# Radiation Reaction Kinetics and Collective QED Signatures

A. Griffith<EMAIL_ADDRESS>K. Qu  N. J. Fisch  Astrophysical Sciences,
Princeton University

###### Abstract

Observing collective effects originating from the interplay between quantum
electrodynamics and plasma physics might be achieved in upcoming experiments.
In particular, the generation of electron-positron pairs and the observation
of their collective dynamics could be simultaneously achieved in a collision
between an intense laser and a highly relativistic electron beam through a
laser frequency shift driven by an increase in the plasma density increase. In
this collision, the radiation of high energy photons will serve a dual
purpose: first, in seeding the cascade of pair generation; and, second, in
decelerating the created pairs for detection. The deceleration results in a
detectable shift in the plasma frequency. This deceleration was previously
studied considering only a small sample of individual pair particles. However,
the highly stochastic nature of the quantum radiation reaction in the strong
field regime limits the descriptive power of the average behavior to the
dynamics of pair particles. Here, we examine the full kinetic evolution of
generated pairs in order to more accurately model the relativistically
adjusted plasma density. As we show, the most effective pair energy for
creating observable signatures occurs at a local minimum, obtained at finite
laser field strength due to the tradeoff between pair deceleration and the
relativistic particle oscillation at increasing laser intensity. For a small
number of laser cycles, the quantum radiation reaction may re-arrange the
generated pairs into anisotropic distributions in momentum space, although, in
the one dimensional simulations considered here, this anisotropy quickly
decreases.

††preprint: AIP/123-QED

##  I Introduction

Reaching the regime where quantum electrodynamics (QED) influences collective
plasma behavior is of wide interest. Di Piazza  _et al._ (2012); Zhang  _et
al._ (2020); Fedotov  _et al._ (2023) New interactions between quantum
electrodynamics and plasma physics with relevance to extreme astrophysical
environments Sironi and Giannios (2014); Turolla, Zane, and Watts (2015);
Uzdensky  _et al._ (2019) might be explored directly at next generation
laboratory experiments. Chen and Fiuza (2023) High field laser experiments
Chen  _et al._ (2009); Sarri  _et al._ (2015) will probe the strong field QED
regime. This regime can be characterized by significant nonlinear quantum
parameter

$\chi=\frac{\gamma}{E_{c}}\sqrt{\left|\mathbf{E}+c^{-1}\mathbf{v}\times\mathbf{B}\right|^{2}-(c^{-1}\mathbf{v}\cdot\mathbf{E})^{2}},$
(1)

for electron velocity $\mathbf{v}$ with corresponding Lorentz factor $\gamma$
and electromagnetic field $\mathbf{E},\mathbf{B}$ scaled to the Schwinger
critical field $E_{c}$. Achieving a $\chi\gg 1$ to probe the strong field
regime requires a combination of a high strength electromagnetic field and
relativistic electrons. This may be achieved with an all optical approach
using the beat wave of two intense lasers, Zhu  _et al._ (2016); Grismayer
_et al._ (2016) or with the collision between an electron beam and a single
intense laser. Abramowicz  _et al._ (2019); Yakimenko  _et al._ (2019); Meuren
_et al._ (2020, 2021) The production of high energy photons, through quantum
synchrotron radiation, and decay, through the Breit-Wheeler process, should
create a cascade of electron-positron pairs. With a large enough density and
volume of electron-positron pairs, experiments can reach the so called QED
plasma regime.

Our study aims to characterize the collective behavior of the QED plasma in
new experiments. To experimentally reveal collective effects, the generated
pairs need to be sufficiently dense and slow. These requirements can be met in
the collision between a high intensity laser and a high energy electron beam.
In this configuration, the generation of a QED plasma and the observation of
collective effects are simultaneously accomplished. As electrons and positrons
increase in density and drop in energy, the driving laser pulse experiences a
frequency shift as the plasma density increases from the added pairs. Qu,
Meuren, and Fisch (2021, 2022) The frequency shift caused by the creation of
new charged particles, analogous to the shift caused by ionization, Wilks,
Dawson, and Mori (1988); Joshi  _et al._ (1990); Savage  _et al._ (1993);
Mendonca (2000); Qu  _et al._ (2018); Qu and Fisch (2019) could serve as an
observable signature of collective QED plasma effects.

The role of the pair slowdown – and of reflection to magnify the contribution
of the pair particles – to producing the frequency signatures was demonstrated
in previous work. Griffith, Qu, and Fisch (2022) This occurs through a
combination of two effects. When pairs have high counter-propagating momenta
in the intense laser field such that $\chi\geq 10^{-1}$, the radiation
reaction may provide a significant average deceleration. When the particles
are reduced to a low enough energy, the effects of Lorentz force on particle
momenta may become significant, and provide a further ponderomotive force in
the presence of a laser gradient to temporarily decelerate and even reflect
pair particles. Previous work Griffith, Qu, and Fisch (2022) focussed on the
expected mean behavior of pair particles experiencing a radiation reaction
slowdown in a strong electromagnetic wave. However, as the contribution of
particles is inversely weighted by their energy; the most decelerated
particles will contribute more significantly. Thus, the plasma frequency, and
thereby the plasma signatures, will be dominated by slow particles. However,
as we address here, capturing the full dynamics of the low energy tail
requires a fuller kinetic picture which captures the stochastic effects of the
quantum radiation reaction across the full range of particle momenta.

The quantum radiation reaction’s effect on pair behavior is qualitatively
different from that of the classical radiation reaction. The importance of the
transition between the two has been demonstrated in great detail. This occurs
as $\chi$ becomes non-negligible. Theoretical results anticipate discrepancies
in emission spectra between classical models, which include the Lorentz-
Abraham-Dirac and Landau-Lifshitz, and QED models, which include the Local
Constant Field Approximation (LCFA) and the Local Monochromatic Approximation
(LMA). Blackburn (2020); Gonoskov  _et al._ (2022); Heinzl, King, and MacLeod
(2020) A number of recent executed and proposed experiments have sought to
distinguish between these models and validate them within the transition from
the classical to the QED regime. Blackburn  _et al._ (2014); Ridgers  _et al._
(2017); Poder  _et al._ (2018); Cole  _et al._ (2018); Li  _et al._ (2017);
Gong  _et al._ (2018) The common experimental configurations used to examine
the radiation reaction typically feature an intense laser set against a
counter-propagating high-energy electron beam. This is a similar
configuration, but at lower $\chi$, than envisioned for the proposed pair
generation experiments. We aim to address the relevance of changes in the
momentum distribution to possible collective signatures.

The different models of the radiation reaction have different implications for
plasma kinetics. In particular, previous work highlighted that, depending on
the regime, the radiation reaction may serve to cluster or disperse particles
in momentum space. Lehmann and Spatschek (2012); Neitz and Di Piazza (2013);
Mackenroth, Neitz, and Di Piazza (2013); Vranic  _et al._ (2016); Niel  _et
al._ (2018) The stochastic nature of the radiation reaction can allow for
qualitatively different particle dynamics. Geng  _et al._ (2019); Gong  _et
al._ (2019) In addition to the dynamics of pair particles in an
electromagnetic wave, the effect of the radiation reaction on particles in a
strong static magnetic field may also shift the momentum distribution. This
has recently been proposed as a possible driver of plasma instabilities.
Bilbao and Silva (2023); Zhdankin, Kunz, and Uzdensky (2023) The interplay of
the radiation reaction in the QED regime and collective plasma behavior is a
developing field, and finding correspondence to detectable signatures in the
laboratory would be of great interest.

In this work, we show how signatures of collective effects in the QED plasma
regime are affected by the distribution of pair momenta. We model pairs
experiencing a radiation reaction driven deceleration as they transition to
the Lorentz force dominated regime. Particles are initialized at quantum
nonlinear parameters higher than previous work to better capture the relevance
to experimental configurations which might generate a high density QED plasma.
The full distribution of initially high momentum particles rapidly slow down
and cluster at low momenta in numerical simulations. Increasing the laser
power to decrease the pair momenta results in a higher effective plasma
density, until a significant enough intensity is reached, at which point the
high oscillatory momentum in the laser dominates and reduces the signature.
Due to the tradeoff between deceleration and transverse momentum, there is a
maximum in signature growth at finite laser intensity. The distribution of
pairs experiences a large initial increase in the spread of momentum in the
parallel direction due to the stochastic nature of the quantum radiation
reaction. However, the widening is short lived, as pairs rapidly cluster at
low momenta when propagating through the laser pulse. A large difference in
parallel and perpendicular momentum spreads, driven by the radiation reaction,
might serve as a source of kinetic instabilities. In one dimension, for the
parameters considered, the radiation reaction cannot serve as a source for a
long-lived, highly anisotropic distribution. However, different pulse
configurations or higher dimensional effects could exhibit the increase in the
spread of parallel particle momenta.

This paper discusses the distributional dynamics for QED signatures. The paper
is organized as follows: In Section II we review the kinetic equation for the
radiation reaction in the strong field regime with a “collision like”
operator. In Section III we detail how this model is numerically applied to
the regime of pair slowdown for increasing signatures. In Section IV we
evaluate the evolution for frequency signatures. In Section V and Subsection
V.1 we examine the relevance of other moments of the distribution. In Section
VI we summarize our findings and further discuss the implications of our
results.

##  II Distributional dynamic equation

As a basis for determining the dynamics of the pair momentum distribution, we
first review a kinetic description of charged particle dynamics including the
quantum radiation reaction. We follow the formalism of Neitz. Neitz and Di
Piazza (2013) The kinetic dynamics are applied to the dynamics of created
pairs, where the dynamics are dominated by a strong externally applied laser
field, while the fields created by the pair plasma are comparatively weak. We
do not consider the separate problem of pair generation, only radiation and
oscillation in the strong externally applied field.

For a homogenous plasma driven by an external laser, the dynamics are
described as function of the laser phase $\phi=\omega_{0}(t-y)$ for a laser
propagating in the $y$ direction. We assume that $t+y$ derivatives vanish,
such that there is no spatial dependence to the distribution function $f$
independent of $t-y$. We note that the distribution of generated pairs, those
of primary interest and at low enough energies to be significantly slowed or
stopped, will be created by the laser, and thus their initial distribution
should indeed depend primarily on $\phi$, as creation will be primarily a
function of the laser phase. The external driving laser is taken to have a
polarization in the $z$ direction such that the pair
$\gamma^{2}=1+p_{y}^{2}+p_{z}^{2}$ and the distribution $f(\phi,p_{y},p_{z})$
evolves according to the dynamical equation

$\frac{p_{-}}{\gamma}\partial_{\phi}f+eE(\phi)\left(\frac{p_{-}}{\gamma}\partial_{p_{z}}f+\frac{p_{z}}{\gamma}\partial_{p_{y}}f\right)=C(f,p_{-},\phi).$
(2)

The momentum coordinate $p_{-}=\gamma-p_{y}$ is substituted for convenience as
it is conserved by the Lorentz force. On the right hand side $C$ is a
collision like operator coming from the stochastic and non-local emission of
high energy photons. Without further approximation Eqn. (2) can be simplified
by transforming all derivatives from the $(p_{z},\ p_{y})$ coordinate system
to the $(p_{z},\ p_{-})$ coordinate system

$\partial_{\phi}f+eE(\phi)\partial_{p_{z}}f=\frac{\gamma}{p_{-}}C(f,p_{-},\phi).$
(3)

The transform into the $p_{-}$ coordinates removes the $\partial_{p_{y}}$
derivative due to the conservation of $p_{-}$ by the Lorentz force when the
electric field is purely a function of $\phi$. The left hand side strictly
conserves $p_{-}$ of particles, and the right hand side conserves the $p_{z}$
of the particles as radiation is taken to strictly occur in the $y$ direction.
Performing a separation of variables on
$f(\phi,p_{y},p_{z})=m(\phi,p_{z})n(\phi,p_{-})$ a normalized version of $C$
taken to be described by the LCFA Neitz and Di Piazza (2013) may be
constructed as

$\displaystyle\frac{\gamma}{p_{-}}\frac{2\pi}{\sqrt{3}\alpha|a(\phi)|}\frac{C(f,p_{-},\phi)}{m(\phi,p_{z})}$
$\displaystyle=\int_{0}^{\infty}dl\
n(\phi,p\beta)\left[(\beta^{-1}+\beta^{-3}\mathcal{K}_{2/3}(l/\beta)-\beta^{-2}\int_{0}^{\infty}dj\mathcal{K}_{1/3}(j+l/\beta)\right]$
$\displaystyle-n(\phi,p)\int_{0}^{\infty}dl\left[(\beta^{-1}+\beta^{-3}\mathcal{K}_{2/3}(l)-\beta^{-2}\int_{0}^{\infty}dj\mathcal{K}_{1/3}(j+l)\right].$
(4)

This is written in units where $\hbar=c=m_{e}=1$, with a normalized vector
potential $\omega_{0}a(\phi)=eE(\phi)$ is taken to assume to follow the slowly
varying envelope approximation. Here $\mathcal{K}_{\nu}$ is the $\nu$’th
modified Bessel function of the second kind and for convenience the expression
$\beta=1+\frac{3}{2}\omega_{0}pa(\phi)$ is used.

The kinetic description may be reduced by separation of variables as the
distribution function vanishes as the momentum goes to infinity. The dynamics
of $m$ may be found by simply integrating over $p_{-}$. Given that right hand
side redistributes particles in $p_{-}$ space and conserves particle number it
follows that

$\partial_{\phi}m+eE(\phi)\partial_{p_{z}}m=0.$ (5)

This equation may be solved through the method of characteristics such that
$m\left(\phi,p_{z}\right)=m\left(\phi_{0},p_{z}+a(\phi)-a(\phi_{0})\right)$.
Integrating over $p_{z}$ gives the separate equation for $n$ which strictly
depends on the radiation reaction, to obtain

$\partial_{\phi}n=\frac{\gamma}{p_{-}}C(n,p_{-},\phi).$ (6)

The particular form of $C$ results in a shift towards lower $p_{-}$ as the
distribution evolves in $\phi$. This operator is non-local in momentum space
at significant $\chi$. As the average $p_{-}$ decreases, the dynamics become
less determined by the stochastic emission of high energy photons, and the
radiation reaction should correspond less to the LCFA. However, the particular
choice of model for small $p_{-}$ (and small $\chi$) over a short duration
should not significantly alter the particle dynamics. Over short times for
particles with $p_{-}$ such that $\chi\ll 1$ the change of the distribution
under any model of the radiation reaction should be minimal.

At small $p_{-}$ when the radiation reaction terminates, the distribution
function over $p_{-}$ will not change significantly, but the distribution over
$p_{y}$ can change significantly. This can be seen in the map from $p_{-}$ and
$p_{y}$,

$p_{y}=\frac{1+p_{z}^{2}-p_{-}^{2}}{2p_{-}},$ (7)

which picks up a significant nonlinearity when $p_{z}$ is no longer $\ll
p_{-}$. As $p_{z}$ scales with $a_{0}$, this will occur when particles have
been decelerated to energy $\gamma\sim a_{0}$. At this energy scale,
oscillations of $p_{z}$ may be strong enough to make $p_{y}$ go from negative
to positive, indicating pair reflection. For significant $p_{-}$, reflecting
pairs requires a strong laser. For pairs to be reflected for all time in one
dimension $p_{-}<1$, which will only be true for a small number of pairs.
However, in higher dimensional simulations, this need not be the case if pairs
are scattered to the side, such that they never experience the backend of the
laser. This physics is not captured by Eqn. (2).

In this paper we consider the impact of the radiation reaction not for the
purposes of the study of it alone, but for its applicability to frequency
shift signatures. We probe the regime where $\chi\gg 1$ which is expected for
the pairs which will be generated in proposed next generation experiments. Our
results differ from those presented in Neitz and Di Piazza, Neitz and Di
Piazza (2013) where $\chi$ is increased up to near unity. Notably, analytical
estimates don’t capture the pair behavior, as even at moderate $\chi$ large
discrepancies emerge between expansions of Eqn. (4) and the full integral
expression. To capture the high $\chi$ regime, the full integro-differential
equation must thus be solved, which becomes non-local in momentum space. To
handle the complexity of the expression, this is performed numerically.

##  III Numerical Implementation

The dynamics of $n$ as described by Eqn. (6) are evolved numerically using a
simple integro-differential solver. The dynamics are determined for a time-
varying scaled laser field which is taken purely to depend on
$\phi=\omega(t-x)$, such that there is a perpendicularly polarized $a(\phi)$.
The non-local collision like operator is pre-evaluated at varying $\chi$ using
Gauss-Laguerre quadrature. The term during the dynamical evolution is
evaluated over a cubic spline which is fit to the precomputed points which
span up to the maximum possible $\chi$ within the simulation. The time-
stepping of $n$ is performed in Julia DifferentialEquation.jl Rackauckas and
Nie (2017) at a thousand inhomogeneously distributed points in $p_{-}$ space.
High momentum packets with an initially Gaussian distribution are initialized
within the laser pulse to correspond to generated pairs. Low $p_{-}$ points
are sampled at higher density to resolve the finer features present at low
momenta, which become apparent at the later points in the simulation. The
$p_{-}$ space is resolved to capture particle momenta ranging from the initial
conditions down to the regime where the radiation reaction becomes weak,
thresholded numerically by $\chi^{2}=(p_{-}a_{0}\omega_{0})^{2}=10^{-5}$.
Pair-pair interactions, and the effects of the pairs upon the driving laser
are not considered beyond the anticipated frequency shift, as both terms are
expected and shown to be higher order.

##  IV Distributional dynamics for QED signatures

To evaluate total pair contributions we computed the pair momentum
distribution evolution in comparison to earlier results on particle slowdown
and reflection. For comparison, pairs are initialized with the same momentum
peak in a driving laser with the same envelope and intensity. For a Gaussian
laser pulse of peak intensity $6\times 10^{22}$ W/cm2, particles are
initialized in the rising edge with an energy of 10 GeV and an energy spread
of 1 GeV. The dynamics of the distribution in $p_{-}$ space are shown in Fig.
1, where particles rapidly slow down and accumulate at low $p_{-}$. For
initially large $\chi$ due to the emission of high energy photons there is an
increase in the spread of energy, however, as the average $\chi$ rapidly
decreases the spread correspondingly decreases.

Figure 1: Evolution of $n(p_{-})$ in momentum space results in particles
moving quickly from high counter-propagating momentum to low counter-
propagating momentum. At short times the spread in momentum space grows, but
as the average particle $\chi$ shrinks, particles quickly cluster in momentum
space.

While changes in the momentum of the particle distribution function are small
compared to the initial energy scale at late times, they are not small
compared to their final energy. This can be seen through a comparison of
early, most intense focus, and late times as shown in Fig. 2. Mean $p_{-}$
still decreases in the back-half of the laser pulse, resulting in an
increasing contribution to the frequency signature. While the spread of the
distribution in $p_{-}$ drops over the duration of the interaction, the spread
relative to the mean energy of the particles increases.

Figure 2: Initial, at focus, and final momentum distributions of particles in
the momentum distribution show a significant change in both the spread and
mean particle energy at late times.

Changes in the evolution across the $p_{-}$ axis can be related to the
evolution of pairs in $p_{y}$ space through the conservation of the canonical
momentum for a strictly transverse vector potential $\mathbf{A}$. As pairs
progress through the laser field and their energy decreases, eventually
oscillations in $p_{y}$ may become significant when $a(\phi)\sim\gamma$. This
transverse oscillation can be significant enough to induce reflection for a
large fraction of the pair particles. In $p_{y}$ space over half of the pairs
can be temporarily reflected in the peak of the laser pulse as shown in
Figure. 3. This requires a significant laser $a_{0}$, simulations at lower
$a_{0}$ will not decrease $p_{-}$ to a sufficient enough level for the Lorentz
force to overcome the initial particle momenta. As the laser progresses in one
dimension the change in the direction of pairs eventually becomes
insignificant as pair particles experience the downramp of the ponderomotive
force in the tail end of the laser pulse.

Figure 3: When the pair distribution is scaled from the $p_{-}$ space to
instead be represented in the $p_{y}$ space, the peaking of the distribution
function is sharpened. A significant fraction of particles are reflected at
the peak of the laser field, but eventually pairs settle to small, but
primarily counter-propagating momenta.

##  V Dynamics of Frequency Shift Signatures

The changing momentum distribution of pairs can be directly related to the
time varying current in the pair plasma. The change in the distribution
function in $p_{-}$ alters the frequency of the electromagnetic wave through a
time varying susceptibility. This susceptibility can be found by examining the
response of the plasma as determined by the kinetic dynamics of Eqn. (2),
which perturbatively changes the electromagnetic wave through an oscillating
current

$\displaystyle\Box\mathbf{A}$ $\displaystyle=\frac{4\pi}{c}\mathbf{J}$ (8)
$\displaystyle\mathbf{J}$ $\displaystyle=e\int
d^{3}p\frac{\mathbf{p}}{\gamma}f.$ (9)

The oscillations of the pairs are reduced by a factor of $\gamma$ in the
relativistic limit. For a wave polarized in the $z$ direction parallel
current, the corresponding $J_{z}$ is

$J_{z}=e\int_{-\infty}^{\infty}\int_{0}^{\infty}dp_{z}dp_{-}\frac{p_{z}}{\gamma}m(\phi_{0},p_{z}+eA(\phi)-eA(\phi_{0}))n(\phi,p_{-}).$
(10)

where the distribution function is represented in $(p_{z},\ p_{-})$ space and
the evolution of $p_{z}$ is determined by the conservation of canonical
momentum. We note that, in comparison to previous work, density is considered
here in the lab frame instead of the pair frame. When considering many pairs,
a single consistent frame must be chosen for pairs of varying momentum. The
expression given in Eqn. (10) in the case where the perpendicular momentum
spread is small, $m(\phi_{0},p_{z})=\delta(p_{z})$, results in an oscillating
transverse current of

$J_{z}=e^{2}A(\phi)\int_{0}^{\infty}dp_{-}\frac{2p_{-}}{1+a(\phi)^{2}+p_{-}^{2}}n(\phi,p_{-}),$
(11)

where $\gamma$ has been substituted by the appropriate relation to $p_{-}$ and
$p_{z}$. The integral given in Eqn. (11) serves as the effective plasma
density

$n_{\textrm{eff}}=\int\frac{n}{\gamma}dp=\int_{0}^{\infty}dp_{-}\frac{2p_{-}}{1+a(\phi)^{2}+p_{-}^{2}}n(\phi,p_{-}).$
(12)

The current is suppressed both by a high counter-propagating momentum, and the
strong oscillations in the passing wave. These factors compete depending on
the regime. Initially pairs have $p_{-}\gg a$, however, $p_{-}$ may be reduced
by the radiation reaction down to the level where they are of similar order.
As the average $p_{-}$ decreases faster with higher $a$, this results in a
balance between which terms dominate within the integral.

The effective current contribution can be calculated for distribution
evolution shown in Figure. 1. For a 10 GeV electron originating in the rising
edge of an electromagnetic wave at peak intensity of $6\times 10^{22}$ W/cm2
in the rising edge this results in a frequency contribution shift changing
with laser phase as shown in Fig. 4. The contributions of pairs rises several
orders of magnitude as $p_{-}$ rapidly decreases. At low enough $p_{-}$ the
gains of decreased momentum begin to saturate, and pairs oscillate around a
slowly decreasing average $\gamma$. After the peak of the laser pulse, as the
laser $a$ decreases, $\gamma$ can continue to decrease as the oscillations
become less significant, and a small amount of further decrease in $p_{-}$
occurs as in Fig. 2.

Figure 4: Changes in the $p_{-}$ momentum distribution increase the effective
plasma density and corresponding frequency as pairs slowdown in the laser. The
density saturates when oscillations in the transverse momentum become
significant enough to impact the average $\gamma$. Figure 5: The distribution
of pairs is evolved at constant $\chi=p_{-}(\phi_{0})a_{0}\omega_{0}$ for
varying $a_{0}$. Higher $a_{0}$ induces a larger increase in the effective
plasma density for lower $a_{0}$. However, at high $a_{0}$, $\gamma$ decreases
due to strong transverse oscillations, resulting in a decreasing effective
density within the laser pulse.

The importance of transverse oscillation becomes apparent when considering
which configurations of pairs are most effective for generating maximal
signature. If the product of the laser strength and pair momentum are kept
constant, such that the initial $\chi$ is constant, pair oscillation may
tradeoff against deceleration. This can be seen in Fig. 5 where a parameter
scan is performed across peak laser $a_{0}$, keeping $\chi$ constant by
decreasing the initial pair momentum. At low $a_{0}$ the laser is not
significant enough to decelerate particles to a low enough $p_{-}$ to increase
their impact. At a high $a_{0}$ transverse oscillations become significant
enough to suppress the signal. The tradeoff between these effects determines
the laser power which maximizes the effective density. A simple model where
the radiation reaction is taken to be decelerate particles up to the point
where $\chi=10^{-1}$, such that $p_{-}(0)=(10\omega_{0}a_{0})^{-1}$ captures
the trend of the dynamics. Assuming that all particles propagate in the laser
for enough time to drop to this level predicts an effective density of the
form

$n_{\textrm{eff}}\approx
n_{0}\frac{2\left(\frac{0.1}{\omega_{0}a_{0}}\right)}{1+a_{0}^{2}+\left(\frac{0.1}{\omega_{0}a_{0}}\right)^{2}}.$
(13)

This curve is plotted in red in Fig. 5, capturing the trend and estimated peak
as found in the full kinetic simulations. This estimate is over-optimistic
with regards to both low and high $a_{0}$, most likely as the $\chi=0.1$
estimate doesn’t capture the radiation reaction dynamics in these limits. We
note that this only optimizes for signature strength given some initial
generated pairs. The efficiency of pair generation, and its dependence on
laser strength and the momentum of the electron beam is a separate
consideration. As pair creation will increase with laser field strength, it is
expected that the total weighted density will plateau instead of decline. Qu,
Meuren, and Fisch (2021) Effective density is of limited use as a single
metric for measuring the impact of generated pairs on the laser frequency. The
precise relation between density changes and laser frequency changes instead
is determined by a more complicated integral over the time varying density.
Qu, Meuren, and Fisch (2021) However, the effective density should serve as a
reasonable, simpler proxy for this quantity.

###  V.1 Distributional Dynamics

To look for possible sources of high anisotropy we characterized the spread of
momentum for pairs originating at different points within the laser - electron
beam collision. An increase and then sudden decrease in the spread is notable
in Fig. 1, and we aim to determine for what range of pairs this behavior
holds. It is known that at significant $\chi$ the effective temperature of
particles may increase as they stochastically emit photons due to the quantum
radiation reaction. Neitz and Di Piazza (2013) In the evolution of the pair
distribution function this can be seen in Fig. 1 where in the rising edge of
the laser pulse the distribution greatly widens over the duration of several
laser cycles. However, in the bulk of the laser as the average $p_{-}$, and
correspondingly $\chi$, decreases the growth in the parallel temperature
rapidly turns to effective cooling. A significant increase in the parallel,
$p_{y}$, temperature, with a static perpendicular temperature, $p_{z}$, could
serve to create a large temperature anisotropy. Temperature anisotropy is a
well known source for plasma instabilities. Weibel (1959); Fried (1959);
Davidson  _et al._ (1972) We characterized wether the quantum radiation
reaction could serve as a unique source for instabilities.

Significant increases in pair parallel, relative to the laser propagation
direction, temperature cannot be sustained within the peak of the driving
laser. This can be seen in Fig. 6, where the distribution of pairs is
initialized at different phase (times) within the pulse. The second moment of
the distribution is calculated in comparison to the initial conditions

$\displaystyle\frac{T_{\parallel}}{T_{0}}$ $\displaystyle=\frac{n_{0}^{-1}\int
p_{y}^{2}n(\phi)dp_{-}-\left(n_{0}^{-1}\int
p_{y}n(\phi)dp_{-}\right)^{2}}{n_{0}^{-1}\int
p_{y}^{2}n(\phi_{0})dp_{-}-\left(n_{0}^{-1}\int
p_{y}n(\phi_{0})dp_{-}\right)^{2}},$ (14) $\displaystyle n_{0}$
$\displaystyle=\int n(\phi)dp_{-}$ (15)

Pairs which experience the high intensity parts of the pulse rapidly slowdown
and cool, as the field strength is high enough that the radiation dominates.
At low enough $p_{-}$ oscillations in $p_{z}$ begin to impact the map from
$p_{-}$ to $p_{y}$, even though the distribution is still strictly converging
in $p_{-}$. Only pairs that are initialized in the tail end of the laser pulse
experience the growth in the temperature associated with high $\chi$, without
the following fall. Pairs will primarily be generated in regions of high
field, so this suggests that in a full simulation coupling both pair
generation and radiation, it would be unlikely for a significant increase in
temperature to be observed across all pairs.

Figure 6: The pair $\langle(p_{y}-\bar{p_{y}})^{2}\rangle$ initially increases
regardless of where pairs are injected in the laser. However, as pairs
transition to low momentum and low $\chi$, this quantity rapidly shrinks. Only
pairs which originate in the weak tail of the laser pulse maintain a large
spread in $p_{y}$.

High parallel temperatures, and thus high temperature anisotropies, could be
maintained in the case where particles exit the high intensity laser field
before settling to low energy. This is present in pairs with originate in the
back half of the laser pulse. Furthermore, if pairs which originate in the
rising edge or peak of the laser are quickly side scattered, decreasing the
high field they experience, then the high parallel spread in momentum might
still remain “imprinted” on the distribution. This effect would not be
captured by the current simulation, which is strictly performed in one
dimension. Side scattering into the perpendicular direction has been noted to
be significant. Griffith, Qu, and Fisch (2022) Through these two mechanisms,
it is possible that the unique properties of the distribution function at
short times could be preserved for observation, or as a source of collective
plasma behavior.

##  VI Summary and Discussion

In conclusion, we provide a full description of pair momentum evolution,
identifying the tradeoff between deceleration and transverse oscillation.
Radiation reaction driven pair deceleration is necessary to produce detectable
frequency shifts from electron-positron pair creation. For the parameters
considered in previous work, Qu, Meuren, and Fisch (2022); Griffith, Qu, and
Fisch (2022) an intense laser can raise the relativistic mass weighted plasma
density orders of magnitude. As the average counter-propagating momentum,
$p_{-}=\gamma-p_{y}$, decreases in the driving laser from the radiation
reaction, the Lorentz force begins to significantly impact the effective
density. The effect of the Lorentz force is not primarily the redirection of
pair particles, but instead an increase in the pair transverse momentum,
$p_{z}$. When $p_{z}$ is on the same order of $p_{-}$, the effective density
is reduced from transverse oscillations, inducing a form of relativistic
transparency. This creates a tradeoff between high laser intensity to achieve
a rapid deceleration of generated pairs, and a low laser intensity to reduce
transverse oscillation.

A naive model for the particle energy, which sets a threshold of $\chi=0.1$
for the radiation reaction, captures the qualitative features of this
tradeoff, as shown in Fig. 5. A laser peak of strength $a_{0}\sim 150$
produces a maximal contribution of pairs to the frequency signature, assuming
a constant initial $\chi$ for the generated pairs. In isolation from the
coupled pair creation problem, this suggests that there is limited benefit
from going to higher intensity if the aim is to produce a more distinct
signature. Well known relativistic behavior in plasma physics restricts the
observability of the new QED plasma behavior which is of interest. However, it
is worth noting that the full coupled problem of pair creation and
deceleration may benefit from a more significant $a_{0}$. The tradeoff
observed for pair deceleration shows that generation cannot be considered in
isolation, and maximizing signatures will require solving the fully coupled
problem.

Changes in the width of the momentum distribution are large over short
timescales, but are quickly quenched in the high $\chi$ limit. Thus, the
changing momentum distribution alone cannot provide a new source for
collective QED plasma behavior. It is known that the radiation reaction can
seed plasma instabilities by creating unstable plasma distributions. Bilbao
and Silva (2023); Zhdankin, Kunz, and Uzdensky (2023) In the high $\chi$
regime, when pairs strictly propagate in one dimension, the distribution
function rapidly expanded and then contracted in parallel momentum. For pairs
which originate with a narrow spread of perpendicular momentum, it might be
the case that a large spread of parallel momentum could serve as a source for
an anisotropic, and thus unstable, distribution.

However, a highly anisotropic momentum distribution cannot be maintained for
pairs which experience a multi-cycle laser pulse with high initial $\chi$. The
initially highly stochastic emission of photons at high $\chi$ quickly
collapses to low momentum as particles lose energy in the strong laser field.
For strongly anisotropic distributions to exist after interacting with the
driving laser pulse, either a distinctly different energy and intensity regime
must be considered, or pairs must be pulled out of the laser pulse before they
evolve to lower and more tightly clustered momenta. The latter case of brief
exposure to high $\chi$ cannot be achieved in one dimension as all particles
pass through the entire wave. However, it can occur in a two-dimensional laser
pulse where side-scattering becomes significant. In a two or three dimensional
simulation, kinetic effects might better correspond to previous work Qu,
Meuren, and Fisch (2022); Qu, Griffith, and Fisch (2023), and could offer an
explanation for the filamentation therein.

Simulations evolved in laser phase and one momentum dimension serve as a
limited starting point in looking for kinetic features in QED experiments. The
addition of spatial variation as well as non-separable momentum coordinates
could open up new, richer behavior. This could be directly addressed through
immediate extension of current work to add additional transverse and
longitudinal components the distribution function in both space and momentum.
A fuller description might lead to the maintenance of kinetic features for
longer durations or allow qualitatively new behavior to develop. To better
understand the stability of QED plasmas, numerically generated distributions
can be analyzed for the presence of unstable modes, and the direct study of
kinetic instabilities could be examined over longer time scales.

###### Acknowledgements.

This research was supported by NSF PHY-2206691.

## References

  * Di Piazza  _et al._ (2012) A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, “Extremely high-intensity laser interactions with fundamental quantum systems,” Reviews of Modern Physics 84, 1177–1228 (2012).
  * Zhang  _et al._ (2020) P. Zhang, S. S. Bulanov, D. Seipt, A. V. Arefiev, and A. G. R. Thomas, “Relativistic plasma physics in supercritical fields,” Physics of Plasmas 27, 050601 (2020).
  * Fedotov  _et al._ (2023) A. Fedotov, A. Ilderton, F. Karbstein, B. King, D. Seipt, H. Taya, and G. Torgrimsson, “Advances in QED with intense background fields,” Physics Reports Advances in QED with Intense Background Fields, 1010, 1–138 (2023).
  * Sironi and Giannios (2014) L. Sironi and D. Giannios, “Relativistic Pair Beams From TeV Blazars: A Source of Reprocessed GeV Emission Rather Than Intergalactic Heating,” The Astrophysical Journal 787, 49 (2014).
  * Turolla, Zane, and Watts (2015) R. Turolla, S. Zane, and A. L. Watts, “Magnetars: The physics behind observations. A review,” Reports on Progress in Physics 78, 116901 (2015).
  * Uzdensky  _et al._ (2019) D. Uzdensky, M. Begelman, A. Beloborodov, R. Blandford, S. Boldyrev, B. Cerutti, F. Fiuza, D. Giannios, T. Grismayer, M. Kunz, N. Loureiro, M. Lyutikov, M. Medvedev, M. Petropoulou, A. Philippov, E. Quataert, A. Schekochihin, K. Schoeffler, L. Silva, L. Sironi, A. Spitkovsky, G. Werner, V. Zhdankin, J. Zrake, and E. Zweibel, “Extreme Plasma Astrophysics,”  (2019), arxiv:1903.05328 [astro-ph, physics:physics] .
  * Chen and Fiuza (2023) H. Chen and F. Fiuza, “Perspectives on relativistic electron–positron pair plasma experiments of astrophysical relevance using high-power lasers,” Physics of Plasmas 30, 020601 (2023).
  * Chen  _et al._ (2009) H. Chen, S. C. Wilks, J. D. Bonlie, E. P. Liang, J. Myatt, D. F. Price, D. D. Meyerhofer, and P. Beiersdorfer, “Relativistic Positron Creation Using Ultraintense Short Pulse Lasers,” Physical Review Letters 102, 105001 (2009).
  * Sarri  _et al._ (2015) G. Sarri, K. Poder, J. M. Cole, W. Schumaker, A. Di Piazza, B. Reville, T. Dzelzainis, D. Doria, L. A. Gizzi, G. Grittani, S. Kar, C. H. Keitel, K. Krushelnick, S. Kuschel, S. P. D. Mangles, Z. Najmudin, N. Shukla, L. O. Silva, D. Symes, A. G. R. Thomas, M. Vargas, J. Vieira, and M. Zepf, “Generation of neutral and high-density electron–positron pair plasmas in the laboratory,” Nature Communications 6, 6747 (2015).
  * Zhu  _et al._ (2016) X.-L. Zhu, T.-P. Yu, Z.-M. Sheng, Y. Yin, I. C. E. Turcu, and A. Pukhov, “Dense GeV electron–positron pairs generated by lasers in near-critical-density plasmas,” Nature Communications 7, 13686 (2016).
  * Grismayer  _et al._ (2016) T. Grismayer, M. Vranic, J. L. Martins, R. A. Fonseca, and L. O. Silva, “Laser absorption via quantum electrodynamics cascades in counter propagating laser pulses,” Physics of Plasmas 23, 056706 (2016).
  * Abramowicz  _et al._ (2019) H. Abramowicz, M. Altarelli, R. Aßmann, T. Behnke, Y. Benhammou, O. Borysov, M. Borysova, R. Brinkmann, F. Burkart, K. Büßer, O. Davidi, W. Decking, N. Elkina, H. Harsh, A. Hartin, I. Hartl, B. Heinemann, T. Heinzl, N. TalHod, M. Hoffmann, A. Ilderton, B. King, A. Levy, J. List, A. R. Maier, E. Negodin, G. Perez, I. Pomerantz, A. Ringwald, C. Rödel, M. Saimpert, F. Salgado, G. Sarri, I. Savoray, T. Teter, M. Wing, and M. Zepf, “Letter of Intent for the LUXE Experiment,”  (2019), arxiv:1909.00860 [hep-ex, physics:physics] .
  * Yakimenko  _et al._ (2019) V. Yakimenko, S. Meuren, F. Del Gaudio, C. Baumann, A. Fedotov, F. Fiuza, T. Grismayer, M. J. Hogan, A. Pukhov, L. O. Silva, and G. White, “Prospect of Studying Nonperturbative QED with Beam-Beam Collisions,” Physical Review Letters 122, 190404 (2019).
  * 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, “On Seminal HEDP Research Opportunities Enabled by Colocating Multi-Petawatt Laser with High-Density Electron Beams,” arXiv:2002.10051 [hep-ph, physics:physics] (2020), arxiv:2002.10051 [hep-ph, physics:physics] .
  * Meuren  _et al._ (2021) S. Meuren, D. A. Reis, R. Blandford, P. H. Bucksbaum, N. J. Fisch, F. Fiuza, E. Gerstmayr, S. Glenzer, M. J. Hogan, C. Pellegrini, M. E. Peskin, K. Qu, G. White, and V. Yakimenko, “MP3 White Paper 2021 – Research Opportunities Enabled by Co-locating Multi-Petawatt Lasers with Dense Ultra-Relativistic Electron Beams,” arXiv:2105.11607 [hep-ph, physics:physics] (2021), arxiv:2105.11607 [hep-ph, physics:physics] .
  * Qu, Meuren, and Fisch (2021) K. Qu, S. Meuren, and N. J. Fisch, “Signature of Collective Plasma Effects in Beam-Driven QED Cascades,” Physical Review Letters 127, 095001 (2021).
  * Qu, Meuren, and Fisch (2022) K. Qu, S. Meuren, and N. J. Fisch, “Collective plasma effects of electron–positron pairs in beam-driven QED cascades,” Physics of Plasmas 29, 042117 (2022).
  * Wilks, Dawson, and Mori (1988) S. C. Wilks, J. M. Dawson, and W. B. Mori, “Frequency Up-Conversion of Electromagnetic Radiation with Use of an Overdense Plasma,” Physical Review Letters 61, 337–340 (1988).
  * Joshi  _et al._ (1990) C. Joshi, C. Clayton, K. Marsh, D. Hopkins, A. Sessler, and D. Whittum, “Demonstration of the frequency upshifting of microwave radiation by rapid plasma creation,” IEEE Transactions on Plasma Science 18, 814–818 (1990).
  * Savage  _et al._ (1993) R. Savage, R. Brogle, W. Mori, and C. Joshi, “Frequency upshifting and pulse compression via underdense relativistic ionization fronts,” IEEE Transactions on Plasma Science 21, 5–19 (1993).
  * Mendonca (2000) J. T. Mendonca,  _Theory of Photon Acceleration_ (CRC Press, 2000).
  * Qu  _et al._ (2018) K. Qu, Q. Jia, M. R. Edwards, and N. J. Fisch, “Theory of electromagnetic wave frequency upconversion in dynamic media,” Physical Review E 98, 023202 (2018).
  * Qu and Fisch (2019) K. Qu and N. J. Fisch, “Laser frequency upconversion in plasmas with finite ionization rates,” Physics of Plasmas 26, 083105 (2019).
  * Griffith, Qu, and Fisch (2022) A. Griffith, K. Qu, and N. J. Fisch, “Particle deceleration for collective QED signatures,” Physics of Plasmas 29, 073104 (2022).
  * Blackburn (2020) T. G. Blackburn, “Radiation reaction in electron–beam interactions with high-intensity lasers,” Reviews of Modern Plasma Physics 4, 5 (2020).
  * Gonoskov  _et al._ (2022) A. Gonoskov, T. G. Blackburn, M. Marklund, and S. S. Bulanov, “Charged particle motion and radiation in strong electromagnetic fields,” Reviews of Modern Physics 94, 045001 (2022).
  * Heinzl, King, and MacLeod (2020) T. Heinzl, B. King, and A. J. MacLeod, “Locally monochromatic approximation to QED in intense laser fields,” Physical Review A 102, 063110 (2020).
  * Blackburn  _et al._ (2014) T. G. Blackburn, C. P. Ridgers, J. G. Kirk, and A. R. Bell, “Quantum Radiation Reaction in Laser–Electron-Beam Collisions,” Physical Review Letters 112, 015001 (2014).
  * Ridgers  _et al._ (2017) C. P. Ridgers, T. G. Blackburn, D. D. Sorbo, L. E. Bradley, C. Slade-Lowther, C. D. Baird, S. P. D. Mangles, P. McKenna, M. Marklund, C. D. Murphy, and A. G. R. Thomas, “Signatures of quantum effects on radiation reaction in laser–electron-beam collisions,” Journal of Plasma Physics 83, 715830502 (2017).
  * Poder  _et al._ (2018) K. Poder, M. Tamburini, G. Sarri, A. Di Piazza, S. Kuschel, C. D. Baird, K. Behm, S. Bohlen, J. M. Cole, D. J. Corvan, M. Duff, E. Gerstmayr, C. H. Keitel, K. Krushelnick, S. P. D. Mangles, P. McKenna, C. D. Murphy, Z. Najmudin, C. P. Ridgers, G. M. Samarin, D. R. Symes, A. G. R. Thomas, J. Warwick, and M. Zepf, “Experimental Signatures of the Quantum Nature of Radiation Reaction in the Field of an Ultraintense Laser,” Physical Review X 8, 031004 (2018).
  * Cole  _et al._ (2018) J. M. Cole, K. T. Behm, E. Gerstmayr, T. G. Blackburn, J. C. Wood, C. D. Baird, M. J. Duff, C. Harvey, A. Ilderton, A. S. Joglekar, K. Krushelnick, S. Kuschel, M. Marklund, P. McKenna, C. D. Murphy, K. Poder, C. P. Ridgers, G. M. Samarin, G. Sarri, D. R. Symes, A. G. R. Thomas, J. Warwick, M. Zepf, Z. Najmudin, and S. P. D. Mangles, “Experimental Evidence of Radiation Reaction in the Collision of a High-Intensity Laser Pulse with a Laser-Wakefield Accelerated Electron Beam,” Physical Review X 8, 011020 (2018).
  * Li  _et al._ (2017) J.-X. Li, Y.-Y. Chen, K. Z. Hatsagortsyan, and C. H. Keitel, “Angle-resolved stochastic photon emission in the quantum radiation-dominated regime,” Scientific Reports 7, 11556 (2017).
  * Gong  _et al._ (2018) Z. Gong, R. H. Hu, H. Y. Lu, J. Q. Yu, D. H. Wang, E. G. Fu, C. E. Chen, X. T. He, and X. Q. Yan, “Brilliant GeV gamma-ray flash from inverse Compton scattering in the QED regime,” Plasma Physics and Controlled Fusion 60, 044004 (2018).
  * Lehmann and Spatschek (2012) G. Lehmann and K. H. Spatschek, “Phase-space contraction and attractors for ultrarelativistic electrons,” Physical Review E 85, 056412 (2012).
  * Neitz and Di Piazza (2013) N. Neitz and A. Di Piazza, “Stochasticity Effects in Quantum Radiation Reaction,” Physical Review Letters 111, 054802 (2013).
  * Mackenroth, Neitz, and Di Piazza (2013) F. Mackenroth, N. Neitz, and A. Di Piazza, “Novel aspects of radiation reaction in the ultrarelativistic quantum regime,” Plasma Physics and Controlled Fusion 55, 124018 (2013).
  * Vranic  _et al._ (2016) M. Vranic, T. Grismayer, R. A. Fonseca, and L. O. Silva, “Quantum radiation reaction in head-on laser-electron beam interaction,” New Journal of Physics 18, 073035 (2016).
  * Niel  _et al._ (2018) F. Niel, C. Riconda, F. Amiranoff, R. Duclous, and M. Grech, “From quantum to classical modeling of radiation reaction: A focus on stochasticity effects,” Physical Review E 97, 043209 (2018).
  * Geng  _et al._ (2019) X. S. Geng, L. L. Ji, B. F. Shen, B. Feng, Z. Guo, Q. Yu, L. G. Zhang, and Z. Z. Xu, “Quantum reflection above the classical radiation-reaction barrier in the quantum electro-dynamics regime,” Communications Physics 2, 1–9 (2019).
  * Gong  _et al._ (2019) Z. Gong, R. H. Hu, J. Q. Yu, Y. R. Shou, A. V. Arefiev, and X. Q. Yan, “Radiation rebound and quantum splash in electron-laser collisions,” Physical Review Accelerators and Beams 22, 093401 (2019).
  * Bilbao and Silva (2023) P. J. Bilbao and L. O. Silva, “Radiation Reaction Cooling as a Source of Anisotropic Momentum Distributions with Inverted Populations,” Physical Review Letters 130, 165101 (2023).
  * Zhdankin, Kunz, and Uzdensky (2023) V. Zhdankin, M. W. Kunz, and D. A. Uzdensky, “Synchrotron Firehose Instability,” The Astrophysical Journal 944, 24 (2023).
  * Rackauckas and Nie (2017) C. Rackauckas and Q. Nie, “DifferentialEquations.jl – A Performant and Feature-Rich Ecosystem for Solving Differential Equations in Julia,” Journal of Open Research Software 5, 15 (2017).
  * Weibel (1959) E. S. Weibel, “Spontaneously Growing Transverse Waves in a Plasma Due to an Anisotropic Velocity Distribution,” Physical Review Letters 2, 83–84 (1959).
  * Fried (1959) B. D. Fried, “Mechanism for Instability of Transverse Plasma Waves,” The Physics of Fluids 2, 337 (1959).
  * Davidson  _et al._ (1972) R. C. Davidson, D. A. Hammer, I. Haber, and C. E. Wagner, “Nonlinear Development of Electromagnetic Instabilities in Anisotropic Plasmas,” The Physics of Fluids 15, 317–333 (1972).
  * Qu, Griffith, and Fisch (2023) K. Qu, A. Griffith, and N. J. Fisch, “Pair filamentation and laser scattering in beam-driven QED cascades,”  (2023), arxiv:2312.02107 [physics] .