# Distributed Transfer Learning with 4th Gen Intel® Xeon® Scalable Processors

Lakshmi Arunachalam, Fahim Mohammad, Vrushabh H. Sanghavi
{lakshmi.arunachalam, fahim.mohammad<EMAIL_ADDRESS>AI
Framework Engineer
Data Center and Artificial Intelligence
Intel Corporation

###### Abstract

In this paper, we explore how transfer learning, coupled with Intel® Xeon®,
specifically 4th Gen Intel® Xeon® scalable processor, defies the conventional
belief that training is primarily GPU-dependent. We present a case study where
we achieved near state-of-the-art accuracy for image classification on a
publicly available Image Classification TensorFlow dataset using Intel®
Advanced Matrix Extensions(AMX) and distributed training with Horovod.

_Keywords_ Artificial Intelligence, Deep Learning Optimization, End-to-End AI
applications, E2E performance optimization, Transfer Learning, Intel® Xeon®

##  1 Introduction

Imagine how kids learn to start coloring with crayons. It may take few days
for them to learn how to hold the crayon, stay within the picture and so on.
They may need lots of crayons and coloring books so that they can get the hang
of it. Then they can easily apply their skills to learn how to use color
pencils, painting, pencil shading or master art. They don’t have to start all
over again from scratch because they already have a foundation of coloring
with crayons. This is what transfer learning is about. Instead of starting
from scratch and needing more time and resources, we can use the skills
already learnt and finetune a little more to learn a similar task and be good
at it.

In the world of machine learning and artificial intelligence, transfer
learning has emerged as a powerful technique. In this blog, we explore how
transfer learning, coupled with Intel® Xeon® Scalable CPUs, specifically 4th
Gen Intel® Xeon® scalable processor, defies the conventional belief that
training is primarily GPU-dependent. We present a case study where we achieved
near state-of-the-art accuracy for image classification on a publicly
available Image Classification TensorFlow dataset [1] using Intel® Advanced
Matrix Extensions(AMX)[2] and distributed training with Horovod.

##  2 Image Classification with Transfer Learning

The basic idea behind transfer learning is to use a pre-trained model, often
trained on a large and diverse dataset, as a starting point. This pre-trained
model has already learned useful features or representations from its original
task, which can be transferred and applied to the new task. The advantage of
transfer learning lies in its ability to significantly reduce the time and
resources needed for training while delivering impressive results. To
illustrate the power of transfer learning, let’s consider a case study of
identifying colorectal cancer tissue types through image classification [3].
We started with the pre-trained ResNet v1.5 [4][5] weights and fine-tuned the
last classification layer using a TensorFlow dataset with 5000 images with
4000 for training. This approach allowed us to build on the knowledge acquired
during pre-training and achieve close to state-of-the-art accuracy of 94.5%
[6] on this dataset. Data augmentation was used as a preprocessing step, and
early stopping criteria with a patience of 10 was employed to stop training
once convergence was reached. The pipeline demonstrated run-to-run variations
of 6-7 epochs, with an average of 45 epochs to achieve convergence. Figure 1
shows the transfer learning pipeline on vision task.

Figure 1: Vision Transfer Learning Pipeline

##  3 Leveraging Intel® Xeon® Scalable CPUs

Traditionally, training deep learning models was GPU-intensive, but with
Intel® Xeon® Scalable CPUs, we witnessed a paradigm shift. By utilizing Intel®
Advanced Matrix Instructions (AMX) with BF16 precision, we achieved remarkable
accuracy of 94.5% with the model converging in just 43 epochs. The entire
training process took less than 5.5 minutes with a single socket, showcasing
the efficiency and speed of Intel® Optimization for TensorFlow. Intel®
Optimization for TensorFlow is powered by Intel® oneAPI Deep Neural Network
Library (oneDNN) [7] [8], which includes convolution, normalization,
activation, inner product, and other primitives vectorized. To achieve the
above performance and accuracy, we used the following settings:

  * •

Use Mixed Precision: Leverage Intel® AMX BF16 precision format by enabling
auto mixed precision in TensorFlow. BF16 offers better precision than FP16
while maintaining higher performance than FP32. In our case study we achieved
similar accuracy with BF16 as with FP32.

  * •

Use numactl: Accessing memory from the local socket is faster than from a
remote socket in NUMA systems. To avoid potential performance issues due to
remote memory access, bind the memory to one socket using the numactl command.
For hyperthreading scenarios, use the command numactl -C 0-55,112-167 -m 0
python train.py to ensure memory is bound to one socket.

  * •

Define run time parameters: Inter-op parallelism involves distributing tasks
across cores to manage system resources efficiently and improve overall system
performance. Intra-op parallelism focuses on optimizing parallel execution
within a single core, breaking tasks into smaller sub-tasks to boost
performance in single-threaded applications. For the case study, the inter-op
parallelism is set to 56 threads (number of cores), and the intra-op
parallelism is set to 56 threads. Additionally, use specific KMP settings as
below

    * –

KMP_SETTINGS = 1

    * –

KMP_BLOCKTIME = 1

    * –

OMP_NUM_THREADS = NUM_CORES (56)

    * –

KMP_AFFINITY = granularity=fine,compact,1,0

##  4 Empowering Multi-Socket Performance with Distributed Training

Intel® Xeon® Scalable CPUs come equipped with two sockets, each having 56
cores. To maximize performance, we employed distributed training with Horovod
[9] and OpenMPI [10] as the backend. Horovod, an open-source distributed
training framework developed by Uber, supports popular deep learning
frameworks like TensorFlow, PyTorch, and MXNet. By leveraging MPI, Horovod
efficiently distributes training data and model parameters across multiple
devices, resulting in faster training times. With all 112 cores, including
hyperthreading, fully engaged, we achieved an impressive training time of
around 3 minutes, comparable to an out-of-the-box training on an NVIDIA A100
Rome GPU. The total training time results are displayed in Figure 2.

In the specified distributed training setup, weak scaling is used, maintaining
the same batch size throughout. The training is performed using Horovod with
two workers on a single Sapphire Rapids system, where each worker is mapped to
one socket of the system. The dataset is divided into halves and assigned to
each worker for processing.

To reduce communication overhead, gradients are averaged every 5 epochs
instead of after each epoch. The training process utilizes the Horovod
optimizer, and a warmup period of 3 epochs is set. The initial learning rate
is set to 0.01, and it is scaled by the number of workers to 0.002. To
optimize intra-op parallelism, the number of threads is set to 54, which is
the number of cores minus 2. This configuration aims to achieve efficient and
effective training while leveraging the computational capabilities of the
Sapphire Rapids system.

To maximize the performance on Intel® Xeon® CPU leverage we followed the
recipe below.

Figure 2: Competitive Perf Results for Vision Transfer Learning Workload.

##  5 Conclusion

Transfer learning has proven to be a game-changer in deep learning, enabling
us to build on existing knowledge and achieve outstanding results with minimal
time and resources. The successful application of transfer learning on Intel®
Xeon® Scalable CPUs, particularly Sapphire Rapids, challenges the GPU-centric
training mindset and offers a compelling alternative for high-performance
image classification tasks. As we continue to explore the possibilities of
leveraging Intel®’s advanced technologies, we look forward to even greater
strides in AI and machine learning.

## Configuration Details

  * •

3rd Gen Intel® Xeon® scalable processor (ICX) Test by Intel® as of 10/21/2022.
1-node, 2x Intel® Xeon® Platinum 8380, 40 cores, HT On, Turbo On, Total Memory
1024 GB (16 slots/ 64 GB/ 3200 MHz [run @ 3200 MHz] ),
SE5C620.86B.01.01.0005.2202160810, 0xd000375, Ubuntu 22.04.1 LTS,
5.15.0-48-generic, n/a, Vision Transfer Learning Pipeline,Intel-tensorflow-
avx512 2.10.0, resnet50v1_5, n/a

  * •

4th Gen Intel® Xeon scalable processor (SPR) Test by Intel® as of 10/21/2022.
1-node, 2x Intel® Xeon® Platinum 8480+ ,56 cores, HT On, Turbo On, Total
Memory 1024 GB (16 slots/ 64 GB/ 4800 MHz [run @ 4800 MHz] ),
EGSDREL1.SYS.8612.P03.2208120629 , 0x2b000041 , Ubuntu 22.04.1 LTS,
5.15.0-48-generic, n/a, Vision Transfer Learning Pipeline, Intel-tensorflow-
avx512 2.10.0, resnet50v1_5, n/a.

  * •

NVIDIA-A100 Test by Intel® as of 10/26/2022. 1-node (DGX-A100), 2xAMD EPYC
7742 64-Core Processor, 64 cores, HT On, Turbo On,, Total 1024GB (16
slots/64GB/3200 MHz) [run @ 3200MHz] ), Nvidia A100 GPU, BIOS 1.1, 0x830104d
,Ubuntu 20.04.2 LTS , 5.4.0-81-generic, n/a, Vision Transfer Learning
Pipeline, Tensorflow 2.10, resnet50v1_5, n/a.

## References

  * [1] TensorFlow Datasets: a collection of ready-to-use datasets.  URL https://www.tensorflow.org/datasets.
  * int [a] Accelerate AI Workloads with Intel AMX, a.  URL https://www.intel.com/content/www/us/en/products/docs/accelerator-engines/advanced-matrix-extensions/ai-solution-brief.html.
  * Kather et al. [2016] Jakob Nikolas Kather, Cleo-Aron Weis, Francesco Bianconi, Susanne M Melchers, Lothar R Schad, Timo Gaiser, Alexander Marx, and Frank Gerrit Z"ollner.  Multi-class texture analysis in colorectal cancer histology.  _Scientific reports_ , 6:27988, 2016.
  * [4] Intel ResNet 50v1.5 models.  URL https://github.com/IntelAI/models/tree/master/benchmarks/image_recognition/tensorflow/resnet50v1_5.
  * He et al. [2015] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep Residual Learning for Image Recognition, 2015.  URL https://arxiv.org/abs/1512.03385.
  * Plumworasawat and Sae-Bae [2023] Sirithep Plumworasawat and Napa Sae-Bae.  Colorectal tissue image classification across datasets.  In _2023 20th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON)_ , pages 1–4, 2023.  doi:10.1109/ECTI-CON58255.2023.10153365.
  * int [b] Intel oneAPI Deep Neural Network Library, b.  URL https://www.intel.com/content/www/us/en/developer/tools/oneapi/onednn.html.
  * int [c] Intel oneAPI AI Analytics Toolkit, c.  URL https://www.intel.com/content/www/us/en/developer/tools/oneapi/toolkits.html.
  * Sergeev and Balso [2018] Alexander Sergeev and Mike Del Balso.  Horovod: fast and easy distributed deep learning in tensorflow, 2018.
  * Gabriel et al. [2004] Edgar Gabriel, Graham E. Fagg, George Bosilca, Thara Angskun, Jack J. Dongarra, Jeffrey M. Squyres, Vishal Sahay, Prabhanjan Kambadur, Brian Barrett, Andrew Lumsdaine, Ralph H. Castain, David J. Daniel, Richard L. Graham, and Timothy S. Woodall.  Open MPI: Goals, concept, and design of a next generation MPI implementation.  In _Proceedings, 11th European PVM/MPI Users’ Group Meeting_ , pages 97–104, Budapest, Hungary, September 2004.


11footnotetext: Université de Lorraine, IECL, UMR 7502, Campus Scientifique,
B.P. 70239, Vandœuvre-lès-Nancy Cedex, F-54506, France22footnotetext: CNRS,
IECL, UMR 7502, Vandœuvre-lès-Nancy, F-54506, France33footnotetext: Inria,
TOSCA team, Villers-lès-Nancy, F-54600, France.
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Uniform convergence to the $Q$-process

Nicolas Champagnat1,2,3, Denis Villemonais1,2,3

###### Abstract

The first aim of the present note is to quantify the speed of convergence of a
conditioned process toward its $Q$-process under suitable assumptions on the
quasi-stationary distribution of the process. Conversely, we prove that, if a
conditioned process converges uniformly to a conservative Markov process which
is itself ergodic, then it admits a unique quasi-stationary distribution and
converges toward it exponentially fast, uniformly in its initial distribution.
As an application, we provide a conditional ergodic theorem.

Keywords: quasi-stationary distribution; $Q$-process; uniform exponential
mixing property; conditional ergodic theorem

2010 Mathematics Subject Classification. 60J25; 37A25; 60B10.

##  1 Introduction

Let $(\Omega,({\cal F}_{t})_{t\geq 0},(X_{t})_{t\geq 0},(\mathbb{P}_{x})_{x\in
E\cup\\{\partial\\}})$ be a time homogeneous Markov process with state space
$E\cup\\{\partial\\}$, where $E$ is a measurable space. We assume that
$\partial\not\in E$ is an absorbing state for the process, which means that
$X_{s}=\partial$ implies $X_{t}=\partial$ for all $t\geq s$,
$\mathbb{P}_{x}$-almost surely for all $x\in E$. In particular,

$\tau_{\partial}:=\inf\\{t\geq 0,X_{t}=\partial\\}$

is a stopping time. We also assume that
$\mathbb{P}_{x}(\tau_{\partial}<\infty)=1$ and
$\mathbb{P}_{x}(t<\tau_{\partial})>0$ for all $t\geq 0$ and $\forall x\in E$.

A probability measure $\alpha$ on $E$ is called a quasi-stationary
distribution if

$\mathbb{P}_{\alpha}(X_{t}\in\cdot\mid t<\tau_{\partial})=\alpha,\quad\forall
t\geq 0.$

We refer the reader to [7, 9, 4] and references therein for extensive
developments and several references on the subject. It is well known that a
probability measure $\alpha$ is a quasi-stationary distribution if and only if
there exists a probability measure $\mu$ on $E$ such that

$\displaystyle\lim_{t\rightarrow+\infty}\mathbb{P}_{\mu}(X_{t}\in A\mid
t<\tau_{\partial})=\alpha(A)$ (1.1)

for all measurable subsets $A$ of $E$.

In [2], we provided a necessary and sufficient condition on $X$ for the
existence of a probability measure $\alpha$ on $E$ and constants $C,\gamma>0$
such that

$\left\|\mathbb{P}_{\mu}(X_{t}\in\cdot\mid
t<\tau_{\partial})-\alpha\right\|_{TV}\leq Ce^{-\gamma
t},\quad\forall\mu\in\mathcal{P}(E),\quad t\geq 0,$ (1.2)

where $\|\cdot\|_{TV}$ is the total variation norm and $\mathcal{P}(E)$ is the
set of probability measures on $E$. This immediately implies that $\alpha$ is
the unique quasi-stationary distribution of $X$ and that (1.1) holds for any
initial probability measure $\mu$.

The necessary and sufficient condition for (1.2) is given by the existence of
a probability measure $\nu$ on $E$ and of constants $t_{0},c_{1},c_{2}>0$ such
that

$\mathbb{P}_{x}(X_{t_{0}}\in\cdot\mid t_{0}<\tau_{\partial})\geq
c_{1}\nu,\quad\forall x\in E$

and

$\mathbb{P}_{\nu}(t<\tau_{\partial})\geq
c_{2}\mathbb{P}_{x}(t<\tau_{\partial}),\quad\forall t\geq 0,\ x\in E.$

The first condition implies that, in cases of unbounded state space $E$ (like
$\mathbb{N}$ or $\mathbb{R}_{+}$), the process $(X_{t},t\geq 0)$ comes down
from infinity in the sense that, there exists a compact set $K\subset E$ such
that $\inf_{x\in E}\mathbb{P}_{x}(X_{t_{0}}\in K\mid t_{0}\tau_{\partial})>0$.
This property is standard for biological population processes such as Lotka-
Volterra birth and death or diffusion processes [1, 3]. However, this is not
the case for some classical models, such as linear birth and death processes
or Ornstein-Uhlenbeck processes.

Many properties can be deduced from (1.2). For instance, this implies the
existence of a constant $\lambda_{0}>0$ such that

$\displaystyle\mathbb{P}_{\alpha}(t<\tau_{\partial})=e^{-\lambda_{0}t}$

and of a function $\eta:E\rightarrow(0,\infty)$ such that $\alpha(\eta)=1$ and

$\displaystyle\lim_{t\rightarrow+\infty}\sup_{x\in
E}\left|e^{\lambda_{0}t}\mathbb{P}_{x}(t<\tau_{\partial})-\eta(x)\right|=0$
(1.3)

as proved in [2, Prop. 2.3]. It also implies the existence and the exponential
ergodicity of the associated $Q$-process, defined as the process $X$
conditioned to never be extinct [2, Thm. 3.1]. More precisely, if (1.2) holds,
then the family $(\mathbb{Q}_{x})_{x\in E}$ of probability measures on
$\Omega$ defined by

$\displaystyle\mathbb{Q}_{x}(\Gamma)=\lim_{t\rightarrow+\infty}\mathbb{P}_{x}(\Gamma\mid
t<\tau_{\partial}),\ \forall\Gamma\in{\cal F}_{s},\ \forall s\geq 0,$ (1.4)

is well defined and the process $(\Omega,({\cal F}_{t})_{t\geq
0},(X_{t})_{t\geq 0},(\mathbb{Q}_{x})_{x\in E})$ is an $E$-valued homogeneous
Markov process. In addition, this process admits the unique invariant
probability measure (sometimes refered to as the doubly limiting quasi-
stationary distribution [5])

$\displaystyle\beta(dx)=\eta(x)\alpha(dx)$

and there exist constants $C^{\prime},\gamma^{\prime}>0$ such that, for any
$x\in E$ and all $t\geq 0$,

$\displaystyle\left\|\mathbb{Q}_{x}(X_{t}\in\cdot)-\beta\right\|_{TV}\leq
C^{\prime}e^{-\gamma^{\prime}t}.$ (1.5)

The measure $\beta$

The first aim of the present note is to refine some results of [2] in order to
get sharper bounds on the convergence in (1.3) and to prove that the
convergence (1.4) holds in total variation norm, with uniform bounds over the
initial distribution (see Theorem 2.1). Using these new results, we obtain in
Corollary 2.3 that the uniform exponential convergence (1.2) implies that, for
all bounded measurable function $f:E\rightarrow\mathbb{R}$ and all $T>0$,

$\displaystyle\left|\mathbb{E}_{x}\left(\frac{1}{T}\int_{0}^{T}f(X_{t})\,dt\mid
T<\tau_{\partial}\right)-\int_{E}f\,d\beta\right|\leq\frac{a\|f\|_{\infty}}{T},$
(1.6)

for some positive constant $a$. This result improves the very recent result
obtained independently by He, Zhang and Zu [6, Thm. 2.1] by providing the
convergence estimate in $1/T$. The interested reader might look into [6] for
nice domination properties between the quasi-stationary distribution $\alpha$
and the probability $\beta$.

The second aim of this note is to prove that the existence of the $Q$-process
with uniform bounds in (1.4) and its uniform exponential ergodicity (1.5) form
in fact a necessary and sufficient condition for the uniform exponential
convergence (1.2) toward a unique quasi-stationary distribution.

##  2 Main results

In this first result, we improve (1.3) and provide a uniform exponential bound
for the convergence (1.4) of the conditioned process toward the $Q$-process.

######  Theorem 2.1.

Assume that (1.2) holds. Then there exists a positive constant $a_{1}$ such
that

$\displaystyle\left|e^{\lambda_{0}t}\mathbb{P}_{x}(t<\tau_{\partial})-\eta(x)\right|\leq
a_{1}\,e^{\lambda_{0}t}\mathbb{P}_{x}(t<\tau_{\partial})e^{-\gamma t},$ (2.1)

where $\lambda_{0}$ and $\eta$ are the constant and function appearing in
(1.3) and where $\gamma>0$ is the constant from (1.2).

Moreover, there exists a positive constant $a_{2}$ such that, for all $t\geq
0$, for all $\Gamma\in\mathcal{F}_{t}$ and all $T\geq t$,

$\displaystyle\left\|\mathbb{Q}_{x}(\Gamma)-\mathbb{P}_{x}(\Gamma\mid
T<\tau_{\partial})\right\|_{TV}\leq a_{2}\,e^{-\gamma(T-t)},$ (2.2)

where $(\mathbb{Q}_{x})_{x\in E}$ is the $Q$-process defined in (1.4).

We emphasize that (2.1) is an improvement of (1.3), since the convergence is
actually exponential and, in many interesting examples, $\inf_{x\in
E}\mathbb{P}_{x}(t<\tau_{\partial})=0$. This is for example the case for
elliptic diffusion processes absorbed at the boundaries of an interval, since
the probability of absorption converges to 1 when the initial condition
converges to the boundaries of the interval. The last theorem has a first
corollary.

######  Corollary 2.2.

Assume that (1.2) holds. Then there exists a positive constant $a_{3}$ such
that, for all $T>0$, all probability measure $\mu_{T}$ on $[0,T]$ and all
bounded measurable functions $f:E\rightarrow\mathbb{R}$,

$\left|\mathbb{E}_{x}\left(\int_{0}^{T}f(X_{t})\mu_{T}(dt)\mid
T<\tau_{\partial}\right)-\int_{E}f\,d\beta\right|\\\ \leq
a_{3}\|f\|_{\infty}\int_{0}^{T}\left(e^{-\gamma^{\prime}t}+e^{-\gamma(T-t)}\right)\mu_{T}(dt).$
(2.3)

This follows from (2.2), the exponential ergodicity of the $Q$-process stated
in (1.5) and the inequality

$\left|\mathbb{E}_{x}\left(\int_{0}^{T}f(X_{t})\mu_{T}(dt)\mid
T<\tau_{\partial}\right)-\int_{E}f\,d\beta\right|\\\
\leq\int_{0}^{T}\left|\mathbb{E}_{x}(f(X_{t})\mid
T<\tau_{\partial})-\mathbb{E}^{\mathbb{Q}_{x}}(f(X_{t}))\right|\,\mu_{T}(dt)\\\
+\int_{0}^{T}\left|\mathbb{E}^{\mathbb{Q}_{x}}(f(X_{t}))-\int_{E}f\,d\beta\right|\,\mu_{T}(dt),$

where $\mathbb{E}^{\mathbb{Q}_{x}}$ is the expectation with respect to
$\mathbb{Q}_{x}$.

In particular, choosing $\mu_{T}$ as the uniform distribution on $[0,T]$, we
obtain a conditional ergodic theorem.

######  Corollary 2.3.

Assume that (1.2) holds. Then there exists a positive constant $a_{4}$ such
that, for all $T>0$ and all bounded measurable functions
$f:E\rightarrow\mathbb{R}$,

$\displaystyle\left|\mathbb{E}_{x}\left(\frac{1}{T}\int_{0}^{T}f(X_{t})\,dt\mid
T<\tau_{\partial}\right)-\int_{E}f\,d\beta\right|\leq\frac{a_{4}\,\|f\|_{\infty}}{T}.$

Considering the problem of estimating $\beta$ from $N$ realizations of the
unconditioned process $X$, one wishes to take $T$ as small as possible in
order to obtain the most samples such that $T<\tau_{\partial}$ (of order
$N_{T}=Ne^{-\lambda_{0}T}$). It is therefore important to minimize the error
in (2.3) for a given $T$. It is easy to check that $\mu_{T}=\delta_{t_{0}}$
with $t_{0}=\gamma T/(\gamma+\gamma^{\prime})$ is optimal with an error of the
order of $\exp(-\gamma^{\prime}\gamma T/(\gamma+\gamma^{\prime}))$. Combining
this with the Monte Carlo error of order $1/\sqrt{N_{T}}$, we obtain a global
error of order

$\frac{e^{\lambda_{0}T/2}}{\sqrt{N}}+e^{-\gamma\gamma^{\prime}T/(\gamma+\gamma^{\prime})}.$

In particular, for a fixed $N$, the optimal choice for $T$ is
$T\approx\frac{\log
N}{\lambda_{0}+2\gamma\gamma^{\prime}/(\gamma+\gamma^{\prime})}$ and the error
is of the order of $N^{-\zeta}$ with
$\zeta=\frac{\gamma\gamma^{\prime}}{2\gamma\gamma^{\prime}+\lambda_{0}(\gamma+\gamma^{\prime})}$.
Conversely, for a fixed $T$, the best choice for $N$ is
$N\approx\exp((\lambda_{0}+2\gamma\gamma^{\prime}/(\gamma+\gamma^{\prime}))T)$
and the error is of the order of
$\exp(-\gamma\gamma^{\prime}T/(\gamma+\gamma^{\prime}))$.

We conclude this section with a converse to Theorem 2.1. More precisely, we
give a converse to the fact that (1.2) implies both (1.5) and (2.2).

######  Theorem 2.4.

Assume that there exists a Markov process $(\mathbb{Q}_{x})_{x\in E}$ with
state space $E$ such that, for all $t>0$,

$\displaystyle\lim_{T\rightarrow+\infty}\sup_{x\in
E}\left\|\mathbb{Q}_{x}(X_{t}\in\cdot)-\mathbb{P}_{x}(X_{t}\in\cdot\mid
T<\tau_{\partial})\right\|_{TV}=0$ (2.4)

and such that

$\displaystyle\lim_{t\rightarrow+\infty}\sup_{x,y\in
E}\left\|\mathbb{Q}_{x}(X_{t}\in\cdot)-\mathbb{Q}_{y}(X_{t}\in\cdot)\right\|_{TV}=0.$
(2.5)

Then the process $(\mathbb{P}_{x})_{x\in E}$ admits a unique quasi-stationary
distribution $\alpha$ and there exist positive constants $\gamma,C$ such that
(1.2) holds.

It is well known that the strong ergodicity (2.5) of a Markov process implies
its exponential ergodicity [8, Thm. 16.0.2]. Similarly, we observe in our
situation that, if (2.4) and (2.5) hold, then the combination of the above
results implies that both convergences hold exponentially.

##  3 Proofs

###  3.1 Proof of Theorem 2.1

For all $x\in E$, we set

$\displaystyle\eta_{t}(x)=\frac{\mathbb{P}_{x}(t<\tau_{\partial})}{\mathbb{P}_{\alpha}(t<\tau_{\partial})}=e^{\lambda_{0}t}\mathbb{P}_{x}(t<\tau_{\partial}),$

and we recall from [2, Prop. 2.3] that $\eta_{t}(x)$ is uniformly bounded
w.r.t. $t\geq 0$ and $x\in E$. By Markov’s property

$\displaystyle\eta_{t+s}(x)$
$\displaystyle=e^{\lambda_{0}(t+s)}\mathbb{E}_{x}\left(\mathbbm{1}_{t<\tau_{\partial}}\mathbb{P}_{X_{t}}(s<\tau_{\partial})\right)$
$\displaystyle=\eta_{t}(x)\mathbb{E}_{x}\left(\eta_{s}(X_{t})\mid
t<\tau_{\partial}\right).$

By (1.2), there exists a constant $C^{\prime}$ independent of $s$ such that

$\displaystyle\left|\mathbb{E}_{x}\left(\eta_{s}(X_{t})\mid
t<\tau_{\partial}\right)-\int_{E}\eta_{s}d\alpha\right|\leq
C^{\prime}\,e^{-\gamma t}.$

Since $\int\eta_{s}d\alpha=1$, there exists a constant $a_{1}>0$ such that,
for all $x\in E$ and $s,t\geq 0$,

$\displaystyle\left|\frac{\eta_{t+s}(x)}{\eta_{t}(x)}-1\right|\leq
a_{1}\,e^{-\gamma t}.$

Hence, multiplying on both side by $\eta_{t}(x)$ and letting $s$ tend to
infinity, we deduce from (1.3) that, for all $x\in E$,

$\displaystyle\left|\eta(x)-\eta_{t}(x)\right|\leq a_{1}\,e^{-\gamma
t}\eta_{t}(x),\,\forall t\geq 0,$

which is exactly (2.1). We also deduce that

$\displaystyle\left(1-a_{1}e^{-\gamma
t}\right)\eta_{t}(x)\leq\eta(x)\leq\left(1+a_{1}e^{-\gamma
t}\right)\eta_{t}(x)$ (3.1)

and hence, for $t$ large enough,

$\displaystyle\frac{\eta(x)}{1+a_{1}e^{-\gamma
t}}\leq\eta_{t}(x)\leq\frac{\eta(x)}{1-a_{1}e^{-\gamma t}}.$ (3.2)

Let us now prove the second part of Theorem 2.1. For any $t\geq 0$,
$\Gamma\in\mathcal{F}_{t}$ and $0\leq t\leq T$,

$\displaystyle\mathbb{P}_{x}\left(\Gamma\mid T<\tau_{\partial}\right)$
$\displaystyle=\frac{\mathbb{P}_{x}\left(\Gamma\cap\\{T<\tau_{\partial}\\}\right)}{\mathbb{P}_{x}(T<\tau_{\partial})}$
$\displaystyle=\frac{e^{\lambda_{0}T}\mathbb{P}_{x}\left(\Gamma\cap\\{T<\tau_{\partial}\\}\right)}{\eta(x)}\,\frac{\eta(x)}{e^{\lambda_{0}T}\mathbb{P}_{x}(T<\tau_{\partial})}.$

We deduce from (2.1) that

$\displaystyle\left|\frac{\eta(x)}{e^{\lambda_{0}T}\mathbb{P}_{x}(T<\tau_{\partial})}-1\right|\leq
a_{1}e^{-\gamma T},$

while, for all $T>\frac{\log a_{1}}{\gamma}$, (3.2) entails

$\displaystyle\left|\frac{e^{\lambda_{0}T}\mathbb{P}_{x}\left(\Gamma\cap\\{T<\tau_{\partial}\\}\right)}{\eta(x)}\right|\leq\frac{\eta_{T}(x)}{\eta(x)}\leq\frac{1}{1-a_{1}e^{-\gamma
T}}.$

Hence, for all $t\geq 0$ and all $T>\frac{\log a_{1}}{\gamma}$,

$\displaystyle\left|\mathbb{P}_{x}\left(\Gamma\mid
T<\tau_{\partial}\right)-\frac{e^{\lambda_{0}T}\mathbb{P}_{x}\left(\Gamma\cap\\{T<\tau_{\partial}\\}\right)}{\eta(x)}\right|\leq\frac{a_{1}e^{-\gamma
T}}{1-a_{1}e^{-\gamma T}}.$ (3.3)

Now, the Markov property implies that

$\displaystyle\mathbb{P}_{x}\left(\Gamma\cap\\{T<\tau_{\partial}\\}\right)=\mathbb{E}_{x}\left(\mathbbm{1}_{\Gamma}\mathbb{P}_{X_{t}}(T-t<\tau_{\partial})\right),$

and we deduce from (3.3) that, for all $T>t+\frac{\log a_{1}}{\gamma}$,

$\displaystyle\left|e^{\lambda_{0}(T-t)}\mathbb{P}_{X_{t}}(T-t<\tau_{\partial})-\eta(X_{t})\right|\leq\frac{a_{1}e^{-\gamma(T-t)}}{1-a_{1}e^{-\gamma(T-t)}}\eta(X_{t}).$

Thus we have

$\left|\frac{e^{\lambda_{0}T}\mathbb{P}_{x}\left(\Gamma\cap\\{T<\tau_{\partial}\\}\right)}{\eta(x)}-\frac{e^{\lambda_{0}t}\mathbb{E}_{x}\left(\mathbbm{1}_{\Gamma}\eta(X_{t})\right)}{\eta(x)}\right|\\\
\begin{aligned}
&\leq\frac{e^{\lambda_{0}t}}{\eta(x)}\mathbb{E}_{x}\left[\mathbbm{1}_{\Gamma}\left|e^{\lambda_{0}(T-t)}\mathbb{P}_{X_{t}}(T-t<\tau_{\partial})-\eta(X_{t})\right|\right]\\\
&\leq\frac{a_{1}e^{-\gamma(T-t)}}{1-a_{1}e^{-\gamma(T-t)}}\frac{e^{\lambda_{0}t}\mathbb{E}_{x}(\eta(X_{t}))}{\eta(x)}\\\
&=\frac{a_{1}e^{-\gamma(T-t)}}{1-a_{1}e^{-\gamma(T-t)}},\end{aligned}$

where we used the fact that
$\mathbb{E}_{x}\eta(X_{h})=e^{-\lambda_{0}h}\eta(x)$ for all $h>0$ (see [2,
Prop. 2.3]). This and (3.3) allows us to conclude that, for all $t\geq 0$ and
all $T>t+\frac{\log a_{1}}{\gamma}$,

$\displaystyle\left|\mathbb{P}_{x}\left(\Gamma\mid
T<\tau_{\partial}\right)-\frac{e^{\lambda_{0}t}\mathbb{E}_{x}\left(\mathbbm{1}_{\Gamma}\eta(X_{t})\right)}{\eta(x)}\right|\leq\frac{2a_{1}e^{-\gamma
T}}{1-a_{1}e^{-\gamma T}}.$

Since
$\mathbb{Q}_{x}(\Gamma)=e^{\lambda_{0}t}\mathbb{E}_{x}\left(\mathbbm{1}_{\Gamma}\,\eta(X_{t})\right)/\eta(x)$
(see [2, Thm. 3.1 (ii)]), we deduce that (2.2) holds true.

This concludes the proof of Theorem 2.1.

###  3.2 Proof of Theorem 2.4

We deduce from (2.4) and (2.5) that there exists $t_{1}>0$ and $T_{1}>0$ such
that, for all $T\geq T_{1}$,

$\displaystyle\sup_{x,y\in E}\left\|\mathbb{P}_{x}(X_{t_{1}}\in\cdot\mid
T<\tau_{\partial})-\mathbb{P}_{y}(X_{t_{1}}\in\cdot\mid
T<\tau_{\partial})\right\|_{TV}\leq 1/2.$

In particular, for all $s\geq 0$ and all $T\geq s+T_{1}$,

$\displaystyle\sup_{x,y\in
E}\left\|\delta_{x}R_{s,s+t_{1}}^{T}-\delta_{y}R_{s,s+t_{1}}^{T}\right\|_{TV}\leq
1/2,$ (3.4)

where, for all $0\leq s\leq t\leq T$, $R_{s,t}^{T}$ is the linear operator
defined by

$\displaystyle\delta_{x}R_{s,t}^{T}f$
$\displaystyle=\mathbb{E}_{x}(f(X_{t-s})\mid T-s<\tau_{\partial})$
$\displaystyle=\mathbb{E}(f(X_{t})\mid X_{s}=x,\ T<\tau_{\partial})$
$\displaystyle=\delta_{x}R_{0,t-s}^{T-s}f,$

where we used the Markov property. Now, for any $T>0$, the family
$(R_{s,t}^{T})_{0\leq s\leq t\leq T}$ is a Markov semi-group. This semi-group
property and the contraction (3.4) classically imply that, for all $T\geq
T_{1}$,

$\displaystyle\sup_{x,y\in
E}\left\|\delta_{x}R_{0,T}^{T}-\delta_{y}R_{0,T}^{T}\right\|_{TV}\leq\left(1/2\right)^{\lfloor
T-T_{1}\rfloor/t_{1}}.$

Then, proceeding as in [2, Section 5.1], we deduce that (1.2) holds true. This
concludes the proof of Theorem 2.4.

## References

  * [1] P. Cattiaux, P. Collet, A. Lambert, S. Martínez, S. Méléard, and J. San Martín.  Quasi-stationary distributions and diffusion models in population dynamics.  Ann. Probab., 37(5):1926–1969, 2009.
  * [2] N. Champagnat and D. Villemonais.  Exponential convergence to quasi-stationary distribution and Q-process.  Probability Theory and Related Fields, 164(1):243–283, 2016.
  * [3] N. Champagnat and D. Villemonais.  Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes.  ArXiv e-prints, Apr. 2017.
  * [4] P. Collet, S. Martínez, and J. Martín.  Quasi-Stationary Distributions: Markov Chains, Diffusions and Dynamical Systems.  Probability and Its Applications. Springer Berlin Heidelberg, 2012.
  * [5] D. C. Flaspohler.  Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains.  Ann. Inst. Statist. Math., 26:351–356, 1974.
  * [6] G. He, H. Zhang, and Y. Zhu.  On the quasi-ergodic distribution of absorbing Markov processes.  ArXiv e-prints, Nov. 2016.
  * [7] S. Méléard and D. Villemonais.  Quasi-stationary distributions and population processes.  Probab. Surv., 9:340–410, 2012.
  * [8] S. P. Meyn and R. L. Tweedie.  Markov chains and stochastic stability.  Communications and Control Engineering Series. Springer-Verlag London, Ltd., London, 1993.
  * [9] E. A. van Doorn and P. K. Pollett.  Quasi-stationary distributions for discrete-state models.  European J. Oper. Res., 230(1):1–14, 2013.


# Optimal photon polarization toward the observation of the nonlinear Breit-
Wheeler pair production

Yunquan Gao  College of Physics and Optoelectronic Engineering, Ocean
University of China, Qingdao, Shandong, 266100, China  Suo Tang
<EMAIL_ADDRESS>College of Physics and Optoelectronic Engineering, Ocean
University of China, Qingdao, Shandong, 266100, China

###### Abstract

We investigate the optimization of the photon polarization to increase the
yield of the Breit-Wheeler pair production in arbitrarily polarized plane wave
backgrounds. We show that the optimized photon polarization can improve the
positron yield by more than $20\%$ compared to the unpolarized case, in the
intensity regime of current laser-particle experiments. The seed photon’s
optimal polarization is resulting from the polarization coupling with the
polarization of the laser pulse. The compact expressions of the coupling
coefficients in both the perturbative and nonperturbative regime are given.
Because of the evident difference in the coupling coefficients for the linear
and circular polarization components, the seed photon’s optimal polarization
state in an elliptically polarized laser background, deviates considerably
from the orthogonal state of the laser polarization.

##  I Introduction

The production of electron-positron pair in the collision of two high-energy
photons, now referred to as the linear Breit-Wheeler process (LBW), was first
proposed in 1930s Breit and Wheeler (1934). The production yield depends not
only on the photons’ dynamical parameters, but also on the relative
polarization of the two photons Breit and Wheeler (1934); Baier and Grozin
(2002); Adam et al. (2021).

With the improvement of the laser intensity, the decay of a single high-energy
photon into a pair of electron and positron in the collision with an intense
laser pulse, which is often referred to as the nonlinear Breit-Wheeler (NBW)
pair production Reiss (1962); Di Piazza et al. (2012); Gonoskov et al. (2021);
Fedotov et al. (2022), has been measured in the multiphoton perturbative
regime via the landmark E144 experiment more than two decades ago Burke et al.
(1997); Bamber et al. (1999) and been broadly studied within different type of
laser fields Nikishov and Ritus (1964); Heinzl et al. (2010); Krajewska and
Kamiński (2012); Titov et al. (2012); Fedotov and Mironov (2013); Titov et al.
(2016); Jansen and Müller (2013); Jansen and M¨¹ller (2017); Titov et al.
(2018); Ilderton (2019, 2020); King (2020); Tang (2021); Tang and King (2021).
The dependence of the NBW process on the polarisation state of the seed photon
has also been partially investigated in the current literature Ivanov et al.
(2005); Katkov (2012); Li et al. (2020); Chen et al. (2022); Wistisen (2020);
Titov and Kämpfer (2020); Seipt and King (2020); Tang (2022), in which the
laser backgrounds are commonly specified with the pure linear and/or circular
polarization, and the production yield could be considerably
improved/suppressed when the polarization of the seed photon is set to be
orthogonal/parallel to that of the background field Titov and Kämpfer (2020);
Seipt and King (2020); Tang (2022). However, in an arbitrarily polarized laser
background, how to assign the photon polarization to acquire the maximal
production yield has not been clearly investigated.

In the LBW process, the polarization dependence of the production is resulting
from the polarization coupling between the two high-energy photon Breit and
Wheeler (1934); Baier and Grozin (2002); Adam et al. (2021). However, how the
polarization of the seed photon couples with that of the laser pulse (or
multiple laser photons) in the NBW process is still not clear. In this paper,
we concentrate on the properties of the polarization coupling between the seed
photon and the laser pulse and reveal the optimal polarization of the seed
photon for the maximal yield of the NBW process in arbitrarily polarized laser
backgrounds. We find that the linear and circular polarization component of
the seed photon couple with the corresponding component of the laser
polarization with the quite different coefficients, and thus in an
elliptically polarized laser pulse, the optimal polarization state of the seed
photon deviates considerably from the orthogonal state of the laser
polarization.

The study of the optimal photon polarization for the maximal production yield
is partly motivated by the upcoming high-energy laser-particle experiment,
_i.e._ , LUXE at DESY Abramowicz et al. (2021); Borysova (2021); Macleod
(2022); Jacobs (2021) and E320 at SLAC Meuren (2019); Naranjo et al. (2021);
Salgado et al. (2021); Meuren et al. (2020) in which beams of photons with the
energy $O(10~{}\textrm{GeV})$ are generated to collide with laser pulses with
the intermediate intensity $\xi\sim O(1)$, and one of their main goals is to
detect the NBW process in the transition regime from the perturbative to the
non-perturbative regime Macleod (2022); Jacobs (2021), where $\xi$ is the
classical nonlinearity parameter for laser intensity. In this planned
intensity regime, the production yield could be enhanced/suppressed
considerably by the photon polarization effect Wistisen (2020); Tang (2022).

The paper is organised as follows. The theoretical model and relevant
parameters are introduced in Sec. II. In Sec. III, we first explore the
perturbative intensity regime and discuss the photon polarization coupling in
the LBW process, and then, we go to the non-perturbative intensity regime to
discuss the polarization coupling between the seed photon and the laser pulse
in the NBW precess in Sec. IV. At the end, we conclude in Sec. V. In following
discussions, the natural units $\hbar=c=1$ is used, and the fine structure
constant is $\alpha=e^{2}\approx 1/137$.

##  II Theoretical model

We consider the typical scenario in the modern-day laser-particle experiments
in which a beam of high-energy photons interacts with an intense laser pulse
in the geometry close to the head-on collision. The laser pulse is modelled as
a plane wave with scaled vector potential $a^{\mu}(\phi)=|e|A^{\mu}(\phi)$
depending only on the laser phase $\phi=k\cdot x$, where
$k^{\mu}=\omega(1,0,0,-1)$ is the laser wave vector, $\omega$ is the central
frequency of the laser pulse and $|e|$ is the charge of the positron. This
plane wave background is a good approximation for collisions between high-
energy particles and weakly focussed pulses Nikishov and Ritus (1964); Di
Piazza (2015, 2016, 2017, 2021). The collision is characterized by the energy
parameter $\eta=k\cdot\ell/m^{2}$ and laser intensity parameter $\xi$, where
$\ell^{\mu}$ is the photon momentum and $m$ is electron rest mass.

The total yield of the NBW pair production from a polarized seed photon is
given as Tang (2022):

$\displaystyle{P}=$
$\displaystyle\frac{\alpha}{(2\pi\eta)^{2}}\int\frac{\mathrm{d}s}{ts}\int\mathrm{d}^{2}\bm{r}\iint\mathrm{d}\phi_{1}\mathrm{d}\phi_{2}~{}e^{i\int_{\phi_{2}}^{\phi_{1}}\mathrm{d}\phi^{\prime}\frac{\ell\cdot\pi_{q}(\phi^{\prime})}{m^{2}\eta
t}}$
$\displaystyle\left\\{h_{s}\bm{\Delta}^{2}/2+1-ih_{s}\Gamma_{3}\bm{w}(\phi_{1})\times\bm{w}(\phi_{2})\right.$
$\displaystyle-\Gamma_{1}\left[w_{x}(\phi_{1})w_{x}(\phi_{2})-w_{y}(\phi_{1})w_{y}(\phi_{2})\right]$
$\displaystyle\left.-\Gamma_{2}\left[w_{x}(\phi_{1})w_{y}(\phi_{2})+w_{y}(\phi_{1})w_{x}(\phi_{2})\right]\right\\}\,,$
(1)

where
$\bm{\Delta}=i\left[\bm{a}^{{\scriptscriptstyle\perp}}(\phi_{1})-\bm{a}^{{\scriptscriptstyle\perp}}(\phi_{2})\right]/m$,
$h_{s}=(s^{2}+t^{2})/(2st)$,
$\bm{w}(\phi)=\bm{r}-\bm{a}^{{\scriptscriptstyle\perp}}(\phi)/m$, and
$\bm{w}(\phi_{1})\times\bm{w}(\phi_{2})=w_{x}(\phi_{1})w_{y}(\phi_{2})-w_{y}(\phi_{1})w_{x}(\phi_{2})$,
$s=k\cdot q/k\cdot\ell$ ($t=1-s$) is the fraction of the light front momentum
taken by the produced position (electron), and
$\bm{r}=(\bm{q}^{{\scriptscriptstyle\perp}}-s\bm{\ell}^{{\scriptscriptstyle\perp}})/m$
denotes the transverse momenta of the positron, and $\pi_{q}(\phi)$ is the
positron’s instantaneous momentum in the laser pulse:

$\pi^{\mu}_{q}(\phi)=q^{\mu}-a^{\mu}(\phi)+\frac{q\cdot a(\phi)}{k\cdot
q}k^{\mu}-\frac{a^{2}(\phi)}{2k\cdot q}k^{\mu}\,.$

The polarization of the seed photon is comprehensively described with the
classical Stokes parameters $(\Gamma_{1},~{}\Gamma_{2},~{}\Gamma_{3})$
Berestetskii et al. (1982); Jackson (1999): $\Gamma_{1}$ ($\Gamma_{2}$) is the
degree of linear polarization indicating the preponderance of the polarization
in the $\varepsilon_{x}$ state ($\varepsilon_{45^{\circ}}$ state) over that in
the $\varepsilon_{y}$ state ($\varepsilon_{135^{\circ}}$ state), and
$\Gamma_{3}$ is the degree of circular polarization giving the preponderance
of the polarization in the $\varepsilon_{{\scriptscriptstyle+}}$ state over
that in the $\varepsilon_{{\scriptscriptstyle-}}$ state. The polarization
basis is given as

$\displaystyle\varepsilon^{\mu}_{x}$
$\displaystyle=\epsilon^{\mu}_{x}-\frac{\ell\cdot\epsilon_{x}}{k\cdot\ell}k^{\mu}\,,~{}~{}~{}\varepsilon^{\mu}_{y}=\epsilon^{\mu}_{y}-\frac{\ell\cdot\epsilon_{y}}{k\cdot\ell}k^{\mu}\,,$
$\displaystyle\varepsilon^{\mu}_{\psi}$
$\displaystyle=\epsilon^{\mu}_{\psi}-\frac{\ell\cdot\epsilon_{\psi}}{k\cdot\ell}k^{\mu}\,,~{}~{}~{}\varepsilon^{\mu}_{\pm}=\epsilon^{\mu}_{\pm}-\frac{\ell\cdot\epsilon_{\pm}}{k\cdot\ell}k^{\mu}\,,$

where $\epsilon^{\mu}_{x}=(0,1,0,0)$, $\epsilon^{\mu}_{y}=(0,0,1,0)$ and
$\epsilon_{\psi}=\epsilon_{x}\cos\psi+\epsilon_{y}\sin\psi$,
$\epsilon_{\pm}=(\epsilon_{x}\pm i\epsilon_{y})/\sqrt{2}$. For fully polarized
photon beams, the Stokes parameters satisfy
$\Gamma_{1}^{2}+\Gamma_{2}^{2}+\Gamma_{3}^{2}=1$ and for partially polarized
photon beams, $\Gamma_{1}^{2}+\Gamma_{2}^{2}+\Gamma_{3}^{2}<1$. The full
definition of the photon Stokes parameters
$(\Gamma_{1},~{}\Gamma_{2},~{}\Gamma_{3})$ can be found in Ref. Tang (2022).

Based on (1), the total yield of the NBW process can be phenomenologically
given as

$\displaystyle\textrm{P}=n_{0}+\Gamma_{1}n_{1}+\Gamma_{2}n_{2}+\Gamma_{3}n_{3}\,,$
(2)

where $n_{0}$ is the unpolarized contribution independent on the photon
polarization $(\Gamma_{1,2,3}=0)$ Tang (2021); Tang and King (2021), and
$n_{1,2,3}$ denote the contributions coupling to the polarization of the seed
photon. As one can simply infer, to maximize the production yield,

$\displaystyle\textrm{P}_{m}=n_{0}+n_{p}\,,$ (3)

the photon polarization should be selected as

$\displaystyle(\Gamma_{1},\Gamma_{2},\Gamma_{3})=(n_{1},n_{2},n_{3})/(n_{1}^{2}+n_{2}^{2}+n_{3}^{2})^{1/2}\,,$
(4)

which prompts the existence of the optimal photon polarization for the
specified laser pulse and collision parameter $\eta$ to achieve the maximal
production yield, where $n_{p}=(n_{1}^{2}+n_{2}^{2}+n_{3}^{2})^{1/2}$ is the
maximal contribution from the photon polarization. However, if reverse the
optimal polarization of the seed photon, _i.e._
$\Gamma_{1,2,3}\to-\Gamma_{1,2,3}$, the pair production would be largely
suppressed.

##  III linear Breit-Wheeler process

One may realize that the polarization contribution $\Gamma_{i}n_{i}$ in (2)
comes from the polarization coupling between the seed and laser photons, and
thus the optimal photon polarization (4) depends on the polarization of the
laser photons. To manifest this polarization coupling effect, we resort to the
perturbative approximation of (1), which is often referred to as the LBW
process, by expanding the integrand in (1), keeping only
$\mathcal{O}(\xi^{2})$ terms and integrating over $s$,

$\displaystyle{P}_{\ell}=$
$\displaystyle\frac{\pi\alpha^{2}\lambdabar^{2}_{e}}{2}\int_{\nu_{*}}^{+\infty}\mathrm{d}{\nu}~{}D(\nu)$
$\displaystyle\left\\{\Xi+\kappa_{c}\Gamma_{3}\varsigma_{3}(\nu)+\kappa_{l}[\Gamma_{1}\varsigma_{1}(\nu)+\Gamma_{2}\varsigma_{2}(\nu)]\right\\}\,,$
(5)

where $\nu_{*}=2/\eta$ is the frequency threshold of the laser photon required
to trigger the pair production,
$D(\nu)=\nu|\tilde{\bm{a}}(\nu)|^{2}/(4\pi^{2}\alpha\lambdabar^{2}_{e}m^{2})$
is the (areal) number density of the laser photon with the frequency
$\nu\omega$, $\lambdabar_{e}=1/m=386.16~{}\textrm{fm}$ is the electron’s
reduced Compton wavelength,
$\tilde{\bm{a}}(\nu)=\int\mathrm{d}\phi[a_{x}(\phi),~{}a_{y}(\phi)]\exp(iv\phi)$,
and

$\displaystyle\begin{aligned}
\varsigma_{1}(\nu)&=\frac{|\tilde{a}_{x}(\nu)|^{2}-|\tilde{a}_{y}(\nu)|^{2}}{|\tilde{\bm{a}}(\nu)|^{2}},&\\\
\varsigma_{2}(\nu)&=\frac{\tilde{a}^{*}_{x}(\nu)\tilde{a}_{y}(\nu)+\tilde{a}_{x}(\nu)\tilde{a}_{y}^{*}(\nu)}{|\tilde{\bm{a}}(\nu)|^{2}},&\\\
\varsigma_{3}(\nu)&=i\frac{\tilde{a}_{x}(\nu)\tilde{a}_{y}^{*}(\nu)-\tilde{a}^{*}_{x}(\nu)\tilde{a}_{y}(\nu)}{|\tilde{\bm{a}}(\nu)|^{2}},\end{aligned}$
(6)

are the classical Stokes parameters of the laser photon $\nu\omega$ Jackson
(1999), satisfying
$\varsigma^{2}_{1}(\nu)+\varsigma^{2}_{2}(\nu)+\varsigma^{2}_{3}(\nu)=1$.
Similar as the seed photon, $\varsigma_{1,2,3}(\nu)$ characterize the
polarization property of the laser photon: $\varsigma_{1}(\nu)$
[$\varsigma_{2}(\nu)$] describes the preponderance of the $\epsilon_{x}$
($\epsilon_{45^{\circ}}$)-linear polarization over the $\epsilon_{y}$
($\epsilon_{135^{\circ}}$)-linear polarization, and $\varsigma_{3}(\nu)$
denotes the preponderance of the $\epsilon_{+}$-circular polarization over the
$\epsilon_{-}$-circular polarization. The parameter

$\displaystyle\Xi=(1-\beta^{2})\left[(3-\beta^{4})\ln\left(\frac{1+\beta}{1-\beta}\right)-2\beta(2-\beta^{2})\right]$
(7)

is the contribution from unpolarized photons Ng and Tsai (1977); Greiner and
Reinhardt (2009), and

$\displaystyle\kappa_{c}$
$\displaystyle=2(1-\beta^{2})\left[\ln\left(\frac{1+\beta}{1-\beta}\right)-3\beta\right]\,,$
(8) $\displaystyle\kappa_{l}$
$\displaystyle=-\frac{(1-\beta^{2})^{3}}{2}\left[\ln\left(\frac{1+\beta}{1-\beta}\right)+\frac{2\beta}{1-\beta^{2}}\right]$
(9)

are, respectively, the circular- and linear-polarization coupling
coefficients, and indicate the amplitude of the contributions from each kind
of polarization coupling between the seed and laser photons, where
$\beta=(1-\nu_{*}/\nu)^{1/2}$ is actually the normalized velocity of the
produced particles in the center-of-mass frame.

In (5), we can clearly see the contributions from the polarization coupling
between the seed and laser photons. To maximize the polarization contribution
in the LBW process, the polarization of the seed photon is optimized, based on
the polarization of the laser photon, as

$\displaystyle(\Gamma_{1},~{}\Gamma_{2},~{}\Gamma_{3})=\hat{\kappa}_{l}\frac{[\varsigma_{1}(\nu),~{}\varsigma_{2}(\nu),~{}\sigma\varsigma_{3}(\nu)]}{[\varsigma_{1}^{2}(\nu)+\varsigma_{2}^{2}(\nu)+\sigma^{2}\varsigma_{3}^{2}(\nu)]^{1/2}}\,,$
(10)

where $\sigma_{l}=\kappa_{c}/\kappa_{l}$, and $\hat{\kappa}_{l}$ is the sign
of $\kappa_{l}$. As we can also see in (5), the two sets of linear
polarization have the identical coupling coefficient $\kappa_{\ell}$, because
of the symmetry by rotating the linear polarization axis for $45^{\circ}$.
This identity results in the orthogonality between the linear polarization
components of the seed and laser photons as
$(\Gamma_{1},~{}\Gamma_{2})\sim-[\varsigma_{1}(\nu),~{}\varsigma_{2}(\nu)]$ as
shown in (10) where $\hat{\kappa}_{l}=-1$ is obtained in Fig. 1.

Figure 1: Comparison between the polarization contributions $\kappa_{c,l}$ and
the unpolarized contribution $\Lambda$ with the change of the parameter
$\beta$ in the linear Breit-Wheeler process. $\beta$ is defined in the text
and can be simply understood as the normalized velocity of the produced
particles in the center-of-mass frame.

In Fig. 1, the unpolarized contribution $\Xi$ and the polarization coupling
coefficients $\kappa_{l,c}$ are presented with the change of the parameter
$\beta$. As shown, the polarization contributions are indeed appreciable
compared with the unpolarized contribution, especially in the low-energy
region $\beta<0.2$, where $\Xi\approx-\kappa_{c}\approx-\kappa_{l}$ and the
energy of the laser photon is close to the frequency threshold
$\nu\to\nu_{*}$. With the proper photon polarization, the production could be
doubled if $\bm{\Gamma}\cdot\bm{\varsigma}(\nu)\to-1$ or completely suppressed
if $\bm{\Gamma}\cdot\bm{\varsigma}(\nu)\to 1$. Similar as the variation of the
unpolarized contribution $\Xi$ with $\beta\in(0,1)$ Greiner and Reinhardt
(2009), the amplitude of the coupling coefficients $\kappa_{c,l}$ increase
from zero at $\beta=0$ to the maximum at around $\beta\approx 0.45$ and then
fall off again to zero at $\beta=1$. In the region of $\beta<0.4$, the two
kind of polarization have the same coupling coefficient,
$\kappa_{c}\approx\kappa_{l}$. This means that, to acquire the maximal
polarization contribution, the seed photon should be fully polarized in the
state orthogonal to that of the laser photon, _i.e._
$(\Gamma_{1},~{}\Gamma_{2},~{}\Gamma_{3})=-[\varsigma_{1}(\nu),~{}\varsigma_{2}(\nu),~{}\sigma\varsigma_{3}(\nu)]$
with $\sigma_{l}\approx 1$ in (10). However, in the higher-energy region with
$\beta>0.4$, the difference between $\kappa_{c}$ and $\kappa_{l}$ becomes
considerable, which implies that the highest production yield is acquired from
the seed photon polarized in the state deviating from the orthogonal state of
the laser photon. Especially in the extremely high-energy region with
$\beta>0.95$ in which $\kappa_{l}$ is close to zero and $\kappa_{c}$ becomes
positive and dominates the polarization contribution, the highest yield
appears when the seed and laser photons have the pure circular polarization
parallel to each other.

We now know that the polarization coupling between the two photons in the LBW
process could contribute considerably to the production yield and the
polarization contributions $n_{1,2,3}$ in (2) are proportional to the Stoke
parameters of the laser photon as $n_{1,2}\sim
D(\nu)\kappa_{l}\varsigma_{1,2}(\nu)$ and $n_{3}\sim
D(\nu)\kappa_{c}\varsigma_{3}(\nu)$ with the coupling coefficients
$\kappa_{l,c}$ depending only on the dynamic parameter $\beta$ in the
perturbative regime $\xi\ll 1$. While in the upcoming laser-particle
experiments Abramowicz et al. (2021); Naranjo et al. (2021), the laser
intensity has increased to the magnitude of $\xi\sim\mathcal{O}(1)$, in which
the Breit-Wheeler pair production is in the transition regime from the
perturbative to the non-perturbative regime, a high number of laser photons
would be involved to satisfy the energy threshold in the center-of-mass frame,
and the NBW process would dominate the pair production. The polarization
contributions would, therefore, come from the polarization coupling with the
laser pulse, _i.e._ multiple laser photons, but not with a single laser
photon, and the coupling coefficients would depend also on the laser intensity
and field ellipticity.

##  IV nonlinear Breit-Wheeler process

In this section, we consider the NBW process stimulated by a high-energy
photon in the collision with the laser pulse in the intermediate intensity
region $\xi\sim\mathcal{O}(1)$. This is the typical setup for the upcoming
laser-particle experiment in LUXE Abramowicz et al. (2021); Jacobs (2021). To
show the polarization effect clearly, we fix the energy parameter $\eta$ and
adjust the relative polarization of the seed photon and laser pulse.

The background laser field is expressed as

$\displaystyle
a^{\mu}(\theta,\phi)=m\xi~{}\textrm{Re}\left\\{\left[0,a_{x}(\theta),a_{y}(\theta),0\right]e^{-i\phi}\right\\}f(\phi),$
(11)

where $\textrm{Re}\left\\{\cdot\right\\}$ means the real part of the argument,
$a_{x}(\theta)=\cos\theta-i\delta\sin\theta$,
$a_{y}(\theta)=\sin\theta+i\delta\cos\theta$. $\delta\in[-1,1]$ characterizes
not only the rotation of laser field: $\delta/|\delta|=1$ means the left-hand
rotation and $\delta/|\delta|=-1$ is right-hand rotation, but also the
ellipticity $|\delta|$ of the laser pulse: $|\delta|=0,~{}1$ corresponds,
respectively, to the linearly and circularly polarized laser background and
$0<|\delta|<1$ gives a laser pulse with the elliptical polarization. The semi-
major axis of the elliptical laser field is along ($\cos\theta,~{}\sin\theta$)
with the deflection angle $\theta\in[-\pi,~{}\pi]$ in the transverse plane.
$f(\phi)$ depicts the envelope of the laser pulse. The polarization of the
laser field could also be described with the classical Stokes parameters
$(\varsigma_{1},~{}\varsigma_{2},~{}\varsigma_{3})$ Jackson (1999) as

$\displaystyle\begin{aligned}
\varsigma_{1}&=\frac{|a_{x}|^{2}-|a_{y}|^{2}}{|a_{x}|^{2}+|a_{y}|^{2}}=\frac{1-\delta^{2}}{1+\delta^{2}}\cos{2\theta}\,,&\\\
\varsigma_{2}&=\frac{a^{*}_{x}a_{y}+a_{x}a^{*}_{y}}{|a_{x}|^{2}+|a_{y}|^{2}}=\frac{1-\delta^{2}}{1+\delta^{2}}\sin{2\theta}\,,&\\\
\varsigma_{3}&=i\frac{a_{x}a^{*}_{y}-a^{*}_{x}a_{y}}{|a_{x}|^{2}+|a_{y}|^{2}}=\frac{2\delta}{1+\delta^{2}}\,.\end{aligned}$
(12)

where $\varsigma^{2}_{1}+\varsigma^{2}_{2}+\varsigma^{2}_{3}=1$. The total
linear polarization degree of the laser pulse is given as
$\varsigma_{l}=(\varsigma^{2}_{1}+\varsigma^{2}_{2})^{1/2}=(1-\delta^{2})/(1+\delta^{2})$,
and the laser’s circular polarization degree is given by $\varsigma_{3}$. The
equivalence between the laser Stokes parameters (12) and those of the laser
photon (6) can be seen when we consider a relatively long laser pulse with the
slowly varying envelope $f^{\prime}(\phi)\approx 0$ and
$|\tilde{f}(\nu+1)|\ll|\tilde{f}(\nu-1)|$ at $\nu\geq 1$ Tang and King (2021).
The frequency components of the laser pulse can be written approximately as
$\tilde{a}^{\mu}(\nu)\approx
m\xi/2~{}\left[0,a_{x}(\theta),a_{y}(\theta),0\right]\tilde{f}(\nu-1)$ and
therefore $\varsigma_{i}\approx\varsigma_{i}(\nu)$ with $i=1,2,3$.

###  IV.1 Numerical results

To show the importance of polarization contributions and their dependence on
the corresponding laser Stokes parameters, we first present the numerical
results for the NBW process stimulated by a $16.5~{}\textrm{GeV}$ photon in
the head-on collision with the laser pulse in the intermediate intensity
region $\xi\sim\mathcal{O}(1)$. The pulse envelope is given as
$f(\phi)=\cos^{2}[\phi/(4\sigma)]$ in $\left|\phi\right|<2\pi\sigma$ and
$f(\phi)=0$ otherwise, where $\sigma=8$. The calculations have been done with
the laser central frequency $\omega=4.65~{}\textrm{eV}$, as an example, which
is the third harmonic of the normal laser with the wavelength
$\lambda=0.8~{}\mu\textrm{m}$. For the detail calculation of (1), one can
refer to the presentation in Ref. Tang (2021) and the analogous calculation in
Ref. King and Tang (2020) for the polarized nonlinear Compton scattering.

Figure 2: The energy spectra of the produced positron via the NBW process in
the head-on collision between a polarized photon and the laser pulse with
different ellipticity: (a) $\delta=1$ circular polarization, (b) $\delta=0.5$
elliptical polarization, and (c) $\delta=0$ linear polarization. The
contributions from an unpolarized photon $n_{0}$ and those $n_{i}$ coupling to
the photon polarization $\Gamma_{i}$ are compared. The energy of the seed
photon is $16.5~{}\textrm{GeV}$. The laser pulse has the intensity $\xi=1$,
central frequency $\omega=4.65~{}\textrm{eV}$ and the deflection angle
$\theta=0$.

In Fig. 2, we present the energy spectra of the produced positrons in the
laser backgrounds with the same intensity $\xi=1$ but different ellipticity
$\delta=1,~{}0.5,~{}0$ in Figs. 2 (a), (b) and (c) respectively. As shown, the
potential contributions coupling to the photon polarization are indeed
appreciable for the total positron yield. For the circularly polarized laser
background, $\delta=1$ in (a) with
$(\varsigma_{1},~{}\varsigma_{2},~{}\varsigma_{3})=(0,~{}0,~{}1)$, the
relative importance of the contribution $n_{3}$, coupling to the circular
polarization $\Gamma_{3}$ of the seed photon, is about $n_{3}/n_{0}\approx
22.3\%$ compared to the unpolarized contribution $n_{0}$. The contributions
$n_{1,2}$ coupling to the photon’s linear polarization are zero, because the
background field has no linear polarization Tang (2022). By increasing the
linear polarization of the background field
$(\varsigma_{1},~{}\varsigma_{2},~{}\varsigma_{3})=(0.6,~{}0,~{}0.8)$ in (b)
with the ellipticity $\delta=0.5$, the polarized contribution $n_{1}$ becomes
important with $n_{1}/n_{0}\approx 27.8\%$, while the importance of the
polarized contribution $n_{3}$ decreases to about $n_{3}/n_{0}\approx 14.5\%$.
For the laser pulse with the full linear polarization in (c) with $\delta=0$
and $(\varsigma_{1},~{}\varsigma_{2},~{}\varsigma_{3})=(1,~{}0,~{}0)$, the
polarized contribution $n_{3}$ becomes zero, and the relative importance of
the polarized contribution $n_{1}$ increases to about $n_{1}/n_{0}\approx
32.6\%$. With the decrease of the laser ellipticity, the harmonic structure
becomes more clear in the energy spectra and appears around
$s_{n>5}=\\{1\pm[1-(2+\xi^{2})/(n\eta)]^{1/2}\\}/2$ when $\delta=0$ Tang
(2022).

In Fig. 2, the contribution $n_{2}$ is always zero with the change of the
laser ellipticity $\delta$. This is because the laser has no polarization
preponderance along the direction of $\theta=\pi/4$, _i.e._ $\varsigma_{2}=0$.
To see the effect of the field deflection angle $\theta$, we plot the
variation of the polarization contributions $n_{i}$ with the change of
$\theta$ in Fig. 3 (a) for $\xi=1$ and $\delta=0.5$. As shown, the
polarization contributions $n_{1,2}$ vary in the trend as
$(n_{1},~{}n_{2})\propto-(\cos 2\theta,\sin 2\theta)$ and $n_{3}$ is unchanged
for different $\theta$. All are in the same trend as the variation of the
corresponding laser Stokes parameters $\varsigma_{1,2,3}$ in (12). However, we
also note that the amplitude of the linearly polarized contribution
$(n^{2}_{1}+n^{2}_{2})^{1/2}$ is constant with the change of $\theta$ shown as
the green dotted lines in Fig. 3 (a). Therefore, the maximized polarization
contribution $n_{p}$ in (3) from the optimized polarization (4) is independent
on the field’s deflection angle $\theta$ as shown in Fig. 3 (b), in which we
also find that the unpolarized contribution $n_{0}$ is unchanged for different
$\theta$. This is because of the azimuthal symmetry of the interaction
geometry. We can thus conclude that, for laser pulses with the fixed
ellipticity $\delta$ and intensity $\xi$, the field’s deflection angle
$\theta$ can only alter the relative value of the linear polarization
contributions $n_{1,~{}2}$ with the constant amplitude
$(n^{2}_{1}+n^{2}_{2})^{1/2}$, but not change the circularly polarized
($n_{3}$) and unpolarized ($n_{0}$) contributions. To show the correlation
between the polarization contribution $n_{i}$ and the corresponding laser
Stokes parameter $\varsigma_{i}$, we fit the numerical results in Fig. 3 (a)
respectively as
$n_{1}:~{}n_{1}(\theta=0)/\varsigma_{1}(\theta=0)\varsigma_{1}$,
$n_{2}:~{}n_{2}(\theta=\pi/4)/\varsigma_{2}(\theta=\pi/4)\varsigma_{2}$, and
$n_{3}:~{}n_{3}(\theta=0)/\varsigma_{3}(\theta=0)\varsigma_{3}$, and find the
precise agreement between the numerical results and data fitting.

Figure 3: Different contributions to the positron yield of the NBW process in
the elliptically polarized laser pulse with $\delta=0.5$ and the deflection
angle $\theta\in[0,\pi]$. (a) The variation of the polarization contributions
$n_{1,2,3}$ with the change of the field deflection angle. The full QED
results (‘cycle’, ‘plus’ and ‘square’) are fitted with the corresponding laser
Stokes parameters as $c_{1,2,3}\varsigma_{1,2,3}$, where
$c_{1}=n_{1}(\theta=0)/\varsigma_{1}(\theta=0)$,
$c_{2}=n_{2}(\theta=\pi/4)/\varsigma_{2}(\theta=\pi/4)$, and
$c_{3}=n_{3}(\theta=0)/\varsigma_{3}(\theta=0)$. The green dotted lines denote
the amplitude of the linear polarization contribution, _i.e._
$\pm(n^{2}_{1}+n^{2}_{2})^{1/2}$. (b) The unpolarised contribution $n_{0}$ and
the maximized polarization contribution $n_{p}$ in (3) from the seed photon
with the optimal polarization in (4). The other parameters are the same as in
Fig. 2. Figure 4: Different contributions to the positron yield of the NBW
process in the laser pulse with different ellipticity $\delta\in[0,~{}1]$, but
the fixed laser power density $I=\xi^{2}(1+\delta^{2})/2=1$ and deflection
angle $\theta=\pi/9$. (a) The unpolarized contribution $n_{0}$ and the
maximized polarization contribution $n_{p}$ from the seed photon with the
optimal polarization in (4). The relative importance $n_{p}/n_{0}$ of the
maximal polarization contribution $n_{p}$ is also plotted and compared with
that of the polarization contribution
$n^{\prime}_{p}=-(\varsigma_{1}n_{1}+\varsigma_{2}n_{2}+\varsigma_{3}n_{3})$
from the photon state orthogonal to the laser polarization. (b) The variation
of the polarization contributions $n_{1,2,3}$ with the change of the laser
ellipticity. The full QED results (‘cycle’, ‘plus’ and ‘square’) are fitted
with the corresponding laser Stokes parameters as
$c_{1,2,3}\varsigma_{1,2,3}$, where
$c_{1,2}=n_{1,2}(\delta=0)/\varsigma_{1,2}(\delta=0)$ and
$c_{3}=n_{3}(\delta=1)/\varsigma_{3}(\delta=1)$. The laser power density $I=1$
corresponds to the real power density $I\approx 3.84\times
10^{19}~{}\textrm{Wcm}^{-2}$. The other parameters are the same as in Fig. 2.

In Fig. 4, we show the variation of the different contributions to the
positron yield with the change of the laser ellipticity $\delta$ for the fixed
deflection angle $\theta=\pi/9$ and laser power density $I=1$, corresponding
to $3.84\times 10^{19}~{}\textrm{W}/\textrm{cm}^{2}$. As shown in Fig. 4 (a),
both the unpolarized contribution $n_{0}$ and the maximized polarization
contribution $n_{p}$ from the optimal polarization (4) [shown in Fig. 5 (b)]
decrease with the increase of the laser ellipticity $\delta$ from $0$ to $1$.
This is because of the decrease of the field intensity
$\xi=[2I/(1+\delta^{2})]^{1/2}$. Simultaneously, the relative importance,
$n_{p}/n_{0}$, of the maximized polarization contribution decreases from about
$31.6\%$ at $\delta=0$ for a linearly polarized laser pulse to about $22.3\%$
at $\delta=1$ for the laser pulse with pure circular polarization. For
comparison, we also plot the importance of the polarization contribution
$n^{\prime}_{p}=-(\varsigma_{1}n_{1}+\varsigma_{2}n_{2}+\varsigma_{3}n_{3})$
from the orthogonal state of the laser polarization, which is clearly smaller
than that from the optimal polarization state especially for the elliptically
polarized laser with $\delta\approx 0.5$. In Fig. 4 (b), we see that the
amplitude of the linear polarization contributions $n_{1,2}$ decrease with the
increase of $\delta$, while the amplitude of the contribution from the
circular polarization, $n_{3}$, increases. These variation are again in the
same trend as the laser Stokes parameters in (12). The difference between the
two linear polarization contributions can be depicted as
$n_{1}/n_{2}\approx\tan 2\theta=\varsigma_{1}/\varsigma_{2}$. The numerical
results in Fig. 4 (b) are respectively fitted as
$n_{1,2}:~{}n_{1,2}(\delta=0)/\varsigma_{1,2}(\delta=0)\varsigma_{1,2}$ and
$n_{3}:~{}n_{3}(\delta=1)/\varsigma_{3}(\delta=1)\varsigma_{3}$, and again, we
see the agreement between the numerical results and data fitting. The slight
difference around $\delta\approx 0.4$ implies the dependence of the
polarization coupling between the seed photon and laser pulse on the laser
ellipticity, as we will see later.

In this section, we investigate the NBW process in the laser pulse with the
ellipticity $\delta\in[0,1]$ and deflection angle $\theta\in[0,\pi]$. For the
laser pulse with the ellipticity $\delta\in[-1,0]$, the laser field would
rotates in the opposite direction as the laser with the ellipticity $-\delta$
(see the expression for $\varsigma_{3}$). The calculations would be consistent
with the above results, except that the polarized contribution $n_{3}$ would
change sign, but keeps the same amplitude. For the laser pulse with the
deflection angle $\theta\in[-\pi,0]$, all the above results would also be the
same except the polarized contribution $n_{2}$ would change sign because of
the odd property of $\varsigma_{2}$. All the calculations have be done for a
relative long laser pulse, and for a ultra-short laser pulse, the conclusion
would be different.

###  IV.2 Polarization coupling coefficients

To manifest the dependence of the polarization contributions on the laser
Stokes parameters, we consider an elliptically polarized monochromatic field
with $f(\phi)=1$ in (11). This is a good approximation for the pulses with
slowly-varying envelope $f^{\prime}(\phi)\approx 0$. After integrating the
transverse momenta in (1), we can acquire the polarization contributions as

$\displaystyle(n_{1},n_{2},n_{3})=\alpha
I(\kappa_{nl}~{}\varsigma_{1},\kappa_{nl}~{}\varsigma_{2},\kappa_{nc}~{}\varsigma_{3})$
(13)

where

$\displaystyle\kappa_{nl}$
$\displaystyle=\frac{1}{\pi\eta}\hat{T}\sin\left(\frac{\vartheta\Lambda}{2\eta
ts}\right)~{}g(\vartheta,\varphi)\,,$ (14a) $\displaystyle\kappa_{nc}$
$\displaystyle=\frac{1}{\pi\eta}\hat{T}\cos\left(\frac{\vartheta\Lambda}{2\eta
ts}\right)h_{s}\left(\textrm{sinc}^{2}\frac{\vartheta}{2}-\textrm{sinc}\vartheta\right)\vartheta$
(14b)

are the coupling coefficients between the polarization of the seed photon and
that of the laser pulse in the NBW process, and

$\displaystyle g(\vartheta,\varphi)=$
$\displaystyle\cos\vartheta+\textrm{sinc}^{2}\frac{\vartheta}{2}-2\textrm{sinc}~{}\vartheta$
$\displaystyle+$
$\displaystyle\frac{1}{\varsigma_{l}}\left(1+\textrm{sinc}^{2}\frac{\vartheta}{2}-2\textrm{sinc}~{}\vartheta\right)\cos
2\varphi\,.$

$\Lambda$ is the Kibble mass and expressed as Brown and Kibble (1964)

$\Lambda=1+I-I\textrm{sinc}^{2}\frac{\vartheta}{2}-I\varsigma_{l}\cos
2\varphi\left(\textrm{sinc}^{2}\frac{\vartheta}{2}-\textrm{sinc}\vartheta\right)$

depending on the laser power density $I=\xi^{2}(1+\delta^{2})/2$ and its
linear polarization degree $\varsigma_{l}$. $\hat{T}$ is the integral operator
given as

$\hat{T}=\int^{1}_{0}\mathrm{d}s\int^{\infty}_{-\infty}\mathrm{d}\varphi\int^{\infty}_{0}\frac{\mathrm{d}\vartheta}{\vartheta}\,,$

with the average phase $\varphi=(\phi_{1}+\phi_{2})/2$ and the interference
phase $\vartheta=\phi_{1}-\phi_{2}$ Dinu et al. (2016); Seipt (2017); Ilderton
et al. (2019).

As we can see, in the NBW process, the polarization contribution $n_{i}$ is
also proportional directly to the corresponding laser Stokes parameter
$\varsigma_{i}$, as shown in Figs. 3 and 4, with the coupling coefficients in
(14) depending not only on the laser power, but also on the field ellipticity.
The two linear polarization components share, again, the same coupling
coefficient because of the symmetry of rotating the linear polarization axis
as discussed in Fig. 3. We put the fine structure constant $\alpha$ out of the
coupling coefficients as the NBW process is a single-vertex process, and $I$
is because of the increase of the contributions with the laser power and in
the perturbative regime, $n_{i}\propto\xi^{2}$ in (5).

Figure 5: (a) The variation of the coupling coefficients
$\kappa_{nl},~{}\kappa_{nc}$ with the change of the field ellipticity. The
ratio $\sigma_{n}=\kappa_{nc}/\kappa_{nl}$ is also plotted with the right
$y$-axis. The coefficient $\kappa_{nl}$ calculated from $n_{1}$ is exactly the
same as that from $n_{2}$. (b) The Stokes parameters of the photon’s optimal
polarization in (3) for different $\delta$. We also show the comparison with
the orthogonal state, $-(\varsigma_{1},\varsigma_{2},\varsigma_{3})$ of the
laser polarization. The same parameters in Fig. 4 are used.

In Fig. 5 (a), we present the dependence of the coupling coefficients
$\kappa_{nl}$ and $\kappa_{nc}$ on the field ellipticity for the lasers with
the fixed power $I=1$ and relatively long duration. As shown, the value of
$\kappa_{nl}$ and $\kappa_{nc}$ vary slightly with the change of the field
ellipticity $\delta$, and there exists significant difference between
$\kappa_{nl}$ and $\kappa_{nc}$ with the ratio $\kappa_{nc}/\kappa_{nl}<1$,
which also changes for different $\delta$.

The dependence of $\kappa_{nl}$ and $\kappa_{nc}$ on the laser power density
is presented in Fig. 6 (a) for the fixed field ellipticity $\delta=0.5$ and
deflection angle $\theta=\pi/8$. As shown, in the low-power density region
$I<10^{-3}$, $\kappa_{nl}$ and $\kappa_{nc}$ are independent on the laser
power $I$ because the LBW process dominates the production, $\kappa_{nl}$ and
$\kappa_{nc}$ can be acquired alternatively from the perturbative result (5)
with $\kappa_{l}$ and $\kappa_{c}$ depending only on the parameter $\beta$.
The value of $\kappa_{nl}$ and $\kappa_{nc}$ are determined by the energy
parameter $\eta$ and the pulse envelope. In this region, the positron yield
increases as $n_{0},n_{p}\propto I$ shown in Fig. 6 (c) because of the single-
photon effect with the high-frequency components from the finite-pulse effect
Tang and King (2021). In the intermediate laser power region,
$10^{-3}<I<10^{-1}$, the coupling coefficients increase as
$\kappa_{nl},\kappa_{nc}\propto I^{3}$ because of the multiphoton perturbative
effect, in which $4=\lceil 2/\eta\rceil$ laser photons are involved in the
production process and the positron yield increase in the trend as
$n_{0},n_{p}\propto I^{4}$ in Fig. 6 (c), where $\lceil x\rceil$ denotes the
minimal integer larger than $x$. With the further increase of the laser power,
$I\gtrsim 0.5$, this $4$-photons channel is forbidden and a higher number of
laser photons, $n=\lceil 2(1+I)/\eta\rceil$, would be involved in the
production process. Therefore, the fully non-perturbative effect would be
dominant. The increase of the coupling coefficients $\kappa_{nl}$ and
$\kappa_{nc}$ become slower, as well as the increase of the positron yield in
Fig. 6 (c). In Fig. 6 (a), we can also see the evident difference between
$\kappa_{nl}$ and $\kappa_{nc}$ in the broad laser power region with the ratio
$\kappa_{nc}/\kappa_{nl}<1$ depending also sensitively on the laser power.
This difference would result in the deviation of the optimal photon
polarization from the completely orthogonal state of the laser polarization.

Figure 6: (a) The variation of the coupling coefficients
$\kappa_{nl},~{}\kappa_{nc}$ with the increase of the laser power density. The
dependence of the ratio $\sigma_{n}=\kappa_{nc}/\kappa_{nl}$ on the laser
power is also presented with the right $y$-axis. (b) The Stokes parameters of
the photon’s optimal polarization with the change of the laser power.
$\Gamma_{1}=\Gamma_{2}$ as the field deflection angle is $\theta=\pi/8$. (c)
The yield from the unpolarized contribution $n_{0}$ and the maximal
polarization contribution $n_{p}$, and the relative importance of the
polarization effect $n_{p}/n_{0}$. In (a) and (b), the pink dotted lines are
the corresponding perturbative results acquired from (5), and the black dotted
lines show the varying trend of the curves. The field ellipticity is
$\delta=0.5$. The other parameters are the same as in Fig. 4.

###  IV.3 Optimal photon polarization

From (14), the optimal polarization of the seed photon (4) can be written as

$\displaystyle(\Gamma_{1},~{}\Gamma_{2},~{}\Gamma_{3})=\hat{\kappa}_{nl}\frac{(\varsigma_{1},~{}\varsigma_{2},~{}\sigma_{n}\varsigma_{3})}{(\varsigma_{1}^{2}+\varsigma_{2}^{2}+\sigma^{2}_{n}~{}\varsigma_{3}^{2})^{1/2}}\,,$
(15)

based on the polarization of the laser pulse, where $\hat{\kappa}_{nl}=-1$ is
the sign of $\kappa_{nl}$ acquired numerically, and
$\sigma_{n}=\kappa_{nc}/\kappa_{nl}$ denotes the difference between the
coupling coefficients $\kappa_{nl}$ and $\kappa_{nc}$. If $\sigma_{n}\neq 1$,
the photon’s optimal polarization state would deviate from the orthogonal
state $-(\varsigma_{1},\varsigma_{2},\varsigma_{3})$ of the laser
polarization.

As shown in Fig. 5 (a), $\sigma_{n}$ is much smaller than $1$ for different
$\delta$. Therefore, the optimal polarization state of the seed photon, for
the maximal yield, is much different from the orthogonal state
$-(\varsigma_{1},\varsigma_{2},\varsigma_{3})$ of the laser polarization as
one can see in Fig. 5 (b), except in the regions around $\delta\approx
0,~{}1$, where the laser is linearly and circularly polarized, respectively.
With the optimized photon polarization in Fig. 5 (b), the production yield
could be enhanced for more than $20\%$ compared to the unpolarized case as
shown in Fig. 4 (a).

In Fig. 6 (b), the optimal polarization state of the seed photon is presented
in a broad laser power region for the specified ellipticity $\delta=0.5$.
Because the field deflection angle is $\theta=\pi/8$, the two linear
polarization components are equal, $\Gamma_{1}=\Gamma_{2}$. Again, because of
the evident difference between $\kappa_{nl}$ and $\kappa_{nc}$ in Fig. 6 (a),
the photon’s optimal polarization state deviates considerably from the
orthogonal state of the laser polarization as shown in Fig. 6 (b). Especially
in the non-perturbative regime $I>0.5$, the circular polarization degree
$|\Gamma_{3}|$ of the optimal polarization decreases rapidly with the increase
of $I$, because of the rapid decrease of the ratio $\kappa_{nc}/\kappa_{nl}$
for larger $I$ in Fig. 6 (a), which means that the contribution from the
circular polarization becomes less important. In the ultra-high intensity
regime $\xi\gg 10$ (not shown in Fig. 6), in which the locally constant field
approximation would work precisely Ilderton et al. (2019); Tang (2022), the
contribution from the circular polarization would be negligible, _i.e._
$k_{nc}\to 0$ and $\Gamma_{3}\to 0$. This is because the formation length of
the NBW process becomes much shorter than the typical length of the field
variation Ritus (1985) and the laser pulse would work as a linearly polarized
field with the direction varying with the laser phase Tang (2022).

With the polarization-optimized seed photon, the positron yield could be
enhanced appreciably as shown in Fig. 6 (c). In the perturbative intensity
region $I<10^{-3}$, the positron yield could be enhanced more than $55\%$ by
the polarization effect compared with the unpolarized case, and in the multi-
photon perturbative region $10^{-3}<I<10^{-1}$, the yield enhancement is about
$34\%$ from the optimized polarization state. With the further increase of the
laser power, even though the relative importance of the polarization
contribution becomes less, the positron yield could still be improved for more
than $16\%$ at $I\lesssim 50$.

##  V Conclusion

The optimization of the photon polarization state to the maximal positron
yield of the Breit-Wheeler pair production is investigated in arbitrarily
polarized plane wave backgrounds for a broad intensity region. Both the
polarization of the photon and the laser pulse are comprehensively described
with the classical Stokes parameters.

The optimal polarization state of the seed photon is resulting from the
polarization coupling with the laser pulse/photon in the production process.
For the laser pulse with the pure linear or circular polarization, the seed
photon’s optimal polarization is the orthogonal state of the laser pulse.
However, because of the evident difference between the coupling coefficients
for the linear and circular polarization components, the seed photon’s optimal
polarization state in elliptically polarized laser backgrounds, deviates
considerably from the orthogonal state of the laser polarization, especially
in the ultrahigh-intensity regime in which the linear-polarization coupling
coefficient is much larger than that of the circular polarization and thus the
seed photon’s optimal polarization would tend to the linear polarization.

With the polarization-optimized seed photon, the positron yield could be
considerably enhanced in a broad intensity region. For the laser intensity
region, $\xi\sim\mathcal{O}(1)$, of current laser-particle experiments, the
yield enhancement from the optimized photon polarization could be more than
$20\%$ compared to the unpolarized case.

##  VI Acknowledgments

The author thank A. Ilderton for helpful suggestions and comments on the
manuscript. The author acknowledge the support from the National Natural
Science Foundation of China, Grant No.12104428. The work was carried out at
Marine Big Data Center of Institute for Advanced Ocean Study of Ocean
University of China.

## References

  * Breit and Wheeler (1934) G. Breit and J. A. Wheeler, Phys. Rev. 46, 1087 (1934), URL https://link.aps.org/doi/10.1103/PhysRev.46.1087.
  * Baier and Grozin (2002) V. N. Baier and A. G. Grozin, arXiv e-prints hep-ph/0209361 (2002), eprint hep-ph/0209361.
  * Adam et al. (2021) J. Adam, L. Adamczyk, J. R. Adams, J. K. Adkins, G. Agakishiev, M. M. Aggarwal, Z. Ahammed, I. Alekseev, D. M. Anderson, A. Aparin, et al. (STAR Collaboration), Phys. Rev. Lett. 127, 052302 (2021), URL https://link.aps.org/doi/10.1103/PhysRevLett.127.052302.
  * Reiss (1962) H. R. Reiss, Journal of Mathematical Physics 3, 59 (1962), URL https://doi.org/10.1063/1.1703787.
  * Di Piazza et al. (2012) A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, Rev. Mod. Phys. 84, 1177 (2012), URL https://link.aps.org/doi/10.1103/RevModPhys.84.1177.
  * Gonoskov et al. (2021) A. Gonoskov, T. G. Blackburn, M. Marklund, and S. S. Bulanov, arXiv e-prints arXiv:2107.02161 (2021), eprint 2107.02161.
  * Fedotov et al. (2022) A. Fedotov, A. Ilderton, F. Karbstein, B. King, D. Seipt, H. Taya, and G. Torgrimsson, arXiv e-prints arXiv:2203.00019 (2022), eprint 2203.00019.
  * Burke et al. (1997) D. L. Burke, R. C. Field, G. Horton-Smith, J. E. Spencer, D. Walz, S. C. Berridge, W. M. Bugg, K. Shmakov, A. W. Weidemann, C. Bula, et al., Phys. Rev. Lett. 79, 1626 (1997), URL https://link.aps.org/doi/10.1103/PhysRevLett.79.1626.
  * Bamber et al. (1999) C. Bamber, S. J. Boege, T. Koffas, T. Kotseroglou, A. C. Melissinos, D. D. Meyerhofer, D. A. Reis, W. Ragg, C. Bula, K. T. McDonald, et al., Phys. Rev. D 60, 092004 (1999), URL https://link.aps.org/doi/10.1103/PhysRevD.60.092004.
  * Nikishov and Ritus (1964) A. Nikishov and V. Ritus, Sov. Phys. JETP 19, 529 (1964).
  * Heinzl et al. (2010) T. Heinzl, A. Ilderton, and M. Marklund, Physics Letters B 692, 250 (2010), ISSN 0370-2693, URL https://www.sciencedirect.com/science/article/pii/S0370269310008968.
  * Krajewska and Kamiński (2012) K. Krajewska and J. Z. Kamiński, Phys. Rev. A 86, 052104 (2012), URL https://link.aps.org/doi/10.1103/PhysRevA.86.052104.
  * Titov et al. (2012) A. I. Titov, H. Takabe, B. Kämpfer, and A. Hosaka, Phys. Rev. Lett. 108, 240406 (2012), URL https://link.aps.org/doi/10.1103/PhysRevLett.108.240406.
  * Fedotov and Mironov (2013) A. M. Fedotov and A. A. Mironov, Phys. Rev. A 88, 062110 (2013), URL https://link.aps.org/doi/10.1103/PhysRevA.88.062110.
  * Titov et al. (2016) A. I. Titov, B. Kämpfer, A. Hosaka, T. Nousch, and D. Seipt, Phys. Rev. D 93, 045010 (2016), URL https://link.aps.org/doi/10.1103/PhysRevD.93.045010.
  * Jansen and Müller (2013) M. J. A. Jansen and C. Müller, Phys. Rev. A 88, 052125 (2013), URL https://link.aps.org/doi/10.1103/PhysRevA.88.052125.
  * Jansen and M¨¹ller (2017) M. J. Jansen and C. M¨¹ller, Physics Letters B 766, 71 (2017), ISSN 0370-2693, URL https://www.sciencedirect.com/science/article/pii/S0370269316308024.
  * Titov et al. (2018) A. I. Titov, H. Takabe, and B. Kämpfer, Phys. Rev. D 98, 036022 (2018), URL https://link.aps.org/doi/10.1103/PhysRevD.98.036022.
  * Ilderton (2019) A. Ilderton, Phys. Rev. D 100, 125018 (2019), URL https://link.aps.org/doi/10.1103/PhysRevD.100.125018.
  * Ilderton (2020) A. Ilderton, Phys. Rev. D 101, 016006 (2020), URL https://link.aps.org/doi/10.1103/PhysRevD.101.016006.
  * King (2020) B. King, Phys. Rev. A 101, 042508 (2020), URL https://link.aps.org/doi/10.1103/PhysRevA.101.042508.
  * Tang (2021) S. Tang, Phys. Rev. A 104, 022209 (2021), URL https://link.aps.org/doi/10.1103/PhysRevA.104.022209.
  * Tang and King (2021) S. Tang and B. King, Phys. Rev. D 104, 096019 (2021), URL https://link.aps.org/doi/10.1103/PhysRevD.104.096019.
  * Ivanov et al. (2005) D. Y. Ivanov, G. Kotkin, and V. Serbo, The European Physical Journal C-Particles and Fields 40, 27 (2005), URL https://doi.org/10.1140/epjc/s2005-02125-1.
  * Katkov (2012) V. Katkov, Journal of Experimental and Theoretical Physics 114, 226 (2012), URL https://doi.org/10.1134/S1063776111160047.
  * Li et al. (2020) Y.-F. Li, R. Shaisultanov, Y.-Y. Chen, F. Wan, K. Z. Hatsagortsyan, C. H. Keitel, and J.-X. Li, Phys. Rev. Lett. 124, 014801 (2020), URL https://link.aps.org/doi/10.1103/PhysRevLett.124.014801.
  * Chen et al. (2022) Y.-Y. Chen, K. Z. Hatsagortsyan, C. H. Keitel, and R. Shaisultanov, arXiv e-prints arXiv:2201.10863 (2022), eprint 2201.10863.
  * Wistisen (2020) T. N. Wistisen, Phys. Rev. D 101, 076017 (2020), URL https://link.aps.org/doi/10.1103/PhysRevD.101.076017.
  * Titov and Kämpfer (2020) A. I. Titov and B. Kämpfer, The European Physical Journal D 74, 218 (2020), URL http://dx.doi.org/10.1140/epjd/e2020-10327-9.
  * Seipt and King (2020) D. Seipt and B. King, Phys. Rev. A 102, 052805 (2020), URL https://link.aps.org/doi/10.1103/PhysRevA.102.052805.
  * Tang (2022) S. Tang, Phys. Rev. D 105, 056018 (2022), URL https://link.aps.org/doi/10.1103/PhysRevD.105.056018.
  * Abramowicz et al. (2021) H. Abramowicz et al., Eur. Phys. J. Spec. Top. 230, 2445 (2021), URL https://doi.org/10.1140/epjs/s11734-021-00249-z.
  * Borysova (2021) M. Borysova, Journal of Instrumentation 16, C12030 (2021), URL https://doi.org/10.1088/1748-0221/16/12/c12030.
  * Macleod (2022) A. J. Macleod, Journal of Physics: Conference Series 2249, 012022 (2022), URL https://doi.org/10.1088/1742-6596/2249/1/012022.
  * Jacobs (2021) R. Jacobs, arXiv e-prints arXiv:2107.10026 (2021), eprint 2107.10026.
  * Meuren (2019) S. Meuren, in _Third Conference on Extremely High Intensity Laser Physics (ExHILP)_ (2019), URL https://conf.slac.stanford.edu/facet-2-2019/sites/facet-2-2019.conf.slac.stanford.edu/files/basic-page-docs/sfqed_2019.pdf.
  * Naranjo et al. (2021) B. Naranjo, G. Andonian, N. Cavanagh, A. D. Piazza, A. Fukasawa, E. Gerstmayr, R. Holtzapple, C. Keitel, N. Majernik, S. Meuren, et al., THPAB270 (2021), ISSN 2673-5490, URL https://jacow.org/ipac2021/papers/thpab270.pdf.
  * Salgado et al. (2021) F. C. Salgado, N. Cavanagh, M. Tamburini, D. W. Storey, R. Beyer, P. H. Bucksbaum, Z. Chen, A. Di Piazza, E. Gerstmayr, Harsh, et al., New Journal of Physics 24, 015002 (2021), ISSN 1367-2630, URL http://dx.doi.org/10.1088/1367-2630/ac4283.
  * Meuren et al. (2020) S. Meuren, P. H. Bucksbaum, N. J. Fisch, F. Fiúza, S. Glenzer, M. J. Hogan, K. Qu, D. A. Reis, G. White, and V. Yakimenko (2020), eprint 2002.10051.
  * Di Piazza (2015) A. Di Piazza, Phys. Rev. A 91, 042118 (2015), URL https://link.aps.org/doi/10.1103/PhysRevA.91.042118.
  * Di Piazza (2016) A. Di Piazza, Phys. Rev. Lett. 117, 213201 (2016), URL https://link.aps.org/doi/10.1103/PhysRevLett.117.213201.
  * Di Piazza (2017) A. Di Piazza, Phys. Rev. A 95, 032121 (2017), URL https://link.aps.org/doi/10.1103/PhysRevA.95.032121.
  * Di Piazza (2021) A. Di Piazza, Phys. Rev. D 103, 076011 (2021), URL https://link.aps.org/doi/10.1103/PhysRevD.103.076011.
  * Berestetskii et al. (1982) V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, _Quantum Electrodynamics (2nd ed.)_ (Butterworth-Heinemann, Oxford, 1982).
  * Jackson (1999) J. D. Jackson, _Classical Electrodynamics (3rd ed.)_ (Wiley, 1999).
  * Ng and Tsai (1977) Y. J. Ng and W.-y. Tsai, Phys. Rev. D 16, 286 (1977), URL https://link.aps.org/doi/10.1103/PhysRevD.16.286.
  * Greiner and Reinhardt (2009) W. Greiner and J. Reinhardt, _Quantum Electrodynamics (4th ed.)_ (Springer Berlin, 2009).
  * King and Tang (2020) B. King and S. Tang, Phys. Rev. A 102, 022809 (2020), URL https://link.aps.org/doi/10.1103/PhysRevA.102.022809.
  * Brown and Kibble (1964) L. S. Brown and T. W. B. Kibble, Phys. Rev. 133, A705 (1964), URL https://link.aps.org/doi/10.1103/PhysRev.133.A705.
  * Dinu et al. (2016) V. Dinu, C. Harvey, A. Ilderton, M. Marklund, and G. Torgrimsson, Phys. Rev. Lett. 116, 044801 (2016), URL https://link.aps.org/doi/10.1103/PhysRevLett.116.044801.
  * Seipt (2017) D. Seipt, arXiv preprint arXiv:1701.03692 (2017).
  * Ilderton et al. (2019) A. Ilderton, B. King, and D. Seipt, Phys. Rev. A 99, 042121 (2019), URL https://link.aps.org/doi/10.1103/PhysRevA.99.042121.
  * Ritus (1985) V. I. Ritus, J. Russ. Laser Res. 6, 497 (1985).