw=black,fill=cg3] table[x=time,y=v2] ;
[draw=black,fill=cg4] table[x=time,y=v3] ;

[row sep=
,col sep=   ]
temp     v1     v2     v3

0        33030   	10735    	6352

25     7139    	2311    	1367

50     1818    	589	    349

75     539     175     104

100     184     60     35

ylabel=# pulses to trigger a bit-flip,
xlabel=Ambient temperature,
ylabel style=at=(-0.004,0.5),anchor=north,
legend style=at=(0.5,0.96), font=,
anchor=north,legend cell align=left,
legend image post style=scale=1.4,
enlarge x limits=abs=1.2cm,
ymajorgrids = true,
bar width=.5cm,

ymode = log,
width=15cm, height=6cm,

ytick=0, 1, 10, 100, 1000, 10000, 100000,
xtick align=inside,
xticklabel style=align=center,
xticklabels =
ylabel style =font=,
xlabel style =font=,
tick label style=font=,
legend entries=10, 30 , 50,
legend columns=6,
[draw=black,fill=cg2] table[x=temp,y=v1] ;
[draw=black,fill=cg3] table[x=temp,y=v2] ;
[draw=black,fill=cg4] table[x=temp,y=v3] ;

Simulation results of the circuit-level simulation framework: (a) impact of the pulse length, (b) impact of the electrode spacing, (c) impact of the ambient temperature, (d) impact of different attack patterns, and (e-h) overview of attack patterns.

In this section, we verify the proposed NeuroHammer attack and demonstrate the flexibility of our simulation framework. Each experiment uses a simple attack pattern in which we attack the cell in the middle of the crossbar.

Pulse length: The simulation uses a crossbar with an electrode spacing of 50, assuming an ambient temperature of 300. The controllers select the attacked cell by applying $V$ and $GND$. All remaining inputs are supplied with $V/2$ to minimize the sneak-path currents. Hence, the bit-flip can only occur in the blue cells based on the voltage drop of $V/2$ compared to the remaining cells, which experience no voltage drop. The results show that the number of required pulses to trigger the error decreases with the pulse length (Fig. <ref>).

Electrode spacing: The second experiment concerns the electrode spacing of the crossbar memory structure. The analysis uses an ambient temperature of 300. The spacing of the electrode is defined by the distance between the electrodes of two adjacent cells. Fig. <ref> illustrates the simulation results of the electrode spacing ranging from 10 to 90, which indicate that the closer the cells are placed, the more vulnerable they are in terms of disturbance errors. Accordingly, we assume that disturbance errors become a serious problem for dense crossbar memory structures, the more the technology node advances.

Ambient temperature: Next, we investigated the effects of the ambient temperature on the occurrence of disturbance errors. Fig. <ref> shows the results that use an electrode spacing of 50. The results indicate a strong impact of the ambient temperature on the number of pulses required to trigger a bit-flip.

§ SECURITY IMPLICATIONS

Several RowHammer-inspired attack scenarios have been presented to threaten both embedded systems and desktop computers. This section illustrates possible attack scenarios in which NeuroHammer can be exploited. Both attacks utilize disturbance errors in dense memory structures, which are deliberately caused by repeatedly hammering certain memory cells. Based on the similarities between both attacks, we briefly describe a common attack scenario using RowHammer to discuss, ultimately, the reuse of attack scenarios for NeuroHammer.

Seaborn et al. <cit.> demonstrated gaining kernel privileges through the DRAM RowHammer bug on a typical laptop. Two requirements enable the attack. First, the adversary has to find the correct address mapping between the physical and virtual memory space to hammer the correct cells. Second, the cache of the system would typically mitigate frequent activation of the same memory location. The authors used the flush instruction to constantly flush the cache, enabling the adversary to frequently activate the same memory location. Ultimately, the attacker gains kernel access to the whole physical memory through a normal process execution on an x86-64 machine by hammering a page table entry to point to an attacker-owned page table.

Considering its advantages, ReRAM has the potential to replace DRAM in modern computing systems. Thus, the mentioned attack scenario is transferable to NeuroHammer. Hereby, the access patterns must be adjusted to match the physical characteristics of the thermal crosstalk in memristive crossbar memories. In general, any attack proven to work with RowHammer could additionally work with NeuroHammer. However, the application domain of DRAM and ReRAM does not completely overlap. Hence, the proposed attack poses a supplementary threat to emerging neuromorphic-based systems, such as neuromorphic machine-learning accelerators.

§ CONCLUSION

This work introduced NeuroHammer, a security threat to emerging non-volatile memories based on thermal crosstalk in dense crossbar structures. We developed a simulation framework to investigate the impact of physical parameters on the effectiveness of the attack. The results show that the NeuroHammer attack deliberately induces bit-flips in memristive crossbar structures. Finally, we discussed the security implications of NeuroHammer by analyzing attack scenarios of the related RowHammer attack. In future work, we plan to verify the existence of the NeuroHammer attack on physical crossbars and explore countermeasures to mitigate the security threat.

With this work, we demonstrated that the fundamental security implication of disturbance errors still persists even in emerging neuromorphic technologies. As neuromorphic hardware has the potential to become a key component of modern computing systems, evaluating its basic security aspects is essential to providing a stepping stone to building secure next-generation devices.


# Enhanced dynamically assisted pair production in spatial inhomogeneous
electric fields with the frequency chirping

Lie-Juan Li  Key Laboratory of Beam Technology of the Ministry of Education,
and College of Nuclear Science and Technology, Beijing Normal University,
Beijing 100875, China  Melike Mohamedsedik  Key Laboratory of Beam Technology
of the Ministry of Education, and College of Nuclear Science and Technology,
Beijing Normal University, Beijing 100875, China  Bai-Song Xie
<EMAIL_ADDRESS>Key Laboratory of Beam Technology of the Ministry of
Education, and College of Nuclear Science and Technology, Beijing Normal
University, Beijing 100875, China  Beijing Radiation Center, Beijing 100875,
China

###### Abstract

Enhanced dynamically assisted pair production in frequency chirped spatially
inhomogeneous electric fields is studied using the Dirac-Heisenberg-Wigner
formalism. The effects of the chirp parameter on the reduced momentum spectrum
and the total particle yield are investigated in detail for the single field
or the dynamically assisted combined fields at various spatial scales. For the
single high frequency field and the combined fields, the interference effect
of momentum spectrum becomes more and more remarkable with the increasing
chirp, meanwhile, in the chirped dynamically assisted case, the reduced total
yield is enhanced significantly when chirping of two fields are present.
Specifically, in the assisted case, the reduced particle number is increased
by more than one order of magnitude at the small spatial scale, while it is
enhanced about two times at the large spatial scale. We also obtain some
optimal chirp and spatial scale parameters for the total yield and the
enhancement factor for various scenarios.

###### pacs:

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

##  I Introduction

Dirac’s equation predicted the existence of positrons Dirac:1928ej ;
Anderson:1933mb  and implied the virtual particles fluctuation inherent in the
quantum electrodynamics (QED) vacuum which further unveiled rich and novel
physical phenomena Dunne:2012vv . Sauter noticed that the QED vacuum could
produce electron-positron ($e^{+}e^{-}$) pairs through tunneling at certain
field strength scales Sauter:1931zz  and Schwinger calculated the pair
production rate $\Gamma\sim\exp(-\pi E_{cr}/E)$ for a constant electric field
with proper-time treatment Schwinger:1951nm , where
$E_{cr}=m^{2}c^{3}/e\hbar\approx 1.3\times 10^{16}\rm{V/cm}$ is the Schwinger
critical field strength corresponding to the laser intensity of $I_{cr}\approx
4.3\times 10^{29}\rm W/\rm cm^{2}$. Since then, pair production under intense
external fields, also known as the Sauter-Schwinger effect, has become a
challenge for both theorists and experimentalists because of its
nonperturbative tunneling nature and the extremely large field strength
required for the observation Gelis:2015kya ; DiPiazza:2011tq .

Fortunately, the development of ultraintense laser technology Heinzl:2008an ;
Marklund:2008gj ; Pike:2014wha  might make the experimental observation of
pair production possible in the near future. Especially the application of
chirped pulse amplification (CPA) technique has greatly improved the laser
intensity Strickland:1985gxr , for example, the current laser intensity has
reached about $10^{22}\rm W/\rm cm^{2}$ 2008Ultra , and the expected intensity
to be $10^{25}\sim 10^{26}\rm W/\rm cm^{2}$ for many planned facilities
Ringwald:2001ib ; Dunne:2008kc . On the other hand, the X-ray free electron
laser (XFEL) system may achieve subcritical field strength as large as
$E\approx 0.01-0.1E_{cr}$ Ringwald:2001ib  so that the vacuum $e^{+}e^{-}$
pair production has become a hot research topic in past decade. So far,
several methods have been developed to investigate pair production in external
fields, such as the worldline instanton technique Gies:2005bz ; Dunne:2006st ;
Schneider:2014mla , quantum Vlasov equation solution method Kluger:1991ib ;
Alkofer:2001ik ; Abdukerim:2017hkh , computational quantum field theory
P2005Creation ; PhysRevLett.111.183204 ; Wang:2019oyk , Dirac-Heisenberg-
Wigner (DHW) formalism Hebenstreit:2011wk ; Kohlfurst:2015zxi ; Xie:re2017 ;
Kohlfurst:2017hbd ; Ababekri:2019qiw ; Kohlf2020Effect , and so on.

Theoretically, many different field configuration, e.g., the temporal and
spatial pulse shaping, have been investigated to reduce the pair creation
threshold. The research on pair production in short laser pulses with subcycle
structure discovered that the momentum spectrum is extremely sensitive to the
corresponding external field parameters Hebenstreit:2009km . Particle self-
bunching effect is identified in pair creation under the spatially
inhomogeneous pulse field Hebenstreit:2011wk . The ponderomotive effect is
found important in the multiphoton pair production process for temporal
oscillating and spatially inhomogeneous electric field Kohlfurst:2017hbd . As
an important step to increase the pair production, Schützhold et al.
Schutzhold:2008pz  proposed dynamically assisted Schwinger mechanism, which is
the combination of a low frequency strong field with a high frequency weak
field. They pointed out that the pair creation rate is enhanced extremely by
the combined field effect. Also, frequency chirp effects are widely studied
because they affect the momentum spectra and the total particle number under
the time dependent field Dumlu:2010vv ; Olugh:2018seh  and spatial
inhomogeneous oscillating electric field Ababekri:2019qiw . Furthermore, many
different combinational field have been studied to enhance the $e^{+}e^{-}$
pair production Orthaber:2011cm ; Ababekri:2019dkl ; Olugh:2019nej .
Especially, the enhanced pair production in time dependent one- and two-color
laser fields by chirping have also been explored Abdukerim:2017hkh ;
Gong:2019sbw .

In this paper, we use the DHW formalism to investigate pair production in
frequency chirped dynamically assisted electric field with spatial
inhomogeneity. We research the reduced momentum spectrum, the reduced total
yield of the created pairs for either low or high frequency one-color field
with different chirp values. Meanwhile, we study chirped dynamically assisted
pair production for the two-color combined fields. Many different momentum
spectrum and pair production number as well as the enhanced factor are
presented for various chirping cases. Throughout this paper natural units
$\hbar=c=1$ are used and all quantities are expressed in terms of the electron
mass $m$.

The paper is organized as follows. In Sec. II, we introduce the background
field that is considered in this work, and review the DHW formalism briefly.
In Sec. III, we show the numerical results for the single electric field with
different chirp values. In Sec. IV, we present the numerical results for
chirped dynamically assisted two-color combined fields and semiclassical
discussion about the results. In Sec. V, we give a brief conclusion and
outlook.

##  II Background field and theoretical formalism

###  II.1 Model of the background electric field

We investigate $e^{+}e^{-}$ pair production for either low or high frequency
one-color field and two-color dynamically assisted combinational fields with
various chirp parameters. Our idealized model is considered as the combination
of a low frequency and strong field $E_{1s}\left(x,t\right)$ with a high
frequency and weak field $E_{2w}\left(x,t\right)$. The form is given as

$\displaystyle E\left(x,t\right)$
$\displaystyle=E_{1s}\left(x,t\right)+E_{2w}\left(x,t\right)$ (1)
$\displaystyle=E_{1s0}\exp\left(-\frac{x^{2}}{2\lambda^{2}}\right)\cos^{4}\left(\frac{t}{\tau}\right)\cos(b_{1}t^{2}+\omega_{1}t)$
$\displaystyle+E_{2w0}\exp\left(-\frac{x^{2}}{2\lambda^{2}}\right)\cos^{4}\left(\frac{t}{\tau}\right)\cos(b_{2}t^{2}+\omega_{2}t),$

where $E_{1s0,2w0}$ denote the strength of the electric field, $b_{1,2}$ are
the corresponding frequency chirp, $\omega_{1}$ and $\omega_{2}$ are the
oscillating frequency of the strong and weak fields, $\tau$ and $\lambda$ are
the characteristic time and spatial length scale, respectively. Note that the
temporal pulse shape function $\cos^{4}(t/\tau)$ is turned on at
$t=-\pi\tau/2$ and turned off at $t=\pi\tau/2$. This pulse duration condition
is applied to all the cases of studied fields in this paper.

In our numerical calculation, a set of parameters of the electric field Eq.
(1) are fixed as

$\displaystyle E_{1s0}=0.3E_{cr},E_{2w0}=0.075E_{cr}=0.25E_{1s0},$ (2)
$\displaystyle\omega_{1}=0.1m,\omega_{2}=0.7m=7\omega_{1},\tau=50m^{-1},$

where $E_{cr}$ is the critical field strength and $m$ the electron mass. Note
that the Keldysh parameter is important for studying the pair creation from
vacuum and defined as $\gamma=m\omega/eE$ Keldysh:1965ojf ; Brezin:1970xf ,
where $E$ and $\omega$ represent the strength and frequency of the background
field, and the Schwinger (tunneling) effect and multiphoton absorption can be
considered by $\gamma\ll 1$ and $\gamma\gg 1$. For the chosen parameters in
this paper, the Keldysh parameter $\gamma_{1s}=m\omega_{1}/eE_{1s0}=1/3$ of
the single field $E_{1s}\left(x,t\right)$ means that the pair production
process is dominated by tunneling effect whereas
$\gamma_{2w}=m\omega_{2}/eE_{2w0}=28/3$ of the field $E_{2w}\left(x,t\right)$
means that the process is dominated by multiphoton absorption. It must be
noted that the combined Keldysh parameter
$\gamma_{C}=m\omega_{2}/eE_{1s0}$=7/3 in this study is the combination of the
strong field strength $E_{1s0}$ and the fast pulse frequency $\omega_{2}$
Schutzhold:2008pz , i.e., the process of pair creation involve both
multiphoton absorption and tunneling effect. We also use the definition of the
time-dependent effective frequency Ababekri:2019qiw
$\omega_{\text{eff}}(t)=\omega+bt$, with $b$ denoting the frequency chirp
parameter. Since that the pair creation process occurs mostly at the dominate
time interval $-\tau\leq t\leq\tau$, so the chirp parameter in the ”normal
regime”, i.e., $\max{b|t|}\sim\omega$ in this interval must be met naturally.
Therefore, in this paper, the values of linear chirp $b$ can be expressed as
$b=\alpha\omega/\tau$ with $0<\alpha<1$. Without losing the generality, we
choose the regime of $0\leq\alpha\leq 0.9$ so that the studied maximum chirp
value is chosen as $b_{1}=0.9\omega_{1}/\tau=0.0018m^{2}$ for
$E_{1s}\left(x,t\right)$. Similarly for $E_{2w}\left(x,t\right)$, the maximum
chirp value is chosen as $b_{2}=0.9\omega_{2}/\tau=0.0126m^{2}$. Furthermore,
for the simplicity of study and convenience of results comparison, we limit
the selection rule for chirping as that the ratio of frequency chirp value of
two fields keeps the same ratio of the original center frequency of two
fields, i.e., $b_{1}/b_{2}=\omega_{1}/\omega_{2}$.

Note that the electric field pulse shape and also the chosen field parameters
are different from that of our previous research Ababekri:2019qiw . Beside
obtaining the rich momentum spectrum, total yield of pair production and
enhancement factor in different fields and chirping cases, moreover, we have
explored some optimal chirp parameters and spatial scales for $e^{+}e^{-}$
pair production via extensive numerical simulation.

###  II.2 Theoretical formalism: DHW formalism

DHW formalism, a relativistic phase-space quantum kinetic approach
Vasak:1987um ; Phase-space str , which has been widely used to explore pair
production within arbitrary electromagnetic Hebenstreit:2011wk ; Olugh:2018seh
; Ababekri:2019dkl ; Kohlf2020Effect . As the specific derivation of DHW has
been given in previous works Phase-space str ; Kohlfurst:2015zxi  and the
detailed introduction is not the purpose of this work, we only present the
basic ideas and the essential points of this method.

We begin with the gauge-covariant density operator of the system based on
Phase-space str

$\hat{\mathcal{C}}_{\alpha\beta}\left(r,s\right)=\mathcal{U}\left(A,r,s\right)\
\left[\bar{\psi}_{\beta}\left(r-s/2\right),\psi_{\alpha}\left(r+s/2\right)\right],$
(3)

where $r$ represents the center-of-mass coordinate and $s$ the relative
coordinate. The Wilson line factor $\mathcal{U}\left(A,r,s\right)$ before the
commutator

$\mathcal{U}\left(A,r,s\right)=\exp\left(\mathrm{i}\ e\
s\int_{-1/2}^{1/2}d\xi\ A\left(r+\xi s\right)\right),$ (4)

is a factor which can guarantee the gauge invariance. Obviously, it is related
to the elementary charge $e$ and the background gauge field $A$. The covariant
Wigner operator is obtained via the Fourier transform of the covariant density
operator from the $s$-space to the $p$-space

$\hat{\mathcal{W}}_{\alpha\beta}\left(r,p\right)=\frac{1}{2}\int d^{4}s\
\mathrm{e}^{\mathrm{i}ps}\ \hat{\mathcal{C}}_{\alpha\beta}\left(r,s\right).$
(5)

In order to make the equation of motion, expressed in terms of Wigner
operator, reasonable for calculation, the background field is treated as that
in mean-field (Hartree) approximation Kohlf2020Effect , i.e.,

$F^{\mu\nu}\left({r}\right)\approx\langle\hat{F}^{\mu\nu}\left({r}\right)\rangle.$
(6)

The Hartree approximation becomes apparent when we consider the vacuum
expectation value of the covariant Wigner operator to get the Wigner function

$\mathbbm{W}\left(r,p\right)=\langle\Phi|\hat{\mathcal{W}}\left(r,p\right)|\Phi\rangle.$
(7)

As a result, the electromagnetic field factors out

$\langle\Phi|F_{\mu\nu}\
\hat{\mathcal{C}}|\Phi\rangle=F_{\mu\nu}\langle\Phi|\hat{\mathcal{C}}|\Phi\rangle\,.$
(8)

Wigner function is in the Dirac algebra, it can be expanded in terms of 16
covariant Wigner coefficients

$\mathbbm{W}=\frac{1}{4}\left(\mathbbm{1}\mathbbm{S}+\textrm{i}\gamma_{5}\mathbbm{P}+\gamma^{\mu}\mathbbm{V}_{\mu}+\gamma^{\mu}\gamma_{5}\mathbbm{A}_{\mu}+\sigma^{\mu\nu}\mathbbm{T}_{\mu\nu}\right)\,.$
(9)

Combined with the above introduction, the equation of motion in terms of
covariant Wigner function can be obtained. However, the numerical solving of
the equation of motion from covariant Wigner function at all spacetime points
is hardly task to perform, therefore, a convenient and powerful treatment is
to obtain the equal-time Wigner function

$\displaystyle\mathbbm{w}\left(\mathbf{x},\mathbf{p},t\right)=\int\frac{dp_{0}}{2\pi}\
\mathbbm{W}\left(r,p\right).$ (10)

When investigating the $1+1$ dimensional spatial inhomogeneous electric field
given in Eq.(1) one finds only $\mathbbm{S}$, $\mathbbm{P}$, and
$\mathbbm{V}_{\mu}(\mu=0,1)$ are nonvanishing, and the corresponding physical
quantities are written in lowercase in the equal-time case. Hence, the
equations of motion can be reduced to $4$ equations Hebenstreit:2011wk ;
Kohlfurst:2015zxi

$\displaystyle D_{t}\mathbbm{s}-2p_{x}\mathbbm{p}=0,$ (11) $\displaystyle
D_{t}\mathbbm{v}_{0}+\partial_{x}\mathbbm{v}_{1}=0,$ (12) $\displaystyle
D_{t}\mathbbm{v}_{1}+\partial_{x}\mathbbm{v}_{0}=-2m\mathbbm{p},$ (13)
$\displaystyle D_{t}\mathbbm{p}+2p_{x}\mathbbm{s}=2m\mathbbm{v}_{1},$ (14)

with the differential operator

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

Corresponding vacuum initial conditions are given Kohlfurst:2015zxi  by

${\mathbbm{s}}_{vac}=-\frac{2m}{\omega}\,,\quad{\mathbbm{v}}_{vac}=-\frac{2{p_{x}}}{\omega}\,,$
(16)

where $\omega=\sqrt{p_{x}^{2}+m^{2}}$ is the energy of a particle. The
particle number density is given Hebenstreit:2011wk ; Kohlfurst:2017hbd

$n\left(x,p_{x},t\right)=\frac{m\mathbbm{s}^{v}\left(x,p_{x},t\right)+p_{x}\mathbbm{v}^{v}\left(x,p_{x},t\right)}{\omega\left(p_{x}\right)},$
(17)

in which
$\mathbbm{s}^{v}\left(x,p_{x},t\right)=\mathbbm{s}-{\mathbbm{s}}_{vac}$ and
$\mathbbm{v}^{v}\left(x,p_{x},t\right)=\mathbbm{v}-{\mathbbm{v}}_{vac}$ are
the modified Wigner components. The momentum space particles number density
could be obtained from $n\left(x,p_{x},t\right)$ by integrating out $x$

$n\left(p_{x},t\right)=\int
dx\,\frac{\mathbbm{s}^{v}\left(x,p_{x},t\right)+p_{x}\,\mathbbm{v}_{1}^{v}\left(x,p_{x},t\right)}{\omega\left(p_{x}\right)}.$
(18)

Accordingly, the total yield of created particles can be written

$N\left(t\right)=\int dxdp_{x}n\left(x,p_{x},t\right).$ (19)

We will investigate the reduced quantities including the reduced particle
number density
$\bar{n}\left(p_{x},t\right)\equiv\frac{n\left(p_{x},t\right)}{\lambda}$ and
reduced total yield of the created particles
$\bar{N}\left(t\rightarrow\infty\right)\equiv\frac{N\left(t\rightarrow\infty\right)}{\lambda}$
to obtain the effect of nontrivial spatial scale $\lambda$.

##  III Numerical results for the single field

In this section, we consider the influence of chirp values on the pair
production for the single field with respect to either
$E_{1s}\left(x,t\right)$ or $E_{2w}\left(x,t\right)$.

###  III.1 Single field $E_{1s}\left(x,t\right)$

Firstly, we investigate the pair production in the one-color strong field
$E_{1s}\left(x,t\right)$, i.e., the Sauter-Schwinger effect is dominant. In
this case, the individual field reads

$\displaystyle E_{1s}\left(x,t\right)$
$\displaystyle=E_{1s0}\exp\left(-\frac{x^{2}}{2\lambda^{2}}\right)\cos^{4}\left(\frac{t}{\tau}\right)\cos(b_{1}t^{2}+\omega_{1}t).$
(20)

The meaning of the corresponding parameters has been introduced in detail in
section II.1. The fixed parameters for this field are $E_{1s0}=0.3E_{cr}$,
$\omega_{1}=0.1m$, and $\tau=50m^{-1}$. The momentum spectrum for different
spatial scales with various chirp values of $b_{1}$ is given in Fig. 1.

Figure 1: (color online). Reduced momentum spectrum for different spatial
scales ($\lambda=1000m^{-1},10m^{-1},2.5m^{-1}$) for one-color strong laser
field $E_{1s}(x,t)$ with various chirp values of $b_{1}[\omega_{1}/\tau]$ as
$0$, $0.1$, $0.5$ and $0.9$, i.e., $b_{1}[m^{2}]$ as $0$, $0.0002$, $0.001$
and $0.0018$, from (a) to (d) respectively. The field parameters are
$E_{1s0}=0.3E_{cr}$, $\omega_{1}=0.1m$ and $\tau=50m^{-1}$. To eyes guidance,
we amplify the value by $10$ times when $\lambda=2.5m^{-1}$.

As shown in Fig. 1(a), i.e., for $E_{1s}\left(x,t\right)$ with vanishing chirp
$b_{1}=0$, at the quasihomogeneous limit $\lambda=1000m^{-1}$, the center of
the reduced momentum spectrum does not corresponding to $p=0$ which is similar
to the homogeneous case in Fig. $2$ of the Ref. Hebenstreit:2009km . This is
because the vector potential value $A(t)\neq 0$ when the external field is
turned off, so that $p_{x}=q-eA(t)$ has a nonzero value
$-A(t\rightarrow\infty)$ corresponding to the momentum peak at $q=0$. In fact
when $b_{1}=0$, and we neglect the spatial variation of field when $\lambda$
is large, then
$A(t)=-\int_{0}^{t}E_{1s0}\cos^{4}\left(t/\tau\right)\cos(\omega_{1}t)dt=-(I_{1}+I_{2}+I_{3})$
can be obtained, where $I_{1}=3/8\int_{0}^{t}\cos(5t/\tau)dt$,
$I_{2}=1/2\int_{0}^{t}\cos(2t/\tau)\cos(5t/\tau)dt$ and
$I_{3}=1/8\int_{0}^{t}\cos(4t/\tau)\cos(5t/\tau)dt$. For $E_{1s0}=0.3E_{cr}$,
$\omega_{1}=0.1m$ and $\tau=50m^{-1}$, we have $A(t)\approx-0.381$ when the
external field is turned off at $t=\pi\tau/2$, and the kinetic momentum
$p_{x}\approx 0.381$. This is approximately consistent with the numerical
result $p_{x}\approx 0.366$ when $\lambda=1000m^{-1}$ shown in Fig. 1(a). In
the chirping cases with different inhomogenous field scale $\lambda$ in Fig.
1, the reason of the momentum peak shifting from zero is the same although the
momentum values of peak position are different because the
$-A(t\rightarrow\infty)$ depend strongly on the $b$ and $\lambda$. For
example, when the spatial scale decreases to $\lambda=10m^{-1}$, momentum
spectrum shifts to the larger momentum values due to the effect of relatively
strong spatial inhomogeneity of the electric field Ababekri:2019qiw . This can
be understood qualitatively as that the electric field strength decreases
rapidly when the spatial scale $\lambda$ is small makes the created particles
with high momenta easier to escape the regime of field existence so that they
are hardly affected by the field. By comparing the results shown in Fig. 1(b)
and (c), we can see that momentum spectrum oscillates significantly with the
increase of chirp in the quasihomogeneous case. The phenomenon could be
understood as the interference effect of particles created by opposite signed
large peaks of the temporal field Dumlu:2010vv . It can be seen from Fig. 1(d)
that the momentum spectrum oscillation is stronger as the chirp $b_{1}$
increases to a large one because of the enhanced interference effect between
created particles. In particular this oscillation behavior is very remarkable
for quasihomogenous case when spatial scale $\lambda=1000m^{-1}$.

On the other hand, by comparing the particle number density of the small
spatial scale, e.g. $\lambda=2.5m^{-1}$, with that of the large spatial scale,
it decreases notably when $b$ is fixed. This is because that the work done by
the electric field is smaller when spatial scale is small so that the
particles created by the electric field are also smaller. Moreover, one sees
that when the chirp is applied, the maximum peak value of the momentum spectra
does not increase apparently in all cases. It indicates that the chirp
parameter we choose has a little effect on the peak value of momentum spectrum
in the case of the strong field $E_{1s}\left(x,t\right)$. There are two
reasons for this result. On the one hand, the contribution of strong fields to
the particles creation is dominated by the field intensity rather than the
frequency, which is usually exponent suppression in subcritical field
strength. On the other hand, in the case of low frequency electric field
studied here, even if in the presence of frequency chirping, the frequency
spectrum is only extended to two or three times of the original center
frequency by the Fourier transform of the electric field Eq. (20). Therefore,
the possible dynamically assisted mechanism is suppressed strongly. Because we
know that the dynamically assisted mechanism is expected to play a key role
only when the chirping leads to about one-order magnitude increasing for
frequency component compare to the original frequency.

In order to understand the momentum spectra behavior, we shall make some
discussions on our numerical results qualitatively by using Wentzel-Kramers-
Brillouin (WKB) method Dumlu:2010ua  in Sec. IV.5 in the following. The
turning points are presented in Fig.15 and Fig.16 for typical fields we
studied in this paper. Note that the plotting of turning points in case of low
frequency strong field studied in this subsection is given in Fig.15.

###  III.2 Single field $E_{2w}\left(x,t\right)$

In this subsection, we investigate the momentum spectrum and total yield in
one-color weak field $E_{2w}\left(x,t\right)$ with various chirp values of
$b_{2}$. In this case, the $n$-photon pair production is dominant and the
model of the external electric field is

$\displaystyle E\left(x,t\right)$
$\displaystyle=E_{2w0}\exp\left(-\frac{x^{2}}{2\lambda^{2}}\right)\cos^{4}\left(\frac{t}{\tau}\right)\cos(b_{2}t^{2}+\omega_{2}t),$
(21)

where the calculation parameters are chosen as $E_{2w0}=0.075E_{cr}$,
$\omega_{2}=0.7m$ and $\tau=50m^{-1}$.

In order to guarantee the reasonability of the chosen parameters, we reproduce
the results of the Fig.3 in Ref. Kohlfurst:2017hbd  as shown in Fig. 2(a),
where the parameters are chosen as field strength $\epsilon=0.5$, the envelop
pulse length $\tau=100m^{-1}$ and frequency $\omega=0.7m$. The reduced
particle momentum spectrum for one-color weak laser $E_{2w}\left(x,t\right)$
with vanishing chirp $b_{2}=0$ for the different spatial scales in our studied
case is shown in Fig. 2(b). Correspondingly, the strength of our field
$E_{2w0}$ =$\epsilon_{2}E_{cr}$ with $\epsilon_{2}=0.075$ that is appear in
the Fig. 2(b).

Figure 2: (color online). Comparison of reduced momentum spectrum for
different spatial scales between the results in Ref. Kohlfurst:2017hbd  and
our results. The left one is the Fig. 3 in Ref. Kohlfurst:2017hbd  with field
parameters $\epsilon=0.5$, $\tau=100m^{-1}$ and $\omega=0.7m$. The right one
is our results for one-color weak laser $E_{2w}\left(x,t\right)$ without
chirping with field parameters $E_{2w0}=0.075E_{cr}$, $\omega_{2}=0.7m$ and
$\tau=50m^{-1}$.

We can see that there is a difference of $5$ orders of magnitude between the
peak value of $\bar{n}$ in Fig. 2(a) and that in Fig. 2(b). Estimating the
value through the relationship between the particle number density
$\bar{n}\left(p_{x},t\right)$ of created particles and the strength
$\epsilon_{2}$ of $E_{2w}\left(x,t\right)$. The expression is
$\bar{n}\propto\epsilon_{2}^{2N_{p}}$, where $N_{p}=3$ denotes the number of
photon during the process of multipohton absorption. Thus, we get
$(\epsilon_{2}/\epsilon)^{6}=(0.075/0.5)^{6}\approx 1.139\times 10^{-5}$,
which is consistent with numerical results of our check example in Fig. 2
comparable to Fig.3 in Ref. Kohlfurst:2017hbd . By the way, this estimation
can be as a test of the correctness of the numerical results performed in this
paper.

From Fig. 2(b), one can see that the peak splitting on the momentum spectrum
at the large spatial scale $\lambda=1000m^{-1}$ because of the interference
effect Dumlu:2011rr . The interference effect is caused by the superposition
of wavefunctions when particles experience different turning points due to the
applied field, i.e., physically this effect is produced by the waves reflected
by the multi-bump structure so that there has an additional phase shift on
reflection. To understand this more in detail, we have depicted the turning
point structure in Fig. 16, along with some discussions in Sec. IV.5.

On the other hand, at the small spatial scales $\lambda=10m^{-1}$ and
$\lambda=2.5m^{-1}$, the observed peak splitting on momentum spectrum can be
understood in terms of ponderomotive forces which is beneficial for the
production of particles by pushing particles out of production region
Kohlfurst:2017hbd . Note that the strength of ponderomotive forces is
inversely proportional to the size of spatial scale $\lambda$, that is, the
smaller the $\lambda$, the more obvious the pushing of the ponderomotive
forces. Consequently, the momentum peaks at spatial scale $\lambda=2.5m^{-1}$
are pushed far away from the center compared with that at $\lambda=10m^{-1}$.

Figure 3: (color online). Reduced momentum spectrum of created $e^{+}e^{-}$
pair for different spatial scales for one-color weak field $E_{2w}(x,t)$ with
different chirp of $b_{2}[\omega_{2}/\tau]$ as $0.1$, $0.3$, $0.5$, $0.9$,
i.e., $b_{2}[m^{2}]$ as $0.0014$, $0.0042$, $0.007$, $0.0126$, from (a) to (d)
respectively. The other parameters are the same as in Fig. 2(b).

In addition, the momentum spectrum of created $e^{+}e^{-}$ pair for different
spatial scales for one-color weak field $E_{2w}(x,t)$ with different chirp
values of $b_{2}$ is shown in Fig. 3. As can be seen in the Fig. 3(a), for
small chirp $b_{2}=0.1\omega_{2}/\tau$, the main peak located around $p=0$ for
different spatial scales. However, there appears also two less pronounced
peaks when $\lambda=1000m^{-1}$. In other words, the spectrum is very
sensitive to chirps even the value of chirp is relatively small. Meanwhile,
the peak is increased compared with the results without chirping ($b_{2}=0$)
shown in Fig. 2(b).

For intermediate chirp $b_{2}=0.3\omega_{2}/\tau$, we can see that there is
not only one main peak but also a number of smaller peaks as shown in Fig.
3(b). When the chirp increases to $b_{2}=0.5\omega_{2}/\tau$, the peaks become
two dominate maxima and also there exist other less pronounced mini-peaks, see
Fig. 3(c). Also, for the large value of chirp $b_{2}=0.9\omega_{2}/\tau$, the
spectrum exhibits incomplete interference, and the rate becomes higher
compared with the other cases due to the increase in effective frequency of
the electric field.

The results show that as the increase of the chirp values, the coherent
interference pattern in the momentum spectrum is changed and it turns into
complex oscillatory. These effects can be qualitatively discussed with the
interference between the complex conjugate pairs of turning points. The number
of turning point pairs that are almost equidistant from the real axis
increases with the increase of the chirp value. Therefore, stronger
interference effect can be observed on the momentum spectrum for large chirp.
This is confirmed by the typical structure of the turning points as an
illustration in Fig. 16.

Figure 4: (color online). Reduced total yield $\bar{N}(t\rightarrow\infty)$
for $E_{2w}\left(x,t\right)$ with different chirp as a function of the spatial
scales. The other parameters are the same as in Fig. 3.

In order to investigate the influence of the chirp values more deeply, the
reduced total yield $\bar{N}(t\rightarrow\infty)$ for $E_{2w}\left(x,t\right)$
with different chirp values as a function of the spatial scale is displayed in
Fig. 4. We can see that the particle number is enhanced with the increasing of
chirp value. For the small chirp values, at least for $b_{2}\leq
0.3\omega_{2}/\tau$, the total yield of created particles quickly decreases
with the shrinking of spatial scale in finite spatial scales. Because the
electric field energy decreases with decreasing of spatial scale, particles
created in the field region also decrease correspondingly. For the
intermediate chirp $b_{2}=0.5\omega_{2}/\tau$, the reduced total yield drops
slowly with decreasing spatial scale. For the large chirp
$b_{2}=0.9\omega_{2}/\tau$, $\bar{N}(t\rightarrow\infty)$ tends to an almost
flat line, because the particles creation process is mainly high energy photon
absorption that is less affected by the spatial scale.

##  IV Numerical results for two-color combinational fields

In this section, we turn to investigate the numerical results of inhomogeneous
dynamically assisted combinational electric fields with several chirped forms.
The model of the field is shown in Eq. (1) and the fixed field parameters are
given in Eq. (2). We explore the reduced momentum spectrum or/and the reduced
total yield of created particles as well as enhancement factor for several
spatial scales, for the following cases: (i) without chirp ($b_{1}=b_{2}=0$),
(ii) chirping is only for $E_{1s}\left(x,t\right)$ ($b_{1}\neq 0,b_{2}=0$),
(iii) chirping is only for $E_{2w}\left(x,t\right)$ ($b_{1}=0,b_{2}\neq 0$),
and (iv) chirping is for both $E_{1s}\left(x,t\right)$ and
$E_{2w}\left(x,t\right)$ ($b_{1}\neq 0,b_{2}\neq 0$).

###  IV.1 Two-color field without chirping

Figure 5: (color online). Reduced momentum spectrum for different spatial
scales for two-color field $E\left(x,t\right)$ = $E_{1s}\left(x,t\right)$
+$E_{2w}\left(x,t\right)$ without chirp $b_{1}=b_{2}=0$. The field parameters
are given in Eq. (2).

We will investigate the pair production for two-color field
$E\left(x,t\right)$ = $E_{1s}\left(x,t\right)$ +$E_{2w}\left(x,t\right)$
without chirp, that is, the chirp parameters $b_{1}=b_{2}=0$. The results for
the momentum spectrum at small and large spatial scales are presented in Fig.
5(a) and (b), respectively. Compared with the results of a single field as
shown in Fig. 1 and Fig. 3, the number density of created $e^{+}e^{-}$ pair
increases significantly because of the dynamically assisted Sauter-Schwinger
mechanism Orthaber:2011cm ; Ababekri:2019dkl . It is noted that the momentum
spectrum peaks obtained by $E\left(x,t\right)$ without chirp are smaller than
that in the single field $E_{2w}\left(x,t\right)$ with the maximum chirp
$b_{2}=0.9\omega_{2}/\tau$ in the last section (Fig. 3(d)) which indicates
that the chirp parameter plays an important role in the momentum distribution.

It can also be seen from the Fig. 5(a), the generation of $e^{+}e^{-}$ pair
does not tend to terminate when the spatial scales decreases further
($\lambda\leq 10m^{-1}$), because the contribution from the high frequency
field does not decrease much by the finite spatial scale.

###  IV.2 Two-color field with chirping only for $E_{1s}\left(x,t\right)$

Figure 6: (color online). Reduced momentum spectrum for different spatial
scales for two-color field $E\left(x,t\right)$ = $E_{1s}\left(x,t\right)$
+$E_{2w}\left(x,t\right)$. The chirping is only for $E_{1s}\left(x,t\right)$
with $b_{1}[\omega_{1}/\tau]$ as $0.1$, $0.3$, $0.5$ and $0.9$, from (a) to
(d) respectively. The fields parameters are given in Eq. (2).

In the case of two-color field $E\left(x,t\right)$ =$E_{1s}\left(x,t\right)$
+$E_{2w}\left(x,t\right)$ with chirping only for $E_{1s}\left(x,t\right)$,
i.e.,
$b_{1}=0.1\omega_{1}/\tau,0.3\omega_{1}/\tau,0.5\omega_{1}/\tau,0.9\omega_{1}/\tau$
and $b_{2}=0$, the reduced momentum spectrum is shown in Fig. 6. As can be
seen in Fig. 6(a) and (b), the small chirp has a little effect on the momentum
spectrum as explained in Sec. III.1. However, when the chirp parameter
increases, such as $b_{1}=0.5\omega_{1}/\tau$ and $b_{1}=0.9\omega_{1}/\tau$,
the momentum spectrum signatures change significantly, especially in the
quasi-homogeneous ($\lambda=1000m^{-1}$) case, which can be seen from Fig.
6(c) and Fig. 6(d). It can be seen that the momentum spectrum has been
significantly changed, even though the height is almost constant. Moreover,
the effects of chirp parameters can be understood in more detail by obtaining
the total particle yield and enhancement factor for various chirp values as a
function of spatial scale $\lambda$.

Figure 7: (color online). Reduced total yield $\bar{N}(t\rightarrow\infty)$
for two-color field $E\left(x,t\right)$ = $E_{1s}\left(x,t\right)$
+$E_{2w}\left(x,t\right)$ with different chirp as a function of the spatial
scales. The chirping is only for $E_{1s}\left(x,t\right)$. The parameters are
the same as in Fig. 6 and the insert is the enlargement of finite spatial
scales.

Fig. 7 shows the total yield of created particles as a function of the spatial
scales for different chirp parameters. It can be seen that for each chirp
parameter, the total yield of created $e^{+}e^{-}$ pair in the quasi-
homogeneous region ($\lambda\textgreater 100m^{-1}$) basically remains a
constant reflecting the uniform approximation for the combined field. However,
for the small spatial scale, the total yield drops rapidly as the field energy
for strong and fast pulse is also decreased. Thus, the result is
understandable and indicates that there is a slight improvement in the total
yield with the increase of chirp value. The same numerical result for finite
spatial scale ($\lambda\textless 4m^{-1}$) is given more clearly in the insert
of Fig. 7.

Figure 8: (color online). Enhancement factor of the reduced total yield
$\bar{N}_{1s+2w}/(\bar{N}_{1s}+\bar{N}_{2w})$ for different chirp as a
function of the various spatial scales. The parameters are the same as in Fig.
6.

In order to investigate whether there is an optimal chirp value or/and spatial
scale for $e^{+}e^{-}$ pair generation for two-color field with chirping only
for $E_{1s}\left(x,t\right)$, we further explore the enhancement factor of
particle production for different chirp values as a function of the spatial
scale. The enhancement factor is defined as the ratio
$\bar{N}_{1s+2w}/(\bar{N}_{1s}+\bar{N}_{2w})$. In this expression,
$\bar{N}_{1s+2w}$ is the reduced total yield of particles created for combined
electric fields $E\left(x,t\right)$=$E_{1s}\left(x,t\right)$
+$E_{2w}\left(x,t\right)$ with chirping only for $E_{1s}\left(x,t\right)$
(chirp of $b_{1}$ [$\omega_{1}/\tau$] as $0.1$, $0.3$, $0.5$ and $0.9$), and
$\bar{N}_{1s}+\bar{N}_{2w}$ denote the direct summation of reduced total yield
for strong fields and weak fields with the certain chirp parameters. We obtain
that the enhancement factor of the total yield for different chirp values, as
shown in Fig. 8.

From Fig. 8, we can see that when the chirp parameter is given, the
enhancement factor decreases with the decrease of the spatial scales in the
extremely small spatial scale range $1.6m^{-1}\textless\lambda\textless
2.4m^{-1}$, and it increases with the decrease of the spatial scale at the
range of $2.4m^{-1}\textless\lambda\textless 100m^{-1}$. For
$\lambda\textgreater 100m^{-1}$, the enhancement factor is almost a constant
for different chirp parameters, which means that the effects of chirp
parameter on the enhancement factor in quasi-homogeneous space are almost
negligible. However, we can get that the enhancement factor increases with the
increase of chirp values at a certain spatial scale. It is noted that the
chirp parameter has a little effect on the total particle number of the
various spatial scales and the momentum spectrum when $\lambda=2.5m^{-1}$ and
$\lambda=10m^{-1}$, but it has a visible effect on the momentum spectrum when
the spatial width increases to $\lambda=1000m^{-1}$, especially in the case of
the large chirp value $b_{1}=0.9\omega_{1}/\tau$.

From the numerical results obtained above and what we have discussed, we can
obtain that $\lambda=2.4m^{-1}$ and $b_{1}=0.9\omega_{1}/\tau$ are the optimal
values of the spatial scale and chirp to get the largest enhancement factor of
created particles in this subsection.

###  IV.3 Two-color field with chirping only for $E_{2w}\left(x,t\right)$

Figure 9: (color online). Reduced momentum spectrum for different spatial
scales for two-color field $E\left(x,t\right)$ = $E_{1s}\left(x,t\right)$
+$E_{2w}\left(x,t\right)$. The chirping is only for $E_{2w}\left(x,t\right)$
with $b_{2}[\omega/\tau]$ as $0.1$, $0.3$, $0.5$ and $0.9$, from (a) to (d)
respectively. The field parameters are given in Eq. (2).

For two-color field $E\left(x,t\right)$=$E_{1s}\left(x,t\right)$
+$E_{2w}\left(x,t\right)$ with chirping only for $E_{2w}\left(x,t\right)$,
i.e., the chirp parameters $b_{2}=\alpha\omega_{2}/\tau$ with $\alpha=0.1$,
$0.3$, $0.5$, $0.9$, the reduced momentum spectrum is exhibited in Fig. 9. The
model of the field is also given as Eq. (1), while the chirp parameters
$b_{1}=0$. Compared with the results of a single weak field
$E_{2w}\left(x,t\right)$, it can be seen that the momentum spectrum in the
present electric field model is more complex and the range of spectrum expands
in momentum space. The complicated reduced momentum spectrum for various
spatial scales is due to the presence of interference effect. Most
importantly, the peak value of momentum spectrum increases with chirp, which
is also understood by the dynamically assisted Sauter-Schwinger mechanism.

It can be seen in Fig. 9(a), the oscillation distribution around the main peak
of the momentum spectrum has a little difference in the quasi-homogeneous case
($\lambda=1000m^{-1}$), when the chirp is small as
$b_{2}=0.1\omega_{2}/\tau=0.0014m^{2}$. For the intermediate chirp parameters
$b_{2}=0.3\omega_{2}/\tau=0.0042m^{2}$ and
$b_{2}=0.5\omega_{2}/\tau=0.007m^{2}$, the oscillation of the momentum spectra
become complex, see Fig. 9(b) and (c). When the chirp is large as
$b_{2}=0.9\omega_{2}/\tau$, the momentum spectrum has significant interference
signatures, as shown in Fig. 9(d). For small space scales $\lambda=2.5m^{-1}$
and $\lambda=10m^{-1}$, the effect of small chirp on the momentum spectrum is
not obvious. These responses of the momentum spectrum to the chirp parameter
means that the interference effect becomes stronger with the increase of chirp
and can also be understood qualitatively with the interference between the
complex conjugate pairs of turning points, see Sec. IV.5. Meanwhile, one sees
that the height of peak is enhanced by at least four times from
$b_{2}=0.1\omega_{2}/\tau$ to $b_{2}=0.9\omega_{2}/\tau$.

Figure 10: (color online). Reduced total yield as a function of the spatial
scales for different setups. $\bar{N}_{1s+2w}$ is the total yield of two-color
field $E\left(x,t\right)$ = $E_{1s}\left(x,t\right)$
+$E_{2w}\left(x,t\right)$. The chirping is only for $E_{2w}\left(x,t\right)$
with $b_{2}=0.9\omega_{2}/\tau$ (solid line), $0.5\omega_{2}/\tau$ (dashed
line) and $0.3\omega_{2}/\tau$ (dotted line). $\bar{N}_{1s}+\bar{N}_{2w}$ is
simple direct summation of two individual field contribution for two kinds of
$b_{2}$ [$\omega_{2}/\tau$] as $0.9$ (dashed dotted line) and $0.5$ (short
dashed line).

It is well known that the dynamically assisted electric field has a great
influence on the generation of $e^{+}e^{-}$ pair including the momentum
spectrum and the total particle number Orthaber:2011cm . Therefore, in Fig. 10
we consider that the reduced total yield of created particles as a function of
the spatial scale for different setups. Note that the total yield
$\bar{N}_{1s+2w}$ is obtained by considering $E\left(x,t\right)$ =
$E_{1s}\left(x,t\right)$ +$E_{2w}\left(x,t\right)$ and the chirping is only
applied to $E_{2w}\left(x,t\right)$ with $b_{2}=0.9\omega_{2}/\tau$,
$0.5\omega_{2}/\tau$ and $0.3\omega_{2}/\tau$, respectively. As a comparison
the yield $\bar{N}_{1s}+\bar{N}_{2w}$ is obtained by a direct summation of
individual yield of each field $E_{1s}\left(x,t\right)$ with $b_{1}=0$ and
$E_{2w}\left(x,t\right)$ with $b_{2}=0.9\omega_{2}/\tau$ and
$b_{2}=0.5\omega_{2}/\tau$, respectively.

From the dashed line and dotted line in the Fig. 10, we can see the total
yield increases slightly when the chirp parameter varying from
$b_{2}=0.3\omega_{2}/\tau$ to $b_{2}=0.5\omega_{2}/\tau$. However compared
with the chosen maximum chirp at $b_{2}=0.9\omega_{2}/\tau$ as shown by the
solid line, the total yield in the latter case increases by at least $2$
times. We think that the role of effective frequency increasing greatly when
the chirp is large would not only provide more energy to created particles but
also create more particles. In particular, it is observed that
$\bar{N}_{1s}+\bar{N}_{2w}$ ($b_{2}=0.9\omega_{2}/\tau$) >$\bar{N}_{1s+2w}$
($b_{2}\leq 0.5\omega_{2}/\tau$) at the finite spatial scale $\lambda\textless
6m^{-1}$. This result indicates that the high frequency field is extremely
sensitive to the chirp, meanwhile, the choice of the appropriate chirp is also
important.

Figure 11: (color online). Enhancement factor of the total number
$\bar{N}_{1s+2w}/(\bar{N}_{1s}+\bar{N}_{2w})$ for different chirp values as a
function of the various spatial scales. The parameters are the same as in Fig.
9.

We also investigate the enhancement factor of total yield for different chirp
values as a function of the various spatial scales and the result is shown in
Fig. 11. It is obvious that the enhancement factor is almost a constant for
each chirp in the quasi-homogeneous region $\lambda\geq 1000m^{-1}$. Moreover,
the variation of enhancement factor is complex in the finite spatial scale.
For relative small chirp values $b_{2}\leq 0.3\omega_{2}/\tau$, the
enhancement factor increases at the spatial scale
$2.4m^{-1}\textless\lambda\textless 100m^{-1}$ and decreases at the spatial
scale $1.6m^{-1}\textless\lambda\textless 2.4m^{-1}$ with the shrinking of the
spatial scale. Thus, we obtain that the enhancement factor reaches maximum at
the extremely spatial scale $\lambda=2.4m^{-1}$. As can be seen from the solid
line ($b_{2}=0.1\omega_{2}/\tau$) and dashed line ($b_{2}=0.3\omega_{2}/\tau$)
in Fig. 11, the larger the chirp parameter is, the larger the enhancement
factor. For the larger chirp value $b_{2}=0.5\omega_{2}/\tau$ and
$b_{2}=0.9\omega_{2}/\tau$, enhancement factor is smaller than that with the
small chirp values. These results can also be qualitatively obtained in Fig.
10.

###  IV.4 Two-color field with chirping for both $E_{1s}\left(x,t\right)$ and
$E_{2w}\left(x,t\right)$

Figure 12: (color online). Reduced momentum spectrum for two-color field
$E\left(x,t\right)$ = $E_{1s}\left(x,t\right)$ +$E_{2w}\left(x,t\right)$ with
chirping for both fields. The chirps are $b_{1}[\omega_{1}/\tau]$ as $0.1$,
$0.3$, $0.5$, $0.9$ and $b_{2}[\omega_{2}/\tau]$ as $0.1$, $0.3$, $0.5$,
$0.9$. The fields parameters are given in Eq. (2).

In the case of two-color field $E\left(x,t\right)$=$E_{1s}\left(x,t\right)$
+$E_{2w}\left(x,t\right)$ with chirping for both $E_{1s}\left(x,t\right)$ and
$E_{2w}\left(x,t\right)$, the model of the external field is given in Eq. (1)
with $b_{1}\neq 0$ and $b_{2}\neq 0$. The reduced momentum spectrum for
various spatial scales is shown in Fig. 12. From Fig. 12(a) and (b) one can
see that the weak interference pattern appears in the momentum spectrum. For
$\lambda=1000m^{-1}$, the large chirp causes a complex oscillation which can
be understood as enhancing of the interference effect of created particles
shown in Fig. 12(c). In case of $b_{1}=0.9\omega_{1}/\tau$ and
$b_{2}=0.9\omega_{2}/\tau$ in Fig. 12(c), we can see that some additional
peaks appear beside the main peak in the momentum spectrum. More importantly,
the peak of the momentum spectrum in this case is higher compared to the
results of the other cases.

Figure 13: (color online). Reduced total yield as a function of the various
spatial scales for two-color field $E\left(x,t\right)$ =
$E_{1s}\left(x,t\right)$ +$E_{2w}\left(x,t\right)$ with four different
chirping cases, $b_{1}=0,b_{2}=0$ (red dashed dotted line), $b_{1}\neq
0,b_{2}=0$ (blue dotted line), $b_{1}=0,b_{2}\neq 0$ (green dashed line), and
$b_{1}\neq 0,b_{2}\neq 0$ (black solid line). The nonzero chirp parameters
$b_{i}$ [$\omega_{i}/\tau$], ($i=1,2$), are chosen as $0.5$ in ($a$) and $0.9$
in ($b$). The other parameters are given in the text.

Fig. 13 shows the reduced total yield of created particles as a function of
the spatial scale for two-color field $E\left(x,t\right)$ =
$E_{1s}\left(x,t\right)$ +$E_{2w}\left(x,t\right)$ with four different
chirping cases, $b_{1}\neq 0,b_{2}\neq 0$ (black solid line),
$b_{1}=0,b_{2}\neq 0$ (green dashed line), $b_{1}\neq 0,b_{2}=0$ (blue dotted
line), and $b_{1}=0,b_{2}=0$ (red dashed dotted line). In Fig. 13(a), the
nonzero chirp parameters are $b_{i}=0.5\omega_{i}/\tau$, where $i=1,2$,
however, in Fig. 13(b) they are chosen as $0.9$.

From Fig. 13(a), it can be seen that for relatively small nonzero chirp
parameters, the total yield has small differences at a certain spatial scale
in the presented four cases. From Fig. 13(b), we can see that the reduced
total yield increases very much when the chirp value is chosen large as
$b_{1}=0.9\omega_{1}/\tau$ and $b_{2}=0.9\omega_{2}/\tau$. When $\lambda\geq
40m^{-1}$, the total yield tends to an almost flat line. In the field
$E\left(x,t\right)$ = $E_{1s}\left(x,t\right)$ +$E_{2w}\left(x,t\right)$, the
reduced total yield does not increase too much when the chirp acts only on
$E_{1s}\left(x,t\right)$, compared with the case without chirp. However, the
reduced total yield increases greatly when the chirp acts on both of the
fields or only on $E_{2w}\left(x,t\right)$, compared with the case that the
chirp acts only on $E_{1s}\left(x,t\right)$. Specifically, the reduced total
yield increases more than one order of magnitude in extremely small spatial
scales $\lambda\approx 1.6m^{-1}$ and increases by $2.38$ and $2.2$ times in
quasi-homogeneous space region $\lambda\geq 100m^{-1}$.

Figure 14: (color online). Enhancement factor of the total number
$\bar{N}_{1s+2w}/(\bar{N}_{1s}+\bar{N}_{2w})$ for different chirp values as a
function of the various spatial scales. The parameters are the same as in Fig.
12.

Fig. 14 shows the enhancement factor of the total yield for different chirp
values as a function of the various spatial scales. It is found that the
enhancement factor is almost a constant for each chirp in the quasi-
homogeneous region. However, the variation of enhancement factor is complex in
the finite spatial scale. We can also obtain that the enhancement factor is
the largest at $\lambda=2.4m^{-1}$ for the small chirp values
$b_{i}[\omega_{i}/\tau]\leq 0.3,(i=1,2)$ and at $\lambda=16m^{-1}$ for the
largest chirp value.

Table 1: Optimal spatial scales related to the optimal enhancement factor at the chosen optimal chirp parameters. Different chirping | Chirp parameter ($\omega_{i}/\tau$), ($i=1,2$) | Spatial scale ($m^{-1}$) | Enhancement factor
---|---|---|---
$b_{1}\neq 0$, $b_{2}=0$ | $b_{1}=0.9$ | $\lambda=2.4$ | $94.756$
$b_{1}=0$, $b_{2}\neq 0$ | $b_{2}=0.3$ | $\lambda=2.4$ | $133.584$
$b_{1}\neq 0$, $b_{2}\neq 0$ | $b_{1}=b_{2}=0.3$ | $\lambda=2.4$ | $132.517$

We collect the optimal chirp parameters and spatial scales related to the
optimal enhancement factors from Fig. 8, Fig. 11 and Fig. 14. The
corresponding chirping scheme and optimal values are given in Table 1. It is
found that the enhancement factor is always the optimal value at the extremely
small spatial scale $\lambda=2.4m^{-1}$ for different optimal chirp parameters
in the study regime. We also find that the enhancement factor is the largest
when the chirp acts only on high frequency field $E_{2w}\left(x,t\right)$
(second row in the table). Compare the result with the first row (a large
chirp parameter acts on the low frequency field in the combinational field),
we find that the effect of large chirp parameter in a low frequency field is
not significant. Interestingly, in third row (a small chirp parameter acts on
both of the fields in the combinational field), the enhancement factor is
smaller than the one in the second row. The physical mechanism behind the
phenomenon needs to study further.

###  IV.5 Semiclassical analysis and discussion

In order to see why the interference effect of the momentum spectrum we
obtained and why the pairs yield is increased by the chirping in some cases,
we employ the semiclassical WKB approach to make some discussion on obtained
numerical results in this subsection. In fact the physical picture is related
to the quantum tunneling problem in an effective scattering potential of
applied background fields, which is reflected in the typical structure of the
turning points.

We know that the WKB approach is a quasiclassical one to study the potential
scattering problem so that it seems to be more effective in the Schwinger
tunneling regime $\gamma\ll 1$, however, it has been reasonably extended to
perturbative multiphoton regime in many works Popov1968QuasiclassicalAF ;
Brezin:1970xf ; Dumlu (2012); Gong:2019sbw ; Ababekri:2019qiw . In fact as
early as in around of 1970’s the WKB associated to the turning points has been
worked for the study of atomic ionization Popov1968QuasiclassicalAF  and also
that of pair production Brezin:1970xf  for both of $\gamma\ll 1$ and
$\gamma\gg 1$. Moreover, recently Dumlu has made a series of study for the
fields in the multiphoton regime Dumlu (2012). For example, he studied the
field with $E_{0}=0.1m^{2}$ and $\omega=0.5m$ and discussed the corresponding
turning points distribution in detail. In fact the applicability of WKB and
the corresponding turning points is seen by that the pair creation rate can be
expressed either as $N\approx\exp(-2K)$ Dumlu:2011rr  or as $N\sim\exp[-(\pi
E_{cr}/2eE)g(\gamma)]$ , see Eq.(48) in Ref.Brezin:1970xf , therefore, $K$ is
associated to $K=-(\pi E_{cr}/2eE)g(\gamma)$. The difference for $\gamma\ll 1$
and $\gamma\gg 1$ is that $g(\gamma)\simeq 1$ when $\gamma\ll 1$ and
$g(\gamma)=(4/\pi\gamma)\ln(2\gamma)+o(1/\gamma)$ when $\gamma\gg 1$, just as
the Eq. (49) of Ref.Brezin:1970xf . In fact the method of stationary phase to
get the pair production rate when $\gamma\gg 1$ in Ref.Brezin:1970xf  is
equivalent to that by the WKB approach with complex turning points structure.

The approximate expression of the particle creation rate for multiple turning
points is (refer to Ref.Dumlu:2011rr )

$\displaystyle N\approx\sum\limits_{t_{i}}e^{-2K_{i}}-\sum\limits_{t_{i}\neq
t_{j}}2\cos(2\theta_{(i,j)})e^{-K_{i}-K_{j}},$ (22)

with

$K_{i}=\left|\int_{t_{i}^{*}}^{t_{i}}{w_{\bm{p}}(t)dt}\right|,$

and

$\theta_{(i,j)}=\left|\int_{Re(t_{i})}^{Re(t_{j})}{w_{\bm{p}}(t)dt}\right|,$

where $t_{i}$ and $t_{j}$ are the solutions of equation $w_{\bf
p}(t)=\sqrt{m^{2}+p_{\perp}^{2}+(p_{x}-eA(t))^{2}}=0$, i.e., the turning
points and $\rm{Re}(t_{i})$ is the real part of point. Note that the turning
points closest to the $t$ real axis tend to dominate the pair production
arising from the first term in Eq.(22), and the oscillatory behavior of the
momentum spectrum, i.e., interference effect is due to the interference terms
$\theta_{(i,j)}$ in Eq. (22).

Since the physical reasons are similar we just choose three representative
situations, i.e., (i) strong field $E_{1s}$ without and with chirping; (ii)
weak filed $E_{2w}$ without and with chirping; and (iii) combined fields
$E(x,t)=E_{1s}+E_{2w}$ with chirping only for $E_{2w}$, to discuss and
understand the numerical results we have obtained above. The corresponding
typical turning point structures are illustrated in Fig. 15 in case of (i) and
Fig. 16 in cases of (ii) and (iii), respectively. To discuss turning points
structure clearly, we depict them at the vanishing momentum $p_{x}=0$.

Figure 15:  Contour plots of $|w_{\bm{p}}(t)|^{2}$ in the complex $t$ plane,
showing the location of turning points for $E_{1s}$ where $w_{\bm{p}}(t)=0$.
These plots are for the vanishing momentum $p_{x}=0$. Form (a) to (b) the
chirp values are $b_{1}=0$ and $b_{1}=0.9\omega_{1}/\tau$. Panel (c) is the
amplified plot of the two set of pairs of turning points at around
$\rm{Re}(t)\sim 30$ in panel (b). The other field parameters are the same as
in Fig. 1. The three dashed lines at $\rm{Re}(t)=0$ and adjacent the real $t$
axis are for eyes guidance.

For the case of $E_{1s}$ without chirp $b_{1}=0$, in the Fig. 15(a), one can
see that there is only a single dominant pair of turning point located the
center of figure, thus there is no interference. This result can be used to
qualitatively explain the corresponding momentum spectrum, as shown in Fig.
1(a). When chirp value $b_{1}=0.9\omega_{1}/\tau$, the distribution of turning
points seems to be more complicated as shown in Fig. 15(b), we find that there
are two dominant complex conjugate pairs of turning points located in the
right real $t$ regime, which can be clearly seen in Fig. 15(c). And they have
almost equidistant from the real axis. It is well known that the interference
effect is strongest when the pairs of different turning points are at a
comparable distance from the real axis, therefore, the obvious interference
patterns appear in the momentum distribution shown in Fig. 1(d). Moreover, in
this chirping case, the two set of pairs of turning points at around
$\rm{Re}(t)\sim 30$ have almost the same distances from real axis at
$|\rm{Im}(t)|=|\rm{Im}(t^{*})|\approx 15$. For the single pair of turning
point without chirping at around $\rm{Re}(t)\sim 0$, the distances from real
axis at $|\rm{Im}(t)|=|\rm{Im}(t^{*})|\approx 5$. These turning points
characteristics indicates two facts that are qualitatively consistent with the
numerical results of Fig. 1(a) and (d) for $b_{1}=0$ and
$b_{1}=0.9\omega_{1}/\tau$ under an approximate homogeneous case of
$\lambda=1000m^{-1}$. On one hand, the closer distance to real axis means that
the particles creation rate in the no-chirping case is large comparable to the
chirping one. Indeed $n\left(p_{x}=0,t\rightarrow\infty\right)=4.67\times
10^{-5}m$ when $b_{1}=0$ is a little larger than
$n\left(p_{x}=0,t\rightarrow\infty\right)=3.43\times 10^{-5}m$ when
$b_{1}=0.9\omega_{1}/\tau$. On the other hand, because the interference effect
is associated to the distances of turning points along the real-axis
direction, so the neighbouring two set of pairs in Fig. 15(c) implies an
obvious interference effect which is revealed in the oscillation behaviour at
around of $p=0$ in momentum spectrum, see Fig. 1(d).

Figure 16: (color online). Contour plots of $|w_{\bm{p}}(t)|^{2}$ in the
complex $t$ plane, showing the location of turning points for different models
of external field where $w_{\bm{p}}(t)=0$. These plots are for the vanishing
momentum $p_{x}=0$. Panel (a): The plot is for $E_{2w}$, with $b_{2}=0$
(black) and $b_{2}=0.9\omega_{2}/\tau$ (red). Panel (b): The red dots are for
$E_{2w}$ (same as that in panel (a)), and the blue dots for
$E(x,t)=E_{1s}+E_{2w}$ with chirping only for $E_{2w}$, i.e., $b_{1}=0$,
$b_{2}=0.9\omega_{2}/\tau$. The other field parameters are given in Eq. (2).
The three dashed lines at $\rm{Re}(t)=0$ and adjacent the real $t$ axis are
for eyes guidance.

In Fig. 16(a), the black dots are turning points of $E_{2w}$ without chirping,
and the red dots are turning points of $E_{2w}$ with large chirp
$b_{2}=0.9\omega_{2}/\tau$. Obviously the turning points located in the
positive real $t$ regime is much more close to real axis as well as more
number in the case of chirping (red) compared to the case of no-chirping
(black). This means that the particle number density is increased due to the
closer of turning points to real $t$ axis and the stronger interference due to
the appearance of more turning points. It is why the stronger interference
effects can be observed on the momentum spectrum as shown in Fig. 3 (d).

As a further comparison, we add $E_{1s}$ to $E_{2w}$ so that the model is a
two-color field , the turning points structure is plotted in Fig. 16(b) with
$b_{1}=0$ and still $b_{2}=0.9\omega_{2}/\tau$, see the blue dots. From Fig.
16(b) one can see that, compared with the one-color chirping field $E_{2w}$
(red dots), for the two-color field, even if the $E_{1s}$ has no chirping but
the $E_{2w}$ chirping is kept, the turning points structure is changed to be
that the dominant one-pair turning point is closer to $t$ real axis very much
and also more points appeared in the right. Therefore, the appearance of more
obvious interference effects in Fig. 9(d) is understandable compared with that
in Fig. 3 (d).

Certainly the other momentum spectra can also be understood by the similar
discussions through the corresponding turning points structure. Since the
physical picture as the semiclassical WKB approximation is simple and valid,
it is not necessary to give a repeat presentation for more situations. What
should be pointed out here is that the delicate understanding of the momentum
signature and pairs yield beyond the WKB picture may be possible and it is
worthy to be studied furthermore in future Kaminski:2018ywj .

##  V Conclusion and outlook

Within the DHW formalism, we investigated the reduced momentum spectrum, the
reduced total yield of the created pairs for either low or high frequency one-
color field with different chirp values. Meanwhile, we studied chirped
dynamically assisted pair production for two-color combinational fields, and
the enhancement factor is obtained in the assisted case. Moreover, we employ
the semiclassical WKB approach and the corresponding turning points structure
to make some qualitative discussion.

For the strong low frequency field, the momentum spectrum has almost the same
peak value and oscillates significantly when the chirp value is the largest,
especially in the quasi-homogeneous region. In the case of the high frequency
weak field, when the chirp increases, the oscillatory pattern of the spectrum
becomes more complicated as the presence of interference effect. While an
incomplete interference is seen in the spectrum when some appropriate chirp is
used. More importantly, the peak value of reduced momentum spectrum and the
total yield increase strongly with the increasing of chirp parameters. We have
examined Ref. Orthaber:2011cm ; Olugh:2019nej  and meanwhile expanded the
dynamically assisted field results in the sense of having chirping. We
obtained that the total yield of produced pair is indeed enhanced
significantly in some cases by appropriate chirp parameters. However, our
studies here exhibit very delicate and subtle effects about the chirping and
the field forms on the pair production problem.

For two-color field with chirping only for $E_{1s}\left(x,t\right)$, we found
that the momentum spectrum depends on the chirp parameters to some extent,
which is more obvious in the quasi-homogeneous region. In addition, the
reduced total yield of created particles and the enhancement factor of total
particle number do not increase greatly with the increase of chirp parameter,
which indicates that the chirp parameter has a little effect on the pair
production in this case. However, in the case of two-color field with chirping
only for $E_{2w}\left(x,t\right)$, the particle number density is sensitive to
the chirp and the reduced momentum spectrum shows a strong nonlinear feature.
The reduced total yield of particles in this electric field increases
significantly, which increases by at least $2$ times for the large chirp value
and by one order of magnitude for the small chirp value, compared with that in
the case of a simple direct summation of two individual field. It is also
noted that, for $E\left(x,t\right)$ with chirping for both
$E_{1s}\left(x,t\right)$ and $E_{2w}\left(x,t\right)$, strong interference
effects are observed in the momentum distribution, so that the high total
yield is achieved. Specifically, in the chirped dynamically assisted scenario,
when the chirps are acting on the two fields the reduced total yield is
enhanced by more than one order of magnitude in the field with a relative
small spatial scale, while it is enhanced at least two times in other case of
field with larger spatial scales or even in the quasi-homogeneous region,
compared with that the chirp is acting only for the low frequency strong
field. Most importantly, we also obtain some optimal chirp parameters and
spatial scales for enhancement factor of the total yield in different
scenarios of the studied combined field.

We believe that these results presented in this paper are useful for having a
better understanding about the effect of the chirp parameter and provide
further insight into the relation between the structure of external field and
pair creation. Our study provides also the theoretical basis of some broader
parameters range for possible experiments in the future by the modeling of
combinational fields that enhance the dynamically assisted pair production.

###### Acknowledgements.

We would like to thank Dr. M. Ababekri and N. Abdukerim for critical reading
of the manuscript. We are also grateful to Dr. Li Wang for useful discussion
and valuable comments. And we are thankful to the anonymous referee for
helpful suggestions to improve the manuscript. This work was supported by the
National Natural Science Foundation of China (NSFC) under Grant No. 11875007
and No. 11935008. The computation was carried out at the HSCC of the Beijing
Normal University.

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