spacing=nonfrench

# Surrey System For DCASE 2022 Task 5: Few-shot Bioacoustic Event Detection
With Segment-Level Metric Learning

###### Abstract

Few-shot audio event detection is a task that detects the occurrence time of a
novel sound class given a few examples. In this work, we propose a system
based on segment-level metric learning for DCASE 2022 challenge few-shot
bioacoustic event detection (task 5). We make better utilization of the
negative data within each sound class to build the loss function, and use
transductive inference to gain better adaptation on the evaluation set. For
the input feature, we find the per-channel energy normalization concatenated
with delta mel-frequency cepstral coefficients to be the most effective
combination. We also introduce new data augmentation and post-processing
procedures for this task. Our final system achieves an f-measure of 68.74 on
the DCASE task 5 validation set, outperforming the baseline performance of
29.5 by a large margin. Our system is fully open-
sourced111https://github.com/haoheliu/DCASE_2022_Task_5.

Index Terms—  few-shot learning, transductive inference, metric learning,
audio event detection

##  1 Introduction

Few-shot learning (FSL) [1] is a machine learning problem that aims to
generalize the system to novel classes after supervised learning with the data
from known classes. Sound event detection [2] is a task that locates the onset
and offset of certain sound classes. By combining the idea of FSL with sound
event detection [3], the system can detect the new sound class with only a few
labeled samples. This technique is useful for audio data labeling, especially
when the user needs to detect a new type of sound.

In the previous DCASE 2021 task 5 challenge, the submitted systems [4, 5, 6,
7] have achieved promising results on the evaluation set. Most of the systems
use a prototypical network [1] as the main architecture. Yang et al. [4]
propose a mutual learning framework to improve the feature extractor. Tang et
al. [5] propose to use embedding propagation to learn a smoother manifold in
the latent space. Data augmentations such as spec-augment and mixup are used
in the system described in [6, 7].

In previous works, the negative segment in each audio file is not fully
utilized. Here negative segment means the audio segments that do not contain
the target sound event. Most metric-learning-based studies [4, 5] optimize the
model only using the labeled positive data, by grouping and separating the
latent prototypes of the same and different positive classes, respectively.
However, the positive segments also need to be distinguishable from the
negative segments within the same audio file. We propose to use negative
segments to enhance the metric learning to address this problem.

This report describes the prototypical-network based system we submitted to
the DCASE challenge 2022 task 5 [8, 9]. Our system builds upon a prototypical
network and proposes new methods for the input features, metric learning,
evaluation data preprocessing, and post-processing.

To increase the generalization ability of our system, we merge the training
set with the animal sound in AudioSet [10]. We conduct studies on a different
combinations of audio features to choose the best feature for this task.
Features includes log-mel spectrogram (MEL), per-channel energy normalization
(PCEN), mel-frequency cepstral coefficients (MFCC), and delta-MFCC
($\Delta$MFCC). We found using PCEN and delta-MFCC together yields the best
average f-measure score on the validation set. To make full use of the labeled
few-shot data, we integrate transductive inference into our metric learning
framework. We treat each evaluation file as a sound class and jointly optimize
it with the training dataset. Our proposed system achieves a f-measure of
68.74 on the DCASE task 5 validation set.

This technical report will be organized as follows. Section 2 provides an
overview of our system. Section 3 introduces the methods we proposed to
improve our system. Section 4 discussed the experimental setup and the four
systems we submitted to the challenge. Section 5 reports the result of the
validation set and the ablation studies. Section 6 summarize this work and
provide a conclusion.

##  2 System Overview

We build our system with a prototypical network, which is widely used for
metric-based few-shot learning. The objective of our system is properly
mapping different sounds into a latent embedding within a high-dimensional
space, where similar sounds have a closer distance.

We use episodic training [11] to optimize our system in an N-way-K-shot way,
in which we set $N=10$ and $K=5$. This means each training batch will select
data from N classes. And for each class, the system will select $2\times K$
segments from the dataset, and treat $K$ segments as support segments and
other $K$ segments as query segments. Then both segments are processed by a
feature extraction network, which outputs latent embeddings. By averaging the
embedding of $K$ support and $K$ query segments for class $i$, we obtain a
query prototype $x^{q}_{i}$ and a support prototypes $x^{s}_{i}$. The system
is optimized by minimizing the distance between query prototypes and support
prototypes for each class. To build a robust latent space and to generalize
better to the new class, we introduce the metric learning with negative
samples and the transductive inference approach in Section 3.2 and 3.3.

During evaluations, only fivethe system uses $K$ labeled positive events to
build positive and negative prototypes, which are the latent representation of
positive and negative segments in one audio file. Here a positive sound event
can contain multiple positive segments. If the negative segment in the labeled
part is too short, we will use a spectrum-similarity-based algorithm (Section
3.6) to identify the possible negative segment in the remaining unlabelled
audio. To classify the unlabelled part, the system will first segment it with
an adaptive segment length and build a query set. Then the query set can be
classified by calculating the distance with prototypes. We introduce the way
we build the positive and negative prototypes in Section 3.6.

Post-processing is important to the performance of the system. We introduce a
splitting-merging-filtering-based post-processing method designed to fully
utilize the information provided in the first $K$ labeled events, such as
minimum negative segment length. We also use various data augmentation
techniques and external training data from AudioSet during model training,
which will be discussed in section 4.

##  3 Methodology

###  3.1 Feature extraction network

Our feature extraction network $f_{\theta}$ is a convolutional neural network
(CNN) based architecture that maps the audio feature $s$ into a latent
embedding $x$:

$f_{\theta}:s\rightarrow x$ (1)

Similar to the architecture proposed by [12], the network $f_{\theta}$
consists of three convolutional blocks with hidden channels of 64, 128, and
64. Each convolutional block consists of three two-dimensional CNN layers with
batch normalization and leaky rectified linear unit activations [13]. As a
common trick in CNN-based network [14, 15], we apply $2\times 2$ max-pooling
after each block. The input and output of each convolutional block have a
residual connection processed by a downsampling CNN layer. In order to
maintain the same output dimension with different input lengths, we apply an
adaptive average pooling at the end of the network. The final output feature
map after adaptive pooling is a $C\times T\times F$ size block, which is also
the final latent embedding.

###  3.2 Segment-level metric learning

In a similar way to [3], we divide the audio features into segments with equal
lengths for metric learning. Each segment has a unique label indicating its
sound class and whether it is a positive or negative event. As illustrated in
the left part of Figure 1, previous methods typically optimize the model only
using the labeled positive classes, by minimizing the distance between the
query and support prototypes that have the same label. However, these systems
only use the labeled positive classes, which are limited in duration. By
comparison, negative samples usually contain a wider variety of sounds and are
easier to collect. In addition, during inference, the system will need to
discriminate between positive and negative samples. This objective is not
directly optimized during training with only positive data.

Figure 1: The left side is the baseline method that performs metric learning
in a 3-way-2-shot way. The baseline method only uses positive classes. The
right side illustrates the improved metric learning with the negative samples
method we used in our system. $x_{i}$ and $\widetilde{x}_{i}$ stand for the
positive and negative samples of class $i$. Transparent color means the
samples are not actually used.

So, to better discriminate between negative and positive segments within the
audio, we propose to perform a segment-level metric learning approach with the
negative segments. In on the probabilities of the positive classes, the query
will also need to calculate the distance with the negative support embeddings.
The comparison between positive and negative segments will provide the model
with more contrastive information on building the latent space. Note that we
do not optimize the probabilities of the negative query prototypes, because
negative samples are not guaranteed to have the same sound for classification.

###  3.3 Transductive inference

We adopt a transductive inference [16] approach during training. Specifically,
our model will be optimized both on the development set and the partially
labeled evaluation set. However, the labeled segment in the evaluation set is
much shorter than in the training set. To balance between these two sets, we
uniformly sample $N$ sound class from both sets during training.

###  3.4 Loss function

We optimize $f_{\theta}$ by maximizing the probability that $x^{q}_{i}$
belongs to class $i$. Specifically, we first calculate the probability
distribution of $x^{q}_{i}$ among $N$ possible classes by Equation 2. Then we
maximize the probability that $x^{q}_{i}$ belongs to class $i$ by summing and
maximizing the diagonal elements of $\mathbf{D}$, given by

$\mathbf{D}_{i,j}=\sqrt{\Sigma((x^{q}_{i}-x^{s}_{j})^{2})},~{}\mathbf{\widetilde{D}}_{i,:}=\textrm{log}(\textrm{Softmax}(\mathbf{D}_{i,:})),$
(2)

$l=\textrm{argmax}_{\theta}(\Sigma(\mathbb{I}\odot\mathbf{\widetilde{D}})),$
(3)

where $l$ is the loss function, $\mathbb{I}$ is the identity matrix, $\odot$
stands for element-wise multiplication, and $\Sigma$ means summing all the
elements in the matrix.

###  3.5 Adaptive segment length

During the inference process, the audio file will be divided into segments
with adaptive segment lengths. The labeled positive segments will be used to
build positive prototypes. The segments after the last labeled positive event,
which are unlabeled segments, will be treated as the query set. Both the query
set and labeled negative segments can be used to build the negative
prototypes.

Sounds of different animal or bird species have drastically different lengths
of vocalization, ranging from 30 milliseconds to 5 seconds. To have robust
detection result, we set different window lengths and hop lengths for each
audio file based on $t_{\max}=\max(t_{1},...,t_{K})$, as shown in Table 1,
where $t_{1},...,t_{K}$ denotes the duration of the $K$ labelled positive
events. We set the hop length as one-third of the window length.

$t_{\max}$ (s) | [0,0.1] | (0.1,0.4] | (0.4,0.8] | (0.8,3.0] | (3.0,+inf)
---|---|---|---|---|---
Length | 8 | $t_{\max}$ |  $t_{\max}$ / 2 |  $t_{\max}$ / 4 |  $t_{\max}$ / 8
Table 1: The window length we use during segmenting the evaluation and
validation audio file for different values of $t_{\max}$.

###  3.6 Positive and negative prototypes

We calculate the positive prototype by simply averaging the embeddings of the
labeled positive segments because we assume positive events do not contain too
much variety. By comparison, building negative prototypes is more tricky for
the following challenges:

  1. 1.

Negative segments can contain many different kinds of sounds. So simply
averaging all the negative embeddings will result in a sub-optimal
representation of negative prototypes.

  2. 2.

The duration of labeled negative data is limited in some test files. In this
case, some negative events appear later in the query set may not be included
in the negative prototype.

Note that the baseline system uses the average embedding of the entire audio
as the negative prototype, under the assumption that the positive event is
sparse. However, this assumption is not true in most of the evaluation files.
Building a negative prototype in this way can lead to a degraded result.

To address the first challenges, we choose to run our evaluation six times,
each with different 30 randomly selected negative segment embeddings. And we
mean the predicted probabilities across time as the final predictions. In this
case, the negative prototype in each run can have a chance to represent
different sound classes.

For the second challenge, we propose a negative sample searching algorithm
(Figure 2) to find more negative segments to build robust negative prototypes.
The algorithm includes a frequency bins weighting step and a frequency pattern
matching step.

Figure 2: The visualization of the negative sample searching algorithm. In the
first row, the left picture is the log-mel spectrogram of an audio clip, and
the picture on the right side is the mel-frequency bins weight. The second row
shows the frequency pattern matching score. The segment with a score under the
red dashed line will be used as negative segment.

The frequency bins weighing operation aims to find the frequency band that is
most likely to contain the target sound event so that we can find the negative
event more accurately. This operation is based on two assumptions. First, we
assume that bioacoustic sound events for a certain class (e.g., owl
vocalization) do not have too many variations in frequency. Localizing the
most important frequency band can therefore assist the model to ignore
interference sounds in other frequency bands. Second, since onset and offset
time mark the start and end of a sound event, we assume that the frequency
bins that have more energy fluctuations before and after one positive event
are the frequency bands that contain target sound events. We calculate
frequency bin weight $w$ by calculating the energy difference between the
positive segments and the negative segments before and after these events, and
then $w$ is normalized between zero and one.

Then we perform frequency pattern matching to locate possible negative
samples. We first multiply the linear mel-spectrogram with the frequency bin
weights. Then we calculate the average of positive time steps as the frequency
pattern template. Finally, we calculate the scale-invariance signal-to-noise
ratio (SISNR) [17] between the template and each time step in the weighted
mel-spectrogram. If the result of a time step is smaller than a threshold, we
will treat it as a negative event. The threshold is calculated using the
minimum SISNR value between the template and positive segments. Here we use
the scale-invariance version of SNR to overcome the situation where the energy
of a positive event changes.

###  3.7 Post processing

We determine the post processing strategy of a sound class based on maximum
length of positive event $t_{\max}$ and minimum length of negative event
$\widetilde{t}_{\min}=\min(\widetilde{t}_{1},...,\widetilde{t}_{K})$, where
$\widetilde{t}_{1},...,\widetilde{t}_{K}$ is the durations of the $K$ labelled
negative segments. We perform the following steps for post processing. Note
that most parameters in the following steps are chosen by experience.

$\widetilde{t}_{\min}~{}(s)$ | [0, 8] | (8, 100] | (100, +inf)
---|---|---|---
Length | 8 | $\widetilde{t}_{\min}/2$ | 100
Table 2: The window length we use during short negative segment detection for different values of $\widetilde{t}_{\min}$. | HB | ME | PB | Overall
---|---|---|---|---
System | Pre (%) | Rec (%) | F (%) | Pre (%) | Rec (%) | F (%) | Pre (%) | Rec (%) | F (%) | Pre (%) | Rec (%) | F (%)
Liu_S1 | 98.87 | 79.31 | 88.01 | 78.95 | 86.54 | 82.57 | 48.18 | 28.70 | 35.97 | 68.90 | 50.84 | 58.51
Liu_S2 | 87.03 | 83.08 | 85.01 | 65.00 | 50.00 | 56.52 | 36.13 | 30.00 | 32.78 | 54.99 | 45.89 | 50.03
Liu_S3 | 94.50 | 59.67 | 73.15 | 65.85 | 51.92 | 58.06 | 35.40 | 17.39 | 23.32 | 55.54 | 32.08 | 40.67
Liu_S4 | 97.95 | 79.46 | 87.74 | 86.27 | 84.62 | 85.44 | 57.52 | 27.66 | 37.36 | 76.56 | 49.54 | 60.16
Liu_S0 | 82.20 | 82.33 | 82.26 | 95.56 | 82.69 | 88.66 | 59.88 | 42.17 | 49.49 | 76.28 | 62.56 | 68.74
Table 3: The precision, recall, and f-measure of each subset in the validation
set. S1,S2,S3,S4 are four systems we finally submitted to the challenge. We do
not submit S0 on considering that the model might overfit to the validation
dataset.

After detecting the onset and offset of short negative segments, we perform
splitting on the positive segment. This operation is only performed when the
length of the detected positive event is longer than $2t_{\max}$ and a
negative segment is detected within the positive event.

We determine the threshold segment length of a sound class with the duration
of the few labeled positive and negative segments $\mathbf{t}$ and
$\mathbf{t^{\prime}}$. During filtering, the minimal segment length is
$0.4*\bar{\mathbf{t}}$. The merging operation will be performed if two
segments have a total duration less than $0.8*\bar{\mathbf{t}}$ and the
negative segment between them is shorter than the minimal duration in
$\mathbf{t^{\prime}}$. The system will consider to perform splitting operation
if the segment length is longer than $2.0*\bar{\mathbf{t}}$. And we design the
negative segment mining approach to determine where to perform the splitting.

The essential reason why the model cannot locate the negative segment between
two adjacent positive sound events is because of the length of the query
window. There is a trade-off during inference. If the window length is not
small enough, the query sample could contain positive and negative samples at
the same time, and the negative part will not be effectively detected.
However, if the window length is too small, the model tends to have too many
false positives that will affect the final score.

To address this problem, we propose negative segment mining operations, which
means detecting the possible negative segment. Since the false positive is not
important in this case, we can use a small query window length that is capable
of detecting small intervals between two positive sound events. The detected
negative intervals will be used for splitting long positive segments if the
segment length is longer than $2.0*\bar{\mathbf{t}}$.

##  4 Experiments

###  4.1 Dataset

Challenge official dataset DCASE 2022 task 5 dataset contains a training set,
a validation set, and an evaluation set. The training and validation set are
both fully labeled. The evaluation set is provided only with the labels of the
first five positive events.

AudioSet AudioSet [10] is a large-scale dataset for audio research [12, 18].
The training set of the DCASE 2022 task 5 challenge only contains 47 different
sound classes. To enhance the latent space with a wider variety of sounds, we
choose to build an extra training set using the strongly labeled part of the
AudioSet dataset. To avoid the domain mismatch problem, we only utilize the
sound labels that are related to animal vocalizations. We only use animal
sounds that do not overlap with other non-animal sounds. We will not use the
data if the total duration of non-overlapping parts is less than two seconds
within the audio because we consider in this case the sound event is not
enough to use.

###  4.2 Evaluation metric

We use the f-measure score, the official evaluation
metric222https://github.com/c4dm/dcase-few-shot-bioacoustic provided by the
organizers of DCASE task 5, as our main evaluation metric. We also report the
accuracy and recall.

###  4.3 Experimental setup

Following the prior works [4], all the audio data are resampled into 22.5 kHz
sampling rate. The input feature of our system is the stack of PCEN [19] and
delta-MFCC [20] features. In the short-time Fourier transform, we set the
window length as 1024 and hop size as 256. We set the mel dimension as 128.
The input length of our model during training is 0.2 seconds, and the output
embedding dimension is 2048. All the experiments use an initial learning rate
of 0.001 with 0.65 exponential decay every 10 epochs. We will stop model
training if the validation accuracy does not improve for 10 consecutive runs.
And the model with the best validation accuracy is used for evaluation. For
model evaluation, we consider the negative segments with a total duration of
fewer than 2.0 seconds are not enough to build the negative prototype. And in
this case, we will use the negative sample searching algorithm mentioned in
Section 3.6.

Different from the baseline that preprocesses and segments the training data
with a certain hop length, we implement a dynamic approach that generates
training data on the fly. During training, we will randomly select the
starting point of the training segment. In this case, the training data can be
fully utilized.

##  5 Result

Method | Precision (%) | Recall (%) | F-measure (%)
---|---|---|---
Template Matching | 2.42 | 18.32 | 4.28
Prototypical [8] | 36.34 | 24.96 | 29.59
Proposed | 76.28 | 62.56 | 68.74
Table 4: Comparison with baseline template matching and prototypical network
methods

The performance of our system on the validation set is reported in Table 4.
The F-measure score of the prototypical network baseline is 29.59. Our system
outperforms the baseline by a large margin with an F-measure score of 68.74,
using a threshold of 0.95. We submit four systems for the DCASE task 5
challenge. The performance of each system on each validation subset is
reported in Table 3. We select systems with different performances to avoid
cherry-picking models that only work well on the validation set. All four
systems use the same configurations except the settings shown in Table 5. We
tried different post-processing methods and training data in these four
systems.

System | Pad | Split | Merge | AudioSet | Threshold | Ens
---|---|---|---|---|---|---
Liu_S1 | ✗ | ✓ | ✗ | ✗ | 0.5 - 0.995 | ✓
Liu_S2 | ✓ | ✗ | ✗ | ✓ | 0.6 | ✗
Liu_S3 | ✓ | ✓ | ✓ | ✗ | 0.95 | ✗
Liu_S4 | ✗ | ✗ | ✗ | ✗ | 0.6 | ✗
Table 5: The setting of each submitted system. Ens stand for performing
ensamble on different thresholds (0.5,0.7,0.9,0.99,0.995).

##  6 Conclusions

This technical report describes the system we submitted to the DCASE 2022
challenge task 5. Our system use segment-level metric learning with negative
segments, negative segment searching, and post-processing. The experimental
result indicates our system can improve the baseline prototypical network by a
large margin.

##  7 ACKNOWLEDGMENT

This research was partly supported by a PhD scholarship from the Centre for
Vision, Speech and Signal Processing (CVSSP), Faculty of Engineering and
Physical Science (FEPS), University of Surrey and BBC Research and
Development, and a Research Scholarship from the China Scholarship Council
(CSC) No. 202006470010. For the purpose of open access, the authors have
applied a Creative Commons Attribution (CC BY) licence to any Author Accepted
Manuscript version arising.

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# Schwinger pair production in inhomogeneous electric fields with symmetrical
frequency chirp

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

###### Abstract

Pair production in inhomogeneous electric fields with symmetrical frequency
chirp is studied numerically using the Dirac-Heisenberg-Wigner formalism. We
investigate high- and low-frequency modes and consider two carrier envelope
phases. Momentum spectrum is sensitive to chirp causing different interference
effect for different spatial scales as well as the carrier phase of the
external field. The reduced particle number is in general enhanced with
increasing chirp. The effect of spatial scale of the field on the reduced
particle number is also examined. It is found that it is enhanced at small
spatial scale but is almost unchangeable at large spatial scales for the
considered field parameters. On the other hand, at small spatial scale, the
reduced particle number is enhanced by one or two orders when chirp is applied
with the exception of cosine low-frequency field which is only a few times
larger. Moreover it is found that the reduced particle number is further
increased by symmetrical chirp at about two times by comparing to the usual
asymmetrical chirp in high frequency field.

###### pacs:

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

##  I Introduction

Electron-positron pair creation from vacuum in intense electromagnetic fields,
known as the Schwinger effect, is a nonperturbative phenomena in quantum
electrodynamics (QED) challenging both theoretically and experimentally
Sauter:1931zz ; Heisenberg:1935qt ; Schwinger:1951nm . As is identified by
Sauter and later formulated by Schwinger within QED, the quantum vacuum decays
into real particles through tunneling when the external field strength reaches
the critical value $E_{\rm{cr}}={m_{e}^{2}c^{3}}/{e\hbar}\approx 1.3\times
10^{18}{\rm V}/{\rm m}$Schwinger:1951nm . However, the corresponding critical
field intensity $I_{\rm{cr}}=4.3\times 10^{29}{\rm W}/{\rm cm^{2}}$ is yet
demanding for the current laser technology and the Schwinger effect still
awaits an experimental verification. Recent theoretical investigations
indicate that vacuum pair production can be achieved through various
enhancement mechanisms via careful shaping of the laser pulses with
intensities one or two orders below the critical value Schutzhold:2008pz ;
Bell:2008zzb ; DiPiazza:2009py ; Bulanov:2010ei , which brings hope for the
detection of the vacuum pair creation with the advanced laser facilities in
the near future Ringwald:2001ib ; Heinzl:2008an ; Marklund:2008gj ;
Pike:2014wha .

Pair creation process, especially the particle momentum distribution, is very
sensitive to external field parameters because of its nonlinear nature
Hebenstreit:2009km ; Dumlu:2010vv . The momentum spectrum of the created pairs
may provide detail about pair creation dynamics as well as the external field
shapeAkkermans:2011yn ; Dumlu:2010ua ; Dumlu:2011rr . Many research focus on
temporal electric fields through changing of one or more field
parametersXie:2017 ; Li:2015cea ; Li:2017qwd ; Hebenstreit:2009km ;
Dumlu:2010vv ; Abdukerim:2013vsa ; Olugh:2018seh , such as field frequency
Olugh:2018seh , carrier phase Abdukerim:2013vsa , and analyze effects of these
arguments on the momentum spectrum of created particles.

Moreover, it is also well known that the chirp plays an important role in
strong field QED. For example, high intensity laser pulse itself, which we
need it for the study of pair production, are obtained by chirped pulse
amplification (CPA) technique Strick  in laboratories. Chirp effects on pair
production have been widely studied in the homogeneous fields Dumlu:2010vv ;
Olugh:2018seh . To our knowledge, current research has been limited on the
asymmetrical chirp effects in the spatially inhomogeneous field set up
Ababekri:2020 . Recently, we have studied symmetrical chirp effects in
homogeneous electric field Wang .

On the other hand, interest for the effect of spatial inhomogeneity on the
vacuum pair production is triggered by the advances in theoretical works and
the numerical computation techniques Hebenstreit:2011wk ; Kohlfurst:2017hbd ;
Ababekri:2019dkl ; Ababekri:2020 ; Kohlfurst:2017git . These works have
provided new features of finite spatial field pair production, such as self-
bunching effect Hebenstreit:2011wk , ponderomotive force Kohlfurst:2017hbd ,
spin field interaction Kohlfurst:2017git  and so on. Recently, Kohlfürst has
studied vacuum pair production in strong electromagnetic field, by considering
both of electric and magnetic field with spatial inhomogeneity and has
revealed the competitive results between the multiphoton and tunneling process
Kohlfurst:2020 ; Kohlfurst:2017git . All these findings emphasize the
importance of considering spatial variations for vacuum pair production in the
strong external field.

In this paper, we study the vacuum pair creation in symmetrical chirped fields
at finite spatial scales using the numerical Dirac-Heisenberg-Wigner (DHW)
method Kohlfurst:2015zxi . We consider oscillating field forms in cosine and
sinusoidal form. We find that momentum spectrum is very sensitive to
symmetrical chirp which exhibits different interference effect for different
spatial scales as well as the carrier phase of field. The reduced particle
number is in general enhanced with increasing chirp. In particular, at small
spatial scale, the reduced particle number is enhanced at most by one or two
orders when chirp is applied. Moreover it is found that the reduced particle
number is further increased by symmetrical chirp at about two times compared
to asymmetrical chirp. We use natural units ($\hbar=c=1$) and express all
quantities in terms of the electron mass $m$.

##  II Model of Background Field

We consider the following spatially inhomogeneous 1+1 dimensional electric
field model

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

where $\epsilon E_{cr}$ is the field strength, $\omega$ is the original center
frequency, $\lambda$ and $\tau$ are the spatial and temporal scales, $b$ is
chirp parameter which introduces a linear variation in the frequency as time
changes and $\varphi$ is the carrier phase. For high frequency field, we
choose frequency $\omega=0.7m$, temporal pulse length $\tau=45m^{-1}$, while
we choose $\omega=0.1m$, $\tau=25m^{-1}$ for low frequency field. For nonzero
chirp parameter $b$, it results in time dependent effective frequency
$\omega_{\text{eff}}(t)=\omega+b|t|$ that it is always larger than field
frequency in the relevant time interval $-\tau\leq t\leq\tau$. In this paper,
the values of chirp $b$ are expressed as the form $b=\alpha\omega/\tau$
($\alpha\geq 0$), we set the theoretical upper limit for the chirp parameter
by letting effective frequency to be around the critical frequency
$\omega_{\text{eff}}(\tau)\sim 1.0m$ for high frequency field. In the low
frequency field, the maximum chirp value is determined by the condition
$\omega_{\text{eff}}(\tau)\tau\sim\mathcal{O}(1)$. These conditions have been
discussed in previous works Ababekri:2020 ; Kohlfurst:2017git .

This idealized standing wave model (1) can be regarded to be constructed by
two counter propagating laser pulses with the needed symmetrical frequency
chirping. Because pair production rate is highest in the vicinity of the focus
of two crossed laser beams and that the effect of the magnetic field is
automatically vanished here. Since the direction of the electric field is
along the $x$\- axis and the particle is created in the field direction, the
study can be simplified as $p_{\perp}=0$.

##  III The DHW formalism

The DHW formalism is one of the quantum kinetic approaches that is widely
adopted to study vacuum pair production in multidimensional electromagnetic
fields Ref. Kohlfurst:2015zxi ; Hebenstreit:2011pm . It enables one to obtain
phase space information about the particles created in vacuum, so that much
can be learned about Schwinger pair production process. However, the
deficiency of this method is the complicated computational process because one
has to solve a set of multidimensional partial differential equations.
Therefore, we choose to rely on numerical simulation method used in the
previous work Kohlfurst:2015zxi .

We focus on pair production in time dependent inhomogeneous electric fields
given in Eq. (1) such that electric fields are inhomogeneous on $x$\- axis,
and the complete DHW equations of motion are reduced to the following $4$
differential equations Kohlfurst:2015zxi

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

with pseudodifferential operator

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

In order to perform the calculation, it is necessary to choose vacuum initial
condition values, we employ them Kohlfurst:2015zxi

${\mathbbm{s}}_{vac}=-\frac{2m}{\Omega}\,,\quad{\mathbbm{v}}_{1\,vac}=-\frac{2{p_{x}}}{\Omega}\,,$
(7)

where $\Omega$ is the energy of single particle and defined as
$\Omega=\sqrt{p_{x}^{2}+m^{2}}$. By subtracting these initial vacuum terms, we
can obtain modified Wigner components

$\mathbbm{w}_{k}^{v}\left(x,p_{x},t\right)=\mathbbm{w}_{k}\left(x,p_{x},t\right)-\mathbbm{w}_{vac}\left(p_{x}\right),$
(8)

where $\mathbbm{w}_{k}$ is the Wigner component in Eqs. (2)-(5) (corresponding
to: $\mathbbm{w}_{0}=\mathbbm{s}$ , $\mathbbm{w}_{1}=\mathbbm{v}_{0}$ ,
$\mathbbm{w}_{2}=\mathbbm{v}_{1}$ and $\mathbbm{w}_{3}=\mathbbm{p}$), and
$\mathbbm{w}_{vac}$ is the corresponding vacuum initial condition Eq. (7).
Total energy of the Dirac particle is
$\varepsilon\left(x,p_{x},t\right)={m\mathbbm{s}^{v}\left(x,p_{x},t\right)+p_{x}\mathbbm{v}_{1}^{v}\left(x,p_{x},t\right)}$,
and the particle number density in phase space can be written as
Hebenstreit:2011wk

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

where ${\Omega\left(p_{x}\right)}$ is energy of the individual particle.
Therefore, the momentum distribution function can be obtained by integration
of Eq. (9) with respect to position

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

Also the total number of created particles could be written

$N\left(t\right)=\int dx\,dp_{x}n\left(x,p_{x},t\right).$ (11)

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

##  IV Numerical results

In this paper, we study the symmetrical frequency chirp effects for both high
frequency and low frequency inhomogeneous fields. We consider two carrier
phase parameters $\varphi=0$ and $\varphi=\pi/2$ representing cosine- and
sine-type oscillating forms respectively. For the convenience of comparison,
we choose similar parameter values as in Ref. Ababekri:2020  where the usual
asymmetrical chirp is studied.

###  IV.1 High frequency field

For high frequency field, we set $\omega=0.7m$, $\tau=45m^{-1}$ for which the
electron-positron pair is predominantly produced by multiphoton absorbtion.
For $b=0$, the external field has a constant frequency
$\omega_{\text{eff}}(t)=\omega$ and we would recover the result in Fig. 1 of
Ref. Ababekri:2020  where the decreasing spatial scale affects the discrete
$n$-photon absorption spectrum Kohlfurst:2017hbd ; Ababekri:2020 .

####  IV.1.1 Momentum distribution when $\varphi=0$

The symmetrical chirp effects on momentum spectrum at different spatial scales
are shown in Fig. 1. At the large spatial scale, $\lambda=1000m^{-1}$, the
momentum spectrum is very sensitive to changing of chirp parameter. Weak
oscillation can be observed even for small chirp, see Fig. 1(a) and (b).
Strong oscillation occurs in the momentum spectrum as chirp increases, see
Fig. 1(c) and (d). For nonzero chirp, the effective frequency
$\omega_{\text{eff}}(t)=\omega+b|t|$ increases with chirp and provides more
energy to create more particles. Consequently, these particles interact with
each other can lead the more obvious interference effect on the momentum
spectrum, this result is similar to that of homogeneous field.

Figure 1: (color online). Reduced momentum spectrum for various chirp values
in high frequency field with different spatial scales. The chirp values are
$b=0.05\omega/\tau\approx 0.00078m^{2}$, $b=0.1\omega/\tau\approx
0.0016m^{2}$, $b=0.2\omega/\tau\approx 0.0031m^{2}$ and
$b=0.5\omega/\tau\approx 0.0078m^{2}$. Other field parameters are
$\epsilon=0.5$, $\omega=0.7m$, $\tau=45m^{-1}$ and $\varphi=0$.

At the small spatial scales, $\lambda=2.5m^{-1}$ and $\lambda=10m^{-1}$, for
small chirp, the oscillation is not obvious, but we observed peak splitting on
momentum spectrum, see Fig. 1(a) and (b). The peak splitting can be explained
by ponderomotive effects which have been reported in Ref. Kohlfurst:2017hbd .
The magnitude of ponderomotive force is inversely proportional to the size of
spatial scale Kohlfurst:2015zxi . Therefore the momentum peaks at spatial
scale $\lambda=2.5m^{-1}$ are more far away from the center compared with the
case of $\lambda=10m^{-1}$ in Fig. 1(a) and (b). However, for large chirp, the
peaks that split by the ponderomotive force are replaced by strong oscillation
as shown in Fig. 1(c) and (d). Because the effective frequency increases
obviously with the large chirp and provides greater contribution to frequency
of external field. So that corresponding effect of ponderomotive force
decreases and strong oscillation occurs on the momentum spectrum. This
oscillation can be understood as the interference effects between created
particles.

Figure 2: (color online). Reduced particle number dependence of spatial scales
for various chirp values in high frequency field. Other field parameters are
the same as in Fig. 1.

In Fig. 2, we also plot the reduced particle number dependence of spatial
scales for various chirp parameters. We can see that reduced particle number
is enhanced with the increasing chirp. This enhancement is most pronounced for
the largest chirp, see in Fig. 2. When chirp is relatively small, i.e., $b\leq
0.2\omega/\tau\approx 0.0031m^{2}$, reduced particle number is enhanced
rapidly as spatial scale increases. Because the electric field energy
increases with widening of spatial scale, particles created in the field
region also increase correspondingly. However, when chirp is very large, i.e.,
$b=0.5\omega/\tau\approx 0.0078m^{2}$, particle number is less affected by
spatial scales. It exhibits a relative flat line in a whole region of spatial
scale of field. At large spatial scales, particle number tends to be a
constant for each chirp parameters as spatial scale increases. At that time,
our field can be regarded as a spatial homogeneous field. We noted that,
compared with the case of $b=0$ (solid line), reduced particle number is
enhanced by one order of magnitude for largest chirp (dash-dot-dotted line).

For a convenient comparison between our results to that of the field with
asymmetrical frequency chirping case in Ref. Ababekri:2020 , it is intuitive
to see their difference for various chirp parameters at fixed spatial scale,
meanwhile, we can also see their difference for various spatial scales with
fixed chirp parameter. As two examples by fixed $\lambda=10m^{-1}$ and fixed
$b=0.5\omega/\tau=0.0078m^{2}$, the comparison data of reduced particle
numbers for our case and asymmetrical frequency chirping case is presented in
Table 1.

Table 1: The reduced particle numbers of symmetrical chirp in present study and asymmetrical chirp in Ref. Ababekri:2020  and the ratio of them, $\bar{N}_{sym}(\varphi=0)$/$\bar{N}_{asym}(\varphi=0)$, for different $b$ when $\lambda=10m^{-1}$ is fixed (the upper part of table), and for different $\lambda$ when $b=0.5\omega/\tau=0.0078m^{2}$ is fixed (the lower part of table). Field parameters are $\epsilon=0.5$, $\omega=0.7m$ and $\tau=45m^{-1}$. $b$ | $\bar{N}_{sym}(\varphi=0)$ | $\bar{N}_{asym}(\varphi=0)$ |  $\bar{N}_{sym}(\varphi=0)$/$\bar{N}_{asym}(\varphi=0)$
---|---|---|---
$0.05\omega/\tau\approx 0.00078m^{2}$ | $0.268$ | $0.1663$ | $1.612$
$0.1\omega/\tau\approx 0.0016m^{2}$ | $0.3274$ | $0.1561$ | $2.097$
$0.2\omega/\tau\approx 0.0031m^{2}$ | $0.3178$ | $0.1791$ | $1.774$
$0.5\omega/\tau\approx 0.0078m^{2}$ | $1.434$ | $0.7474$ | $1.919$
$\lambda$ | $\bar{N}_{sym}(\varphi=0)$ | $\bar{N}_{asym}(\varphi=0)$ | $\bar{N}_{sym}(\varphi=0)/\bar{N}_{asym}(\varphi=0)$
$2.5m^{-1}$ | $1.3639$ | $0.7059$ | $1.93$
$10m^{-1}$ | $1.434$ | $0.7474$ | $1.919$
$1000m^{-1}$ | $1.2601$ | $0.7268$ | $1.73$

Compared to the case of asymmetrical frequency chirp, the reduced particle
number is enhanced at about two times in our symmetrical chirp case. The
enhancement is most pronounced for $b=0.1\omega/\tau\approx 0.0016m^{2}$ and
$\lambda=2.5m^{-1}$.

####  IV.1.2 Momentum distribution when $\varphi=\pi/2$

Figure 3: (color online). Reduced momentum spectrum for various chirp
parameters in the high frequency field with different spatial scales. The
field parameters are the same as in Fig. 1 except $\varphi=\pi/2$.

When $\varphi=\pi/2$, we show the symmetrical chirp effects on momentum
spectrum at different spatial scales in Fig. 3. For small chirp, see Fig. 3(a)
and (b), we obtain almost the same results in both phase cases, see also Fig.
1(a) and (b). At large spatial scale $\lambda=1000m^{-1}$, however, some
delicate differences can be observed on momentum spectrum when $b$ is large,
see Fig. 3(c) and (d) in comparison with the results of $\varphi=0$ in Fig.
1(c) and (d). For example, when $b=0.2\omega/\tau\approx 0.0031m^{2}$, the
main peak value corresponding to $p=0$ is split by strong oscillation, see
Fig. 3(c). Moreover it reaches larger peak value
$\bar{n}(t\rightarrow\infty,p)=1.9m$ for $b=0.5\omega/\tau\approx
0.0078m^{2}$, see Fig. 3(d). On the other hand, however, at small spatial
scales $\lambda=2.5m^{-1}$ and $\lambda=10m^{-1}$, while we can still observe
oscillation on the momentum spectrum, there is no obvious increasing of peak
values compared with the former case of $\varphi=0$.

Figure 4: The ratio of reduced particle numbers
$\bar{N}(\varphi=\pi/2)$/$\bar{N}(\varphi=0)$ for different cases of chirp
$b\leq 0.2\omega/\tau\approx 0.0031m^{2}$ (solid line) and
$b=0.5\omega/\tau\approx 0.0078m^{2}$ (dashed line).

The ratio of reduced particle numbers
$\bar{N}(\varphi=\pi/2)$/$\bar{N}(\varphi=0)$ for different cases of chirp
$b\leq 0.2\omega/\tau\approx 0.0031m^{2}$ and $b=0.5\omega/\tau\approx
0.0078m^{2}$ are shown in Fig. 4. It is found that when $b\leq
0.2\omega/\tau\approx 0.0031m^{2}$, the ratio for different chirp is almost
equal to $1$, see the solid line of figure. We have checked it to be valid at
different spatial scales for chirp when $b=0$, $b=0.05\omega/\tau\approx
0.00078m^{2}$, $b=0.1\omega/\tau\approx 0.0016m^{2}$ as well as
$b=0.2\omega/\tau\approx 0.0031m^{2}$. On the other hand, however, when
$b=0.5\omega/\tau\approx 0.0078m^{2}$, the particle number
$\bar{N}(\varphi=\pi/2)$ is enhanced a little as spatial scale increases.

The asymmetrical chirp effect when $\varphi=\pi/2$ has not been considered in
Ref. Ababekri:2020 . For comparison, we also perform some numerical
calculations in the asymmetrical chirping field. The comparison data of
reduced particle numbers when $\varphi=\pi/2$ is given in Table 2. We find
that compared to the case of asymmetrical frequency chirp, the reduced
particle number is enhanced at about two times in our symmetrical chirping
case. The largest enhancement still occurs for $b=0.1\omega/\tau\approx
0.0016m^{2}$ and $\lambda=2.5m^{-1}$.

Table 2: The reduced particle numbers of symmetrical chirp in present study and asymmetrical chirp in Ref. Ababekri:2020  and the ratio of them, $\bar{N}_{sym}(\varphi=\pi/2)$/$\bar{N}_{asym}(\varphi=\pi/2)$, for different $b$ when $\lambda=10m^{-1}$ is fixed (the upper part of table), and for different $\lambda$ when $b=0.5\omega/\tau=0.0078m^{2}$ is fixed (the lower part of table). Other field parameters are the same in Table 1. $b$ | $\bar{N}_{sym}(\varphi=\pi/2)$ | $\bar{N}_{asym}(\varphi=\pi/2)$ |  $\bar{N}_{sym}(\varphi=\pi/2)$/$\bar{N}_{asym}(\varphi=\pi/2)$
---|---|---|---
$0.05\omega/\tau\approx 0.00078m^{2}$ | $0.2682$ | $0.1663$ | $1.61$
$0.1\omega/\tau\approx 0.0016m^{2}$ | $0.3279$ | $0.1561$ | $2.1$
$0.2\omega/\tau\approx 0.0031m^{2}$ | $0.3185$ | $0.1773$ | $1.8$
$0.5\omega/\tau\approx 0.0078m^{2}$ | $1.5702$ | $0.8304$ | $1.89$
$\lambda$ | $\bar{N}_{sym}(\varphi=\pi/2)$ | $\bar{N}_{asym}(\varphi=\pi/2)$ | $\bar{N}_{sym}(\varphi=\pi/2)/\bar{N}_{asym}(\varphi=\pi/2)$
$2.5m^{-1}$ | $1.3596$ | $0.7085$ | $1.92$
$10m^{-1}$ | $1.5702$ | $0.8304$ | $1.89$
$1000m^{-1}$ | $1.3825$ | $0.8124$ | $1.7$

###  IV.2 low frequency field

In this section, we choose $\omega=0.1m$ and $\tau=25m^{-1}$ to study
symmetrical chirp effects on the momentum spectrum and reduced particle number
in low frequency fields with $\varphi=0$ and $\varphi=\pi/2$.

####  IV.2.1 Momentum distribution when $\varphi=0$

Figure 5: (color online). Reduced momentum spectrum for various chirp
parameters in the low frequency field with different spatial scales. The chirp
values are $b=0$, $b=0.5\omega/\tau=0.002m^{2}$, $b=1.0\omega/\tau=0.004m^{2}$
and $b=1.5\omega/\tau=0.006m^{2}$. Other field parameters are $\epsilon=0.5$,
$\omega=0.1m$, $\tau=25m^{-1}$ and $\varphi=0$.

When $b=0$, see Fig. 5(a), momentum spectrum is same as in Fig. 6(a) of Ref.
Ababekri:2020 . At large spatial scale $\lambda=500m^{-1}$, when $b=0$ , a
very weak oscillation can be observed. For small chirp, we can see stronger
oscillation on the two sides of momentum spectrum in Fig. 5(b). For the large
chirp, very strong oscillation appears, meanwhile, momentum spectrum shrinks
and shifts to positive direction, see Fig. 5(c) and (d). These oscillations
can be understood as interference effect of particles created by opposite
signed large peaks of the temporal field Hebenstreit:2009km .

When the spatial scale is narrowed to $\lambda=10m^{-1}$, when $b=0$, it has
no oscillation on the momentum spectrum. For small chirp, we can observed a
weak oscillation on the left side, meanwhile, momentum spectrum is broadened
to negative field region, see dash-dotted line in Fig. 5(b). This is because
particles created with certain momentum leave the field region and miss the
deceleration by the negative field peak Ababekri:2020 . For large chirp, the
strong oscillation occurs on momentum spectrum, it is caused by the
interference effect of created particles from opposite field peaks, see Fig.
5(c) and (d).

When $\lambda=2m^{-1}$, for small chirp, we have not observed obvious
oscillation on the momentum spectrum. For large chirp, a weak oscillation
appeared, see dashed line in Fig. 5(c) and (d). Because, the spatial scale is
very small, the work done by the electric field is reduced correspondingly.
Consequently, particles created by the electric field are also decreased, so
that no pronounced interference can be observed on momentum spectrum.

In the low frequency field, we also make a comparison between the reduced
particle number with our case and that with asymmetrical frequency chirp. We
show the comparison data of reduced particle numbers in Table 3. Compared to
the case of asymmetrical frequency chirp, the reduced particle number is
almost unchanged in our symmetrical chirping case.

Table 3: The reduced particle numbers of symmetrical chirp in present study and asymmetrical chirp in Ref. Ababekri:2020  and the ratio of them, $\bar{N}_{sym}(\varphi=0)$/$\bar{N}_{asym}(\varphi=0)$, for different $b$ when $\lambda=2m^{-1}$ is fixed (the upper part of table), and for different $\lambda$ when $b=0.5\omega/\tau=0.002m^{2}$ is fixed (the lower part of table). Field parameters are $\epsilon=0.5$, $\omega=0.1m$, and $\tau=25m^{-1}$. $b$ | $\bar{N}_{sym}(\varphi=0)$ | $\bar{N}_{asym}(\varphi=0)$ | $\bar{N}_{sym}(\varphi=0)/\bar{N}_{asym}(\varphi=0)$
---|---|---|---
$0$ | $0.0018$ | $0.0018$ | $1$
$0.5\omega/\tau=0.002m^{2}$ | $0.0018$ | $0.0018$ | $1$
$1.0\omega/\tau=0.004m^{2}$ | $0.0020$ | $0.0019$ | $1.052$
$1.5\omega/\tau=0.006m^{2}$ | $1.0025$ | $0.0022$ | $1.136$
$\lambda$ | $\bar{N}_{sym}(\varphi=0)$ | $\bar{N}_{asym}(\varphi=0)$ | $\bar{N}_{sym}(\varphi=0)/\bar{N}_{asym}(\varphi=0)$
$2m^{-1}$ | $0.0018$ | $0.0018$ | $1$
$10m^{-1}$ | $0.0135$ | $0.0140$ | $0.96$
$500m^{-1}$ | $0.0144$ | $0.0149$ | $0.97$

####  IV.2.2 Momentum distribution when $\varphi=\pi/2$

The symmetrical chirp effects on momentum spectrum at different spatial scales
with $\varphi=\pi/2$ are shown in Fig. 6 where the remarkable difference in
the momentum spectrum compared with that of $\varphi=0$ shown in Fig. 5 could
be noticed. When $b=0$, see Fig. 6(a), momentum spectrum is same as in Fig.
8(a) of Ref. Ababekri:2020 .

Figure 6: (color online). Reduced momentum spectrum for various chirp values
in the low frequency field with different spatial scales. The field parameters
are the same as in Fig. 5 except $\varphi=\pi/2$.

At large spatial scale $\lambda=500m^{-1}$, when $b=0$, we can see the bell-
shaped momentum spectrum in the negative region with oscillation which can be
understood as the interference effect between two separated pair production
events that happen at different times Hebenstreit:2009km , see Fig. 6(a). For
small chirp, momentum spectrum shrinks and shifts to positive direction, see
solid line in Fig. 6(b). For large chirp, the bell-shaped momentum spectrum is
replaced by strong oscillation, see Fig. 6(c) and (d). In the next section, we
will discuss some of these oscillation effects in the semiclassical picture.

When the spatial scale is decreased to $\lambda=10m^{-1}$, chirp effects on
momentum spectrum become stronger than that of $\varphi=0$. For $b=0$, pair is
created by two opposite signed field peaks and leave field region in two
directions Ababekri:2020 , so that form the two peaks on momentum spectrum,
see Fig. 6(a) which is same as Fig. 8(a) of Ref. Ababekri:2020 . For small
chirp, i.e., $b=0.5\omega/\tau=0.002m^{2}$, the two momentum peaks are still
remained, it is different from the result for asymmetrical chirped field where
one peak vanishes when the same $b=0.5\omega/\tau=0.002m^{2}$ is applied, see
Fig. 8(b) in Ref. Ababekri:2020 . However, for large chirp, these peaks are
replaced by interference pattern of created particles, see Fig. 6(c) and (d).

At the small spatial scale $\lambda=2m^{-1}$, two momentum peaks can also be
observed, but the peak values are very small compared with the case of
$\lambda=10m^{-1}$. We think it is because that, in the small spatial scale
$\lambda=2m^{-1}$, the work done by the electric field would be so greatly
reduced and very few of the particles are created through tunneling.

Figure 7: (color online). Reduced particle number dependence of spatial scales
for various chirp parameters in the low frequency field with different carrier
phase parameter. The carrier phase parameters are $\varphi=0$ for (a) and
$\varphi=\pi/2$ for (b), respectively. Other field parameters are the same as
in Fig. 5.

Now let us to see the reduced particle number dependence of spatial scales for
various chirp parameters, shown in the Fig. 7. When $\varphi=0$, the reduced
particle number is not sensitive to small chirp. For small chirp
$b=0.5\omega/\tau$, for instance, the created particle number is almost
similar to the case of $b=0$ without chirping, see the solid and dashed lines
in Fig. 7(a). However the reduced particle number enhancement occurs for large
chirp. Moreover, when $\varphi=\pi/2$, compared with the case of $\varphi=0$,
particle number enhancement is more sensitive to increasing chirp. We can
observe an obvious enhancement on the reduced particle number even for the
small chirp, see Fig. 7(b). At small spatial scales, for both of two cases,
the reduced particle number is enhanced rapidly as spatial scale enlarges.
Compared to the case of $b=0$ without chirping, the reduced particle number is
increased by two orders of magnitude for largest chirp when $\varphi=\pi/2$,
see solid and dotted line in Fig. 7(b). At large spatial scale, reduced
particle number tends to be a constant no matter how spatial scale enlarges.
Furthermore, for the same chirp, the reduced particle number in the field with
carrier phase $\varphi=\pi/2$ is always smaller than that of the field with
$\varphi=0$.

When $\varphi=\pi/2$, we also compare our results to that of asymmetrical
frequency chirp in Ref. Ababekri:2020  and the comparison data of reduced
particle numbers for two cases is shown in Table 4.

Table 4: The reduced particle numbers of symmetrical chirp in present study and asymmetrical chirp in Ref. Ababekri:2020  and the ratio of them, $\bar{N}_{sym}(\varphi=\pi/2)$/$\bar{N}_{asym}(\varphi=\pi/2)$, for different $b$ when $\lambda=2m^{-1}$ is fixed (the upper part of table), and for different $\lambda$ when $b=0.5\omega/\tau=0.002m^{2}$ is fixed (the lower part of table). Other field parameters are the same in Table 3. $b$ | $\bar{N}_{sym}(\varphi=\pi/2)$ | $\bar{N}_{asym}(\varphi=\pi/2)$ | $\bar{N}_{sym}(\varphi=\pi/2/\bar{N}_{asym}(\varphi=\pi/2)$
---|---|---|---
$0$ | $1.88\times 10^{-4}$ | $1.88\times 10^{-4}$ | $1$
$0.5\omega/\tau=0.002m^{2}$ | $6.26\times 10^{-4}$ | $3.15\times 10^{-4}$ | $1.987$
$1.0\omega/\tau=0.004m^{2}$ | $1.1\times 10^{-3}$ | $5.61\times 10^{-4}$ | $1.96$
$1.5\omega/\tau=0.006m^{2}$ | $1.9\times 10^{-3}$ | $9.26\times 10^{-4}$ | $2.05$
$\lambda$ | $\bar{N}_{sym}(\varphi=\pi/2)$ | $\bar{N}_{asym}(\varphi=\pi/2)$ | $\bar{N}_{sym}(\varphi=\pi/2)/\bar{N}_{asym}(\varphi=\pi/2)$
$2m^{-1}$ | $6.26\times 10^{-4}$ | $3.15\times 10^{-4}$ | $1.987$
$10m^{-1}$ | $0.0079$ | $0.0049$ | $1.612$
$500m^{-1}$ | $0.0085$ | $0.0053$ | $1.603$

This table clearly demonstrates that the reduced particle number in our
symmetrical frequency chirping field is increased about two times of
asymmetrical frequency chirp case. The largest enhancement occurs for
$b=1.5\omega/\tau=0.006m^{2}$ and $\lambda=2m^{-1}$.

##  V Discussions

The symmetrical chirp effects on the momentum spectrum and reduced particle
number can also be discussed in terms of the turning point structure of
potential in semiclassical regime by employing complex Wentzel-Kramers-
Brillouin (WKB) scattering approach. The scattering potential is determined by
solving the equation $\omega_{\bf
p}(t)=\sqrt{m^{2}+p_{\perp}^{2}+(p_{x}-eA(t))^{2}}=0$ for the turning points
which appear as complex conjugate pairs when the potential is real
Dumlu:2010vv . When a single complex turning point dominates, the WKB results
for the particle creation rate is given as heading ; landau

$N\approx e^{-2K},\qquad K=\left|\int_{t_{1}}^{t_{2}}\omega(t)\,dt\,\right|,$
(12)

where $t_{1}$ and $t_{2}$ are the dominant (closest to the real axis) turning
points of the corresponding scattering potential. In the low frequency field
with symmetrical chirp, more complex conjugate pairs appear with increasing
chirp as demonstrated for various chirp values in Fig. 8 for $p_{\perp}=0$.

Figure 8: (color online). Contour plots of $|\omega_{\bf p}(t)|^{2}$ in the
complex $t$ plane, showing the turning point distribution where $\omega_{\bf
p}(t)=0$ for $p_{\perp}=0$, $p_{x}=0$. The black dots are turning point for
the case of $b=0$ without chirping, the red dots from left to right are the
complex conjugate turning points for various chirp parameter:
$b=0.5\omega/\tau=0.002m^{2}$(left panel),
$b=1.0\omega/\tau=0.004m^{2}$(center panel) and
$b=1.5\omega/\tau=0.006m^{2}$(right panel), respectively. Other field
parameters are $\epsilon=0.5$, $\omega=0.1m$, $\tau=25m^{-1}$ and
$\varphi=\pi/2$.

The multiple turning point generalizations of the pair production rate Eq.
(12) is

$N\approx\sum_{i=1}^{n}e^{-2K_{i}}\pm\sum_{i\neq
j}^{n}2\cos(2\alpha)\,e^{-K_{i}-K_{j}},$ (13)

where $+$ and $-$ signs represent the scalar and spinor cases respectively.
The other quantities are defined as

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle
L-\sum_{i=1}^{n}\sigma(K_{i}),$ $\displaystyle L$ $\displaystyle=$
$\displaystyle\left|{\mathcal{R}e}\left(\int_{t_{2}}^{t_{3}}\omega(t)\,dt\right)\right|,$
$\displaystyle\sigma(K)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{K}{\pi}\left(\ln\left(\frac{K}{\pi}\right)-1\right)+{\rm
Arg}\,\Gamma\left(\frac{1}{2}-i\frac{K}{\pi}\right)\right].$

The second term in Eq. (13) is the interference term and is responsible for
the oscillations observed on the momentum spectrum. The turning point pair
numbers are increased with increasing chirp (see Fig. 8) so that one can
observe stronger interference effect on momentum spectrum for large chirp in
Fig. 6(c) and (d). On the other hand, the symmetry of turning point is
destroyed by increasing chirp, which results in an asymmetry more and more on
the momentum spectrum in Fig. 6(c) and (d). In addition, the dominant
contribution to pair production rate comes from the terms of involving turning
points which are closest to the real axis and may cause enhancement. In Fig.
8, we can also observe that the central pair of turning points (central pair
on the right side) are more closest to real axis with increasing chirp.
Therefore the corresponding pair production rate in Eq. (13) is always
enhanced with the increasing chirp.

There exists various (competing) pair creation mechanisms in our numerical
results corresponding to different chirp values. In low frequency field
without chirping, pair production process is dominated by the tunneling
mechanism. With increasing chirp, the field frequency contains many high-
frequency components which are higher than its own original center frequency.
In this way, the symmetrical chirp of low frequency field have played a role
of the dynamically assisted Sauter-Schwinger mechanism such that it can
produce more particles. In high frequency field, however, multiphoton
absorption overtakes the process and the effect of dynamically assisted
mechanism would become very weak. From this point of view, the chirp effect
may be more helpful for pair production in the low frequency field.

##  VI Summary

We have numerically studied the symmetrical chirp effects on the momentum
spectrum and total particle number in two oscillating forms (cosine and sine)
with spatial inhomogeneity. We have examined impacts of different field
parameters, such as field central frequency and its chirp, spatial scale, and
oscillating form on pair creation. Our results are summarized briefly as
follows.

For high frequency field, the interference effect in the momentum spectrum is
strengthen as chirp increases, and the total particle number is also enhanced
greatly with increasing chirp. Also, the enhancement due to large chirp is
more pronounced at smaller spatial scales and the momentum spectrum is more
sensitive to the chirp in the cosine oscillation compared to the sinusoidal
one.

For low frequency field, tunneling dominates the particle creation process and
chirp plays crucial role. Small chirp has stronger interference effect on the
momentum spectrum at larger spatial scales, while large chirp always causes
obvious interference effects regardless of the spatial scale. It is noticed
that the particle number increases much as chirp becomes large and, at small
spatial scale, is enhanced by two orders of magnitude for large chirp.

More importantly, by comparing with the result of the usual asymmetrical
frequency chirp, we have found that the total particle number is about two
times larger in the asymmetrical chirp case with the exception of the low
frequency cosine oscillation form where it is almost the same.

These results indicate that the chirp symmetry and carrier phase play very
important role on pair production in spatially inhomogeneous electric fields.
While we have only considered two typical cases of the carrier phase, we
speculate that more detailed analysis on the phase effect with symmetrical
chirp may reveal further novel aspects about Schwinger pair creation.

###### Acknowledgements.

We thank M. Ababekri for critical reading of the manuscript. We are also
grateful to Li Wang and N. Abdukerim for fruitful discussions and their help
with the numerical calculation. 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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