# Peak in the critical current density in (CaxSr1-x)3Rh4Sn13 tuned towards
the structural quantum critical point

Xinyou Liu  Wei Zhang  Kwing To Lai  Department of Physics, The Chinese
University of Hong Kong, Shatin, Hong Kong SAR, China  K. Moriyama  Department
of Chemistry, Kyoto University, Kyoto 606-8502, Japan  J. L. Tallon  Robinson
Institute, Victoria University of Wellington, P. O. Box 600, Wellington, New
Zealand  K. Yoshimura  Department of Chemistry, Kyoto University, Kyoto
606-8502, Japan  Swee K. Goh<EMAIL_ADDRESS>Department of Physics, The
Chinese University of Hong Kong, Shatin, Hong Kong SAR, China

###### Abstract

(CaxSr1-x)3Rh4Sn13 is a rare system that has been shown to display an
interesting interplay between structural quantum criticality and
superconductivity. A putative structural quantum critical point, which is
hidden beneath a broad superconducting dome, is believed to give rise to
optimized superconducting properties in (CaxSr1-x)3Rh4Sn13. However, the
presence of the superconducting dome itself hinders the examination of the
quantum critical point through electrical transport, as the transport
coefficients vanish in the superconducting state. Here, we use critical
current density to explore within the superconducting dome. Our measurements
reveal a large enhancement of the critical current density at the zero-
temperature limit when the system is tuned towards the structural quantum
critical point.

A quantum critical point (QCP) is a location in parameter space where a
second-order quantum phase transition occurs at zero temperature. Its
prominence in condensed matter physics is tied to the observation of
superconductivity in a large variety of materials in the vicinity of the
putative QCP Paschen and Si (2021); Vojta (2018); Keimer and Moore (2017);
Paglione and Greene (2010); Dai (2015); Gupta  _et al._ (2021); Frachet  _et
al._ (2020); Gupta  _et al._ (2021); Frachet  _et al._ (2020); Tallon  _et
al._ (1999); Cho  _et al._ (2018); Hashimoto  _et al._ (2012); Analytis  _et
al._ (2014); Putzke  _et al._ (2014); Shibauchi  _et al._ (2014); Gegenwart
_et al._ (2008); Park  _et al._ (2006); Mathur  _et al._ (1998); Gruner  _et
al._ (2017). Examples where the presence of the QCP has been proposed include
cuprates Tallon  _et al._ (1999); Gupta  _et al._ (2021); Frachet  _et al._
(2020), iron pnictides Analytis  _et al._ (2014); Hashimoto  _et al._ (2012);
Shibauchi  _et al._ (2014); Putzke  _et al._ (2014), charge density wave (CDW)
systems Gruner  _et al._ (2017); Cho  _et al._ (2018), and heavy-fermion
systems Mathur  _et al._ (1998); Park  _et al._ (2006); Gegenwart  _et al._
(2008), in which a dome-shaped dependence of the superconducting transition
temperature $T_{c}$ is found. However, such a superconducting dome naturally
masks the QCP. Usually, to reveal the exact pressure or composition where the
QCP is located requires the extrapolation of the relevant phase transition
temperature to zero, or the application of a strong magnetic field to suppress
superconductivity. To probe the QCP directly, a physical parameter which does
not vanish in the superconducting state is required.

Critical current density $J_{c}$ is just such a parameter that does not vanish
inside the superconducting dome. $J_{c}$ in the zero-temperature limit,
$J_{c}(0)$, is particularly suitable for detecting the presence of a QCP, as
it is a singular point which exists precisely at the absolute zero
temperature. In the heavy-fermion system Sn-doped CeRhIn5, an
antiferromagnetic QCP can be reached by applying a hydrostatic pressure of
1.35 GPa Seo  _et al._ (2015). Indeed, at the antiferromagnetic QCP, an
enhancement of $J_{c}$ was reported Jung  _et al._ (2018), and the location of
the QCP is consistent with that determined by other physical properties of Sn-
doped CeRhIn5 through extrapolations of the antiferromagnetic phase transition
from finite temperatures Seo  _et al._ (2015); Mendonça Ferreira  _et al._
(2008). In high-$T_{c}$ cuprates, $J_{c}(0)$ has also been shown to peak at
the critical doping Naamneh  _et al._ (2014); Talantsev  _et al._ (2014).
Therefore, $J_{c}(0)$ offers a powerful means to probe the true ground-state
property of the system along the $T=0$ axis inside a superconducting dome.

To establish the extent to which $J_{c}(0)$ can detect a QCP, it is important
to look beyond a magnetic QCP, because a nonmagnetic QCP can also support the
optimization of superconductivity in an analogous manner. Quasiskutterudite
“Remeika” compounds M3T4Sn13, where M is an alkaline-earth metal and T is a
transition metal, are promising systems to explore this relationship Remeika
_et al._ (1980); Kase  _et al._ (2011); Goh  _et al._ (2015); Yu  _et al._
(2015); Cheung  _et al._ (2018); Lue  _et al._ (2016); Klintberg  _et al._
(2012); Chen  _et al._ (2016); Veiga  _et al._ (2020); Terasaki  _et al._
(2021); Biswas  _et al._ (2014, 2015); Krenkel  _et al._ (2022); Zhou  _et
al._ (2012); Sarkar  _et al._ (2015); Yang  _et al._ (2010); Luo  _et al._
(2018); Hu  _et al._ (2017); Kaneko  _et al._ (2020). Sr3Rh4Sn13 is a cubic
system which shows a second-order structural transition at $T^{*}=138$ K and a
superconducting transition at $T_{c}=4.2$ K Kase  _et al._ (2011); Goh  _et
al._ (2015); Yu  _et al._ (2015); Cheung  _et al._ (2018); Lue  _et al._
(2016). Upon the substitution of Ca to form the series (CaxSr1-x)3Rh4Sn13,
$T^{*}$ can be suppressed rapidly, and it extrapolates to 0 K at $x_{c}\sim
0.9$, giving rise to the proposal of a structural QCP at $x_{c}$ Goh  _et al._
(2015); Yu  _et al._ (2015); Cheung  _et al._ (2018); Terasaki  _et al._
(2021); Krenkel  _et al._ (2022). Concomitant with the suppression of $T^{*}$,
$T_{c}$ follows a dome-shaped dependence on $x$ with a maximum $T_{c}$ of
$\sim$7.9 K at the calcium-rich end of the series. Further experimental
support for the structural QCP at $x_{c}$ includes a linear-in-$T$ resistivity
from $T_{c}$ to $\sim$40 K, a minimum Debye temperature, and a complete
softening of the relevant phonon mode at the M point of the Brillouin zone in
the zero-temperature limit Goh  _et al._ (2015); Yu  _et al._ (2015); Cheung
_et al._ (2018). Moreover, strong-coupling superconductivity is stabilized
near $x_{c}$, as benchmarked by large values of $2\Delta_{sc}/k_{B}T_{c}$ and
$\Delta C/\gamma T_{c}$ that are both beyond the BCS weak-coupling limit Goh
_et al._ (2015); Yu  _et al._ (2015); Cheung  _et al._ (2018); Terasaki  _et
al._ (2021). The related series (CaxSr1-x)3Ir4Sn13 behaves similarly Klintberg
_et al._ (2012), but $T^{*}$ only extrapolates to 0 K when Ca3Ir4Sn13 is
subjected to a hydrostatic pressure of 1.8 GPa Klintberg  _et al._ (2012);
Biswas  _et al._ (2015). Thus (CaxSr1-x)3Rh4Sn13 offers a platform to study a
structural QCP without the need of applying pressure.

In this manuscript, we report our $J_{c}$ data for the entire substitution
series (CaxSr1-x)3Rh4Sn13, straddling the putative structural QCP. With the
availability of $J_{c}$ data down to 50 mK a reliable extraction of $J_{c}(0)$
for all samples can be achieved. Our $J_{c}(0)$ reveals a significant
enhancement when the system is tuned to $x_{c}$, offering compelling evidence
for the existence of a structural QCP in (CaxSr1-x)3Rh4Sn13.

Figure 1:  (a) Temperature dependence of the electrical resistivity for
(CaxSr1-x)3Rh4Sn13 across the entire substitution series. The traces are
offset vertically for clarity. The arrows indicate $T^{*}$, the temperature at
which the second-order structural transition occurs. Zero resistivity for each
curve is marked by the horizontal dashed line. (b) The low-temperature zoom-in
of $\rho(T)$ for $x=0$, $x=0.5$, $x=x_{c}=0.9$, and $x$ = 1 samples.

Single crystals of (CaxSr1-x)3Rh4Sn13 were synthesized by the Sn-flux method
as described elsewhere Yang  _et al._ (2010). Electrical transport
measurements were carried out in a Physical Property Measurement System (PPMS)
from Quantum Design and in a dilution fridge manufactured by Bluefors.
Temperature-dependent electrical resistivity was measured with a conventional
four-probe configuration. Dupont 6838 silver epoxy and gold wires with
diameters of 10 $\mu$m or 25 $\mu$m were used for making the electrical
contacts. $I$-$V$ curves were measured using the same contacts by a Keithley
2182A nanovoltmeter combined with a Keithley 6221 current source in the pulsed
delta mode. The duration of the pulsed current was 11 ms, and the pulse
repetition time was 0.1 s. These pulse sequence parameters not only minimize
Joule heating, but also ensure a practical measurement time SUP .

Figure 1(a) shows the temperature dependence of the electrical resistivity
$\rho(T)$ of (CaxSr1-x)3Rh4Sn13 from $x$ = 0 to $x$ = 1 upon cooling from 300
K. All $\rho(T)$ traces are progressively offset vertically for clarity, with
the zero resistivity for each trace marked by the horizontal dashed line. A
prominent anomaly in $\rho(T)$ is recorded in the $x$ = 0 sample at $T^{*}$ =
137 K, which corresponds to a second-order structural phase transition from a
high-temperature crystal structure with a space group $Pm\bar{3}n$ to a low-
temperature structure $I4\bar{3}d$ below $T^{*}$. To determine $T^{*}$, we use
the minimum in the first derivative of resistivity, and $T^{*}$ decreases
rapidly with an increasing Ca fraction $x$. In the $x$ = 0.75 sample, $T^{*}$
is 26.6 K, but it becomes difficult to trace $T^{*}$ further with higher $x$
values. Figure 1(b) shows the low-temperature region of $\rho(T)$. The
superconducting transition temperature $T_{c}$, defined as the midpoint of the
transition, experiences a significant enhancement from 4.5 K in the $x=0$
sample to 7.9 K in the $x=1$ sample. Furthermore, a quasilinear $\rho(T)$ is
observed from $T_{c}$ to 40 K near $x=x_{c}=0.9$, which has been attributed to
the softening of phonon modes at the structural QCP. The curvature of
$\rho(T)$ is gradually recovered when $x$ moves away from $x_{c}$. These
observations are consistent with previous reports Goh  _et al._ (2015); Yu
_et al._ (2015); Cheung  _et al._ (2018).

Figure 2:  $V$-$I$ characteristics at various temperatures for (a) $x=0$
(sample S4) and (b) $x_{c}=0.9$ (sample S2). The calculated first derivative
of $V(I)$ (d$V$/d$I$) for (c) $x=0$ and (d) $x_{c}=0.9$. The long arrows show
the direction of an increasing temperature. The peak positions in d$V$/d$I$
define the corresponding $I_{c}$. The vertical arrows illustrate the $I_{c}$
at 3.5 K for $x=0$, and at 6.5 K for $x_{c}$.

We proceed to measure the critical current $I_{c}$ for 13 samples spanning the
entire substitution series. For $x$ = 0, 0.65, 0.9, and 1, two samples from
the same batch have been measured for each composition (see Supplemental
Material SUP ). For each sample, we measure the voltage at a fixed temperature
when a pulsed current is applied perpendicular to the cross section of the
specimen with current leads covering the whole width. $V$-$I$ curves at
various temperatures for $x=0$ (sample S4) are shown in Fig. 2(a). For a
typical trace, a drastic increase of the voltage from zero is recorded when
the current exceeds a threshold, indicating a recovery from the
superconducting state to the normal state. To quantify the value of $I_{c}$,
we calculate the first derivative of $V(I)$. Figure 2(c) displays d$V$/d$I$
versus $I$ for the same $x=0$ sample, and the peak positions are defined as
the corresponding $I_{c}$. When the temperature is reduced, $I_{c}$ increases
significantly and saturates at the zero-temperature limit. As we will argue
later, this saturation is consistent with a dominant nodeless order parameter.
Figures 2(b) and (d) show the $V$-$I$ curves and d$V$/d$I$ versus $I$ for
$x=x_{c}=0.9$ (sample S2), respectively. However, we point out that $I_{c}$ at
the zero-temperature limit, $I_{c}(0)$, is noticeably higher in the $x=0.9$
sample, despite the fact that the $x=0.9$ sample has a smaller cross-sectional
area than that of the $x=0$ sample.

Figure 3:  (a) Temperature dependence of $J_{c}$ and (b) $J_{c}$ normalized by
$J_{c}(0)$ as a function of the reduced temperature $t=T/T_{c}$ for all
samples. The normalized $J_{c}(t)$ collapse onto a universal curve, which can
be described by the red solid curve governed by Eq.(1) with $\alpha=0.5$ and
$\beta=0.2$. Blue dashed and green dotted lines represent the curve for
$\delta T_{c}$-pinning and $\delta l$-pinning mechanisms, respectively
Griessen  _et al._ (1994). (c) Ca content $x$ dependence of $J_{c}(0)$ and (d)
$\widetilde{J_{c}}(0)$ for all the samples. The solid triangles in (c) and (d)
denote the $J_{c}(0)$ and $\widetilde{J_{c}}(0)$ = $J_{c}(0)\times b$ of the
two $x$=0.9 samples. Note that $x$ = 0.75 (sample S2) is from the same batch
as $x$ = 0.75 (sample S1), whose $\rho(T)$ is shown in Fig. 1. The error bars
in $x$ are determined following a scheme described in Ref. SUP .

For a sample with a width of $2a$ and a thickness of $2b$,
$J_{c}=I_{c}/(2a\times 2b)$, as conventionally defined. Figure 3(a) shows the
temperature dependence of $J_{c}$ for all 13 samples. We note the discrepancy
in the $J_{c}$ values for the two $x=0.9$ samples, which we are able to
resolve below. Nevertheless, the overall temperature dependencies for all
samples are similar, despite their different $T_{c}$. To visualize the
similarity, we plot in Fig. 3(b) $J_{c}$ normalized by $J_{c}(0)$ as a
function of the reduced temperature $t=T/T_{c}$. Indeed, all $J_{c}$ data
collapse onto the same curve when plotted in this manner, hinting at a
universal mechanism governing the temperature dependence of $J_{c}$. We find
empirically that the normalized $J_{c}$ can be accurately described by

$\frac{J_{c}(t)}{J_{c}(0)}=(1-t^{2})^{\alpha}(1+t^{2})^{\beta},$ (1)

with $\alpha=0.5$ and $\beta=0.2$ (red solid curve in Fig. 3(b)). In fact, Eq.
(1) with $(\alpha,\beta)=(7/6,5/6)$ is frequently employed to described
$J_{c}(t)$ when the so-called $\delta T_{c}$-pinning is effective Griessen
_et al._ (1994). For $\delta l$-pinning (where $l$ is the mean free path),
$(\alpha,\beta)=(5/2,-1/2)$ Xiang  _et al._ (2013). The curves due to $\delta
l$\- and $\delta T_{c}$-pinnings are added to Fig. 3(b), and both mechanisms
are incompatible with our data. This is not surprising because, in the absence
of the applied magnetic field, the vortex physics does not play a role. Hence,
the pinning mechanisms are irrelevant to our discussion.

We adopt the viewpoint that our $J_{c}(T)$ reflects, to a certain extent, the
temperature dependence of $\lambda^{-2}$, where $\lambda$ is the London
penetration depth. This viewpoint emerges from a recent discovery that the
self-field critical current, i.e. a transport $J_{c}$ without an applied
magnetic field, is fundamentally determined by $\lambda$. As demonstrated in
Refs. Talantsev and Tallon (2015); Talantsev  _et al._ (2017), the self-field
$J_{c}$ is given by

$J_{c}=\frac{\phi_{0}}{4\pi\mu_{0}\lambda^{3}}\left(\ln\left(\frac{\lambda}{\xi}\right)+0.5\right)\times\frac{\lambda}{b}\tanh\left(\frac{b}{\lambda}\right),$
(2)

where $\phi_{0}$ is the flux quantum, $\mu_{0}$ is the vacuum permeability,
and $\xi$ is the coherence length.

For our samples, the thickness $2b$ is of the order of several tens of
$\mu{\rm m}$ – the thinnest sample has $2b=17~{}\mu{\rm m}$ SUP . On the other
hand, $\lambda$ in the zero-temperature limit, $\lambda(0)$, has been reported
to be $\sim$290 nm in the related compound Sr3Ir4Sn13 Biswas  _et al._ (2014).
Assuming that $\lambda(0)$ in (CaxSr1-x)3Rh4Sn13 is of the same order of
magnitude, then $b/\lambda(0)\gg 1$. Equation (2) in the ‘thick limit’ becomes

$J_{c}=\frac{\phi_{0}}{4\pi\mu_{0}\lambda^{2}}\left(\ln\left(\frac{\lambda}{\xi}\right)+0.5\right)\times\frac{1}{b}.$
(3)

Near $T_{c}$, $\lambda$ diverges and can potentially complicate the analysis,
because $\lambda$ will eventually exceed $b$. However, given the thickness of
our samples SUP , this would occur very near $T_{c}$. In the Bardeen-
Schrieffer-Cooper weak-coupling limit, $\lambda$ will only reach
$10\lambda(0)$ at $T/T_{c}=0.995$. Here, our sample thickness is roughly two
orders of magnitude larger than $\lambda(0)$. Therefore, the condition
$b/\lambda(0)\gg 1$ is essentially valid for all samples at all practical
$T<T_{c}$.

From Eq. (3), one can immediately see that $J_{c}$ depends on a sample-
specific extrinsic factor (the half sample thickness $b$), while all other
factors are either constants or material parameters. This provides a hint to
reconcile the drastic difference between the $J_{c}$ values of the two $x=0.9$
samples mentioned above: The difference is simply due to the different
thicknesses. To illustrate the validity of this hypothesis, the critical
current density at the zero-temperature limit of all samples before and after
taking this factor into account is shown in Figs. 3(c) and (d). Indeed, when
the thickness is taken into account, $\widetilde{J_{c}}(0)=J_{c}(0)\times b$
of the two $x=0.9$ samples become much closer: The $\widetilde{J_{c}}(0)$
values are $\pm 3.74\%$ from the mean value, while before correcting for the
thickness, the $J_{c}(0)$ values are $\pm 24.16\%$ from the mean value (see
the solid triangles in Figs. 3(c) and (d)). In fact, $\widetilde{J_{c}}(0)$
over the entire range of the Ca content $x$ becomes less scattered than that
of $J_{c}(0)$, as can be seen by comparing Fig. 3(d) with Fig. 3(c). To
further verify the $1/b$ dependence of $J_{c}$, we measure another piece of
$x=0.5$ crystal, which we polish to three different thicknesses, all in the
thick limit. As presented in the Supplemental Material SUP , this measurement
with the same starting crystal confirms the applicability of $1/b$ scaling of
$J_{c}$.

Despite the confirmation of the $1/b$ scaling, the measured $J_{c}$ values are
lower than the $J_{c}$ values calculated from Eq. (3). Take, for instance,
$x=0.9$ (sample S2) with $2b=17~{}\mu$m. Assuming $\lambda(0)\sim 290$ nm and
$\lambda/\xi\sim 30$ Biswas  _et al._ (2014), the calculated $J_{c}$ is
$715\times 10^{3}$ mA/mm2, about 60 times larger than the measured values
($12\times 10^{3}$ mA/mm2). One possibility is the existence of weakly linked
superconducting domains in our samples. Thus, the cross-sectional area through
which the transport current flows used for converting from $I_{c}$ to $J_{c}$
is too large. In the presence of weak links, $J_{c}^{\rm
WL}=\hbar/(2\mu_{0}de\lambda^{2}$), where $d$ is some characteristic length
scale associated with the weak-link morphology Tallon  _et al._ (2006);
Ambegaokar and Baratoff (1963). Therefore, $J_{c}^{\rm WL}$ is also
proportional to $\lambda^{-2}$. If the domains are sufficiently large (such
that $b\gg\lambda$) and close to each other (so that $d$ is exceedingly
small), $J_{c}^{\rm WL}$ could be too large to be the current limiting
mechanism, and the mechanism giving rise to Eq. (3) prevails. This probably
explains the observed $1/b$ dependence, and without the precise knowledge of
the domain size, the change in $J_{c}$ can only be associated with the change
in $\lambda^{-2}$ with an indeterminate proportionality factor.

The realization that the self-field $J_{c}$ is proportional to $\lambda^{-2}$
offers a means to evaluate the superconducting gap. Since the ratio
$\lambda/\xi$ enters the logarithm, whose temperature dependence is weak,
$J_{c}(t)/J_{c}(0)$ is dominated by $\lambda^{-2}(t)/\lambda^{-2}(0)$, which
reflects the temperature dependence of the superfluid density. As Fig. 3(b)
depicts, $J_{c}(t)/J_{c}(0)$ has a weak temperature dependence at low
temperatures. Below $t=0.2$, $J_{c}(t)/J_{c}(0)$ only deviates from the zero-
temperature value by at most $1.5\%$. Therefore, this rapid saturation of
$J_{c}$ as temperature decreases suggests a nodeless order parameter,
consistent with the consensus on this material system Luo  _et al._ (2018);
Krenkel  _et al._ (2022); Yu  _et al._ (2015); Kase  _et al._ (2011); Zhou
_et al._ (2012); Sarkar  _et al._ (2015).

Figure 4:  A contour map of $\widetilde{J_{c}}$ constructed using our
experimental data superimposed on the $T$-$x$ phase diagram. Open circles
denote $T^{*}$, while solid circles denote $T_{c}$. The dashed line is the
$T^{*}(x)$ from Ref. Goh  _et al._ (2015).

We now present our $J_{c}$ data in relation to the $T$-$x$ phase diagram,
compactly summarized in Fig. 4. The colour map indicates the magnitude of
$\widetilde{J_{c}}$. Again, we use $\widetilde{J_{c}}$ to eliminate the effect
of the sample thickness. In the Ca-rich end, $\widetilde{J_{c}}$ is generally
larger. At $T=0$, $\widetilde{J_{c}}(0)$ is gradually enhanced with an
increasing $x$, and it peaks at $x=0.9$, before decreasing again. Note that
the enhancement in $\widetilde{J_{c}}(0)$ cannot be attributed simply to the
enhancement in $T_{c}$ – moving from $x=0.5$ to $x=0.9$, $T_{c}$ only
increases by 10% while $\widetilde{J_{c}}(0)$ rises by 400$-$500%.
Incidentally, the maximum $\widetilde{J_{c}}(0)$ occurs close to the
composition where $T^{*}(x)$ extrapolates to 0 K. Therefore our data provide
compelling evidence for the existence of a structural quantum critical point
hidden beneath the superconducting dome. Taking together the transport,
thermodynamic, and inelastic scattering results Goh  _et al._ (2015); Yu  _et
al._ (2015); Cheung  _et al._ (2018), the existence of a structural quantum
critical point in (CaxSr1-x)3Rh4Sn13 at around $x=0.9$ is now firmly
established.

Our finding of a peak in the critical current is reminiscent of the recent
discovery by Jung et al. Jung  _et al._ (2018) in Ce-based heavy-fermion
compounds, in which a magnetic QCP hidden beneath the superconducting phase
was also revealed by an enhancement of the critical current. In addition, the
hole-doped high-$T_{c}$ cuprate superconductor Y0.8Ca0.2Ba2Cu3Oy also features
a sharp peak in $J_{c}$ at the critical hole doping $p=0.19$ Talantsev  _et
al._ (2014); Talantsev and Tallon (2015), where a quantum critical point was
proposed to be located. In (CaxSr1-x)3Rh4Sn13, the peak appears sharper than
in both heavy-fermion compounds and some cuprates Talantsev  _et al._ (2014);
Talantsev and Tallon (2015); Jung  _et al._ (2018). Other examples of a sharp
quantum-critical like peak in $J_{c}$ do in fact exist for cuprates Tallon
_et al._ (1999); Naamneh  _et al._ (2014). Such a strong similarity across
three very distinct classes of superconductors deserves further investigation.
Taken together, these works suggest that the critical current is indeed a
suitable diagnostic probe to locate the quantum critical point masked by the
superconducting dome. As a future direction, the measurement of $J_{c}$ in a
magnetic field will enable the discussion of pinning mechanisms, potentially
revealing new physics associated with the suppression of structural order. For
instance, magnetization measurements in
$\text{Ba}{({\text{Fe}}_{1-x}{\text{Co}}_{x})}_{2}{\text{As}}_{2}$ also show a
peak in $J_{c}$ close to the optimum doping Prozorov  _et al._ (2009): The in-
field measurements led to the proposal that the $J_{c}$ peak is related to
pinning on structural domains.

The peak in the critical current density would imply a dip in the London
penetration depth (see Eqs. (2) and (3)). This is counter-intuitive for our
system. When the electron-phonon coupling is taken into account, the London
penetration depth $\lambda$ is modified by the electron-phonon coupling
constant $\lambda_{ep}$ according to
$\lambda^{2}=m_{e}(1+\lambda_{ep})/(\mu_{0}e^{2}\rho_{s})$, where $m_{e}$ is
the electron mass, $e$ is the electronic charge, and $\rho_{s}$ is the
superfluid density. If the superfluid density does not depend strongly on the
Ca content, one would expect $\lambda$ to peak near the structural QCP instead
Hashimoto  _et al._ (2012); Joshi  _et al._ (2020), because of the stronger
electron-phonon coupling there Goh  _et al._ (2015); Yu  _et al._ (2015);
Cheung  _et al._ (2018). We tentatively attribute the dip in $\lambda$ to a
strongly enhanced $\rho_{s}$ at $x_{c}$. At present, this opposite trend
remains a puzzle. In the sister compound Ca3Ir4Sn13, a sudden increase in
$\lambda(0)^{-2}$, and hence a decrease in $\lambda(0)$, has also been
detected by muon spin relaxation ($\mu$SR) measurements beyond the pressure-
induced structural QCP Biswas  _et al._ (2015), although a peak-like anomaly
is not seen in Ca3Ir4Sn13 under pressure. Such a sensitive change in
$\lambda(0)$ is seldom reported in typical BCS superconductors. Therefore, the
observations in both Rh and Ir series indicate that structural instabilities
may play an essential role in enhancing superconducting properties in this
class of compounds. Further investigations may bring new insights to explain
the intriguing behaviour of $\lambda$ in superconducting stannides near a
structural QCP.

In summary, we have probed the $T$-$x$ phase diagram of the Remeika series
(CaxSr1-x)3Rh4Sn13 via critical current measurements. The critical current
density is significantly larger for $x=0.9$ samples, the composition at which
a putative structural quantum critical point has been identified using other
techniques. Some of these techniques require the removal of the
superconductivity by a magnetic field, or a long-range extrapolation of the
normal state properties to the zero-temperature limit. Thus, the singular
behaviour of the critical current density at zero field is valuable for
revealing the existence of a quantum critical point masked by the
superconducting dome. Given that the transport critical current is relatively
straightforward to measure, our scheme can have a wider impact in identifying
the quantum phase transition in a wide range of tunable quantum materials. For
a superconductor to be useful for application, it must support a large
transport current flow. As such, tremendous efforts have been devoted to the
enhancement of the critical current density. Our work, as well as the studies
on cuprates and Ce-based heavy fermion systems Naamneh  _et al._ (2014);
Talantsev  _et al._ (2014); Jung  _et al._ (2018), suggests that a delicate
tuning of neighbouring instabilities could be an avenue for this endeavour.

###### Acknowledgements.

We acknowledge useful discussions with Yajian Hu and Tuson Park. This work is
financially supported by the Research Grants Council of Hong Kong
(A-CUHK402/19, CUHK 14300117), CUHK Direct Grant (4053461, 4053408, 4053463,
4053410) and Grant-in-Aids for Scientific Research from Japan Society for the
Promotion Science (18KK0150 and 20J20353).

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#  Polarization of recoil photon in non-linear Compton process

A. I. Titov  Bogoliubov Laboratory of Theoretical Physics, Joint Institute for
Nuclear Research, Dubna 141980<EMAIL_ADDRESS>
###### Abstract

The polarization of recoil photon ($\gamma^{\prime}$) in the non-linear
Compton process $e+\vec{L}\to\vec{\gamma}^{\prime}+e^{\prime}$ in the
interaction of a relativistic electron with a linearly polarized laser beam
($\vec{L}$) is studied within the Furry picture in the lowest-order, tree-
level S matrix element. In particular, we consider the asymmetry of
differential cross sections ${\cal A}$ for two independent axes describing the
Compton process equal to the intrinsic spin variable ${\xi}^{f}_{3}$, that
determines the polarization properties of $\gamma^{\prime}$. The sign and
absolute value of the asymmetry determine the direction and degree of
$\gamma^{\prime}$ polarization. We have analyzed the process in a wide range
of laser intensity that covers existing and future experiments. Our results
provide additional knowledge for studying non-linear multi-photon effects in
quantum electrodynamics and can be used in planning experiments at envisaged
laser facilities.

non-linear Compton scattering, multi-photon processes, photon polarization

###### pacs:

12.20.Ds, 13.40.-f, 23.20.Nx

##  I Introduction

The dynamics of the interaction of an ultra-relativistic electron beam with a
super-intense laser attracts much attention from theory and experiment in the
hope of shedding light on open problems of non-linear quantum electrodynamics.
An excellent analysis of theoretical achievements and possible expectations
from new facilities is given in a recent review AM_Review , see also ADPiazza
. Important parts of these studies are the non-linear Compton (nlCo) and non-
linear Breit-Wheeler (nlBW) processes, where a probe particle, electron or
photon, respectively, interacts with a high-intensity background
electromagnetic field (laser). Both processes were thoroughly researched
theoretically in the past and then re-considered and improved RitusGroup ;
Ritus-79 ; Serbo2004 ; Seipt:2020diz ; TitovPRA2013 ; DiPiazza:2020wxp ;
Ilderton:2020dhs ; Piazza2018 ; Heinzl2020 ; TitovPEPAN ; Granz:2019sxb ;
Kampfer2023 . In most cases, the field of a high-intensity optical laser is
considered as a background field. For example, the famous SLAC experiment
E-144 E-144 , envisaged European LUXE LUXE_exp  and E-320 at FACET II/SLAC
E_320 ; E_320add ; E_320add23  projects. For X-ray photon beam experiments
(XFEL), see XFEL . In nlBW, the formation of an electron-positron pair using
an optical laser pulse of several eV requires an external photon with a
frequency of tens of GeV. This may be a recoil photon ($\gamma^{\prime}$)
arising from the interaction of a relativistic electron with an optical laser
beam in a non-linear multiphoton Compton process. Consideration of this
process with an emphasis on $\gamma^{\prime}$-polarization is the main goal of
this work.

We use the following notation. The electromagnetic field intensity is
described by the dimensionless parameter $\xi=|e|{\cal E}/(m\omega)$, where
$\cal{E}$ and $\omega$ are the central electric field strength and the central
laser frequency, respectively; $-|e|$ and $m$ stand for electron charge and
mass, respectively. The four-vectors $p=(E_{e},\vec{p})$ and
$p^{\prime}=(E^{\prime}_{e},\vec{p}^{\prime})$ are the four-momenta of
incoming and outgoing bare electrons, respectively; $q=(q_{0},\vec{q})$ and
$q^{\prime}=(q^{\prime}_{0},\vec{q}^{\prime})$ are the four-momenta for
dressed electrons LL4 , respectively, with e.g., $q=p-k(\xi^{2}m^{2}/4k\cdot
p)$, where $k=(\omega,\mathbf{k})$ is the four-momentum of the beam photon
with $\mathbf{k}=\mathbf{z}\omega$.

The four-momentum of the outgoing photon with frequency $\omega^{\prime}$ is
$k^{\prime}=(\omega^{\prime},\mathbf{k}^{\prime})$ with
$\mathbf{k}^{\prime}=\omega^{\prime}(\mathbf{x}\sin\theta\cos\varphi+\mathbf{y}\sin\theta\sin\varphi+\mathbf{z}\cos\theta)$,
where $\theta$ and $\varphi$ are the corresponding polar and azimuthal angles,
respectively. We also use the quantum non-linearity parameter $\chi=\xi(k\cdot
p)/m^{2}$. It is assumed that the laser beam polarization is directed along
the $\mathbf{x}$ axis. We use natural units with $c=\hbar=1$, and
$e^{2}/4\pi=\alpha\approx 1/137.036$. Following the LUXE experiment design, we
are limited to unpolarized electrons.

In case of linearly polarized initial photons, the yield of the $e^{+}e^{-}$
pair in nlBW process depends on the kinematics (square of the total energy in
c.m s., $s$), $\xi$, and mutual polarization of the initial photons. For
example, at ultra-high field intensity with $\xi\gg 1$ the asymmetry of the
electron-positron yield
${\cal{A}}=(W_{\perp}-W_{\parallel})/(W_{\perp}+W_{\parallel})$, when the
polarization of the initial photons are mutually perpendicular $W_{\perp}$ or
parallel $W_{\parallel}$, varies from 1/3 to 1/5 Ritus-79 , depending on
$\kappa=\xi s/2m^{2}$. At low field strength with $\xi\lesssim 1$, the
asymmetry $\cal{A}$ exhibits non-monotonic behavior, varying from zero to
unity depending on the initial kinematic conditions TK2020 . Therefore, the
mutual polarization of two linearly polarized initial photons is an important
component of the nlBW process. Since nlCo is a possible source of high energy
photons, studying the polarization of Compton recoil photons is useful. In
addition, it has independent significance as a source of additional
information about the dynamics of non-linear multiphoton processes.

In the nlCo process, the polarization density matrix for the recoil photon
$\rho^{f}$ is expressed through the amplitude Akhiezer_Berestetsky

$\displaystyle M=\sum\limits_{a}{e^{\prime}_{a}}^{*}M(a)~{},$ (1)

where ${e^{\prime}_{a}}$ is the polarization unit vector for the photon in the
final state as

$\displaystyle\rho^{f}_{ab}=\frac{M(a)M^{*}(b)}{\sum\limits_{a}|M(a)|^{2}}~{}.$
(2)

The corresponding Stoks parameters $\xi^{f}_{i}$ read

$\displaystyle\xi^{f}_{i}={\rm Sp}(\rho^{f}\sigma_{i})~{},$ (3)

where $\sigma_{i}$ is the Pauli matrix. Thus, the parameter $\xi^{f}_{3}$ is
equal to the asymmetry

$\displaystyle\xi^{f}_{3}={\cal
A}\equiv\frac{|M(1)|^{2}-|M(2)|^{2}}{|M(1)|^{2}+|M(2)|^{2}}~{}.$ (4)

The sign of ${\cal A}$, plus or minus, indicates the direction of
$\gamma^{\prime}$-polarization relative to the ${e}^{\prime}_{1}$ or
${e^{\prime}}_{2}$ axes, respectively. The degree of polarization
${\cal{P}}_{12}$ relative to axes 1 and 2 is related to the asymmetry as

$\displaystyle{\cal P}_{12}=\frac{1\pm{\cal A}}{2}~{}.$ (5)

Our study is close to previous work Serbo2004  and recent Seipt:2020diz .
Compared to Serbo2004 , we now pay special attention to the kinematics of the
LUXE and other future experiments and concentrate on the photon polarization
in a wide region of $\xi$. We have analyzed the impact of a finite number of
oscillations in the pulse describing the background field. The difference from
Seipt:2020diz  lies in some technical details: such as the choice of
polarization axes, the use of the asymmetry ${\cal A}$ as an important
observable, and the difference in calculation of transition amplitudes for the
finite e.m. pulse discussed in text. In addition, we pay special attention to
the azimuthal angle distributions of the cross sections and asymmetries, which
is important for determination of $\gamma^{\prime}$ polarization. Our study
seems to be useful as an additional and independent analysis of alignment and
degrees of $\gamma^{\prime}$ polarization, which seems to be in demand in
connection with planned experiments on electron-positron pair production in
electron-laser interactions.

Methodologically, this is close to TK2020 , where the crossed nlBW process
with linearly polarized photons in the initial state in a wide range of e.m.
strength is considered.

Our work is organized as follows. Section II provides the basic formulas for
cross sections and asymmetries. The cases $\xi\leq 1$ and $\xi\gg 1$ are
discussed in Sections III and IV, respectively. Our summary is given in
Section V.

##  II Cross sections and asymmetries

As mentioned above, we consider nlCo as a process involving a large number of
laser photons simultaneously. Our consideration is based on the Furry picture.
Here we present the basic formulas for cross sections and asymmetry, which are
used below. We analyze them as the functions of the frequency
$\omega^{\prime}$ and azimuthal angle $\varphi$ over a wide range of $\xi$.
The energy of the incoming electron $E_{e}=16.5$ GeV and frequency of the
optical laser pulse $\omega=1.55$ eV with $\chi=0.196\,\xi$ are chosen
according to the assumed values in the LUXE experiment LUXE_exp .

For simplicity and without loss of generality, in this section, we use the
plane wave approximation where the background field has a form of an
monochromatic plane wave (an infinite plane wave) $A_{\mu}=a_{\mu}\cos\phi$
with $\phi=k\cdot x$ and $a_{\mu}=(0,\xi m/|e|,0,0)$. The effect of a finite
pulse with a limited number of oscillations in the pulse is discussed in Sect.
III C.

The transition matrix element reads

$\displaystyle
S=\frac{i}{\sqrt{2\omega^{\prime}2q_{0}2q^{\prime}_{0}}}\sum\limits_{\ell=1}^{\infty}M_{\ell}(2\pi)^{4}\delta^{4}(q+\ell
k-q^{\prime}-k^{\prime})~{},$ (6)

where $\ell$ is the number of laser beam photons involved in the process, and

$\displaystyle
M_{\ell}=\bar{u}_{p^{\prime}}[{e\\!\\!\\!/^{\prime}}^{*}A_{0}(\ell)+(\frac{e}{2k\cdot
p}{e\\!\\!\\!/^{\prime}}^{*}k\\!\\!\\!/a\\!\\!\\!/+\frac{e}{2k\cdot
p^{\prime}}a\\!\\!\\!/k\\!\\!\\!/{e\\!\\!\\!/^{\prime}}^{*})A_{1}(\ell)$
$\displaystyle\qquad\qquad-\frac{e^{2}a^{2}(k\cdot{e^{\prime}}^{*})}{2k\cdot
p\,k\cdot p^{\prime}}k\\!\\!\\!/A_{2}(\ell)]u_{p},$ (7)

where $u_{p}$ is the Dirac spinor, normalized as $(\bar{u}u)=2m$. The basic
functions $A_{m}(\ell)\equiv A_{m}(\ell,\alpha^{\prime},\beta)$ are determined
as RitusGroup

$\displaystyle
A_{m}(\ell\alpha^{\prime}\beta)=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}d\phi\cos^{m}(\phi)\,\,{\rm
e}^{i\ell\phi-i\alpha^{\prime}\sin\phi+i\beta\sin 2\phi}$ (8)

with

$\displaystyle\alpha^{\prime}$ $\displaystyle=$ $\displaystyle
z\cos\varphi,\quad\beta=\xi^{3}u/8\chi,\quad u=kk^{\prime}/kp^{\prime}~{},$
(9) $\displaystyle z$ $\displaystyle=$
$\displaystyle\frac{2\ell\xi}{\sqrt{1+\xi^{2}/2}}\sqrt{\frac{u}{u_{l}}(1-\frac{u}{u_{l}})},\,\,u_{l}=\frac{2l\chi}{\xi(1+\xi^{2}/2)}.$

The four vectors $e^{\prime}_{i}$ are chosen as LL4

$\displaystyle e^{\prime}_{i}=(0,\mathbf{e}^{\prime}_{i})~{},$ (11)

where $\mathbf{e}^{\prime}_{1,2}$ are mutually orthogonal and orthogonal to
$\mathbf{k}^{\prime}$: $\mathbf{e}^{\prime}_{1,2},\perp\mathbf{k}^{\prime}$.
The axes $\mathbf{e}^{\prime}_{i}$ are selected according to
Akhiezer_Berestetsky

$\displaystyle{\mathbf{e}}^{\prime}_{1}=\frac{[\mathbf{k},\mathbf{k}^{\prime}]}{|[\mathbf{k},\mathbf{k}^{\prime}]|},\qquad{\mathbf{e}}^{\prime}_{2}=\frac{[\mathbf{k}^{\prime},{\mathbf{e}^{\prime}}_{1}]}{|\mathbf{k}^{\prime}|}~{},$
(12)

leading to

$\displaystyle{\mathbf{e}}^{\prime}_{1}$ $\displaystyle=$
$\displaystyle-{\mathbf{x}}\sin\varphi+{\mathbf{y}}\cos\varphi$
$\displaystyle{\mathbf{e}}^{\prime}_{2}$ $\displaystyle=$
$\displaystyle-{\mathbf{x}}\cos\theta\cos\varphi-{\mathbf{y}}\cos\theta\sin\varphi+{\mathbf{z}}\sin\theta~{}.$
(13)

The physical meaning of ${\mathbf{e}}^{\prime}_{i}$ becomes clear at backward
scattering with $\cos\theta=-1$. In a coplanar reference frame with
$\varphi=0$, ${\mathbf{e}}^{\prime}_{1}$ is parallel to the $\mathbf{y}$ axis,
i.e. perpendicular to the laser beam polarization, and
${\mathbf{e}}^{\prime}_{2}$ is parallel to the $\mathbf{x}$ axis or parallel
to the laser beam polarization. This means that positive (negative) sign of
${\cal A}$ corresponds to the alignment of $\gamma^{\prime}$ polarization
perpendicular (parallel) to the laser beam polarization. In the reference
frame with $\varphi=\pi/2$, the positive (negative) sign of ${\cal A}$
corresponds to alignment of $\gamma^{\prime}$ polarization parallel
(perpendicular) to the laser beam polarization.

The choice of ${\mathbf{e}}^{\prime}_{i}$ in the form of (12)
Akhiezer_Berestetsky  seems to be attractive for its clarity and physical
transparency. However, Refs. Seipt:2020diz  and Rumyantsev  use expressions
for ${\mathbf{e}}^{\prime}_{i}$ different from (12). On the other hand, in the
exact QED evaluation of Compton scattering in a strong magnetic field
performed in Mushtukov , ${\mathbf{e}}^{\prime}_{i}$ is chosen in the form of
(13). In coplanar geometry with $\varphi=0$ (which gives the dominant
contribution to the total cross section) all variants are equivalent. Some
differences between different variants may appear on finite $\varphi$, which
can be the subject of experimental study of this issue.

The yield of $\gamma^{\prime}$ production for the fixed polarization is equal
to the square of the sum of the matrix elements $\sum|M_{\ell}|^{2}/2VT$
multiplied by the phase space factor of outgoing particles. The yield
expressed in $\varphi$ and $\omega^{\prime}$ is calculated by integrating the
phase space factor over $d^{3}q^{\prime}$

$\displaystyle\delta^{4}(\ell
k+q-q^{\prime}-k^{\prime})\frac{d^{3}q^{\prime}d^{3}k^{\prime}}{q^{\prime}_{0}k^{\prime}_{0}}\to\frac{d\varphi
d\omega^{\prime}}{|\vec{q}-\ell k_{0}|}~{}.$

The corresponding cross sections are determined by multiplying the yield by
the flux factor ${q_{0}}\omega/{p\cdot k}$ $\times$ $\rho^{-1}_{\gamma}$,
where $\rho_{\gamma}={m^{2}\xi^{2}\omega}N_{0}/{4\pi\alpha}$ is the
electromagnetic laser field energy density divided by $\omega$: $\langle{\it
E_{e.m.}}\rangle/\omega$. For monochromatic background the flux parameter
$N_{0}=1/2$. For a pulse with a “finite” number of oscillations, $N_{0}$
depends on the wave envelope (cf. Sect. III C).

In case of monochromatic background the matrix element squared for the fixed
${\mathbf{e}^{\prime}}_{1}$ and ${\mathbf{e}^{\prime}}_{2}$ is calculated in a
standard way, using the gauge transformation Ritus-79 ; Greiner
$e^{\prime}\to\bar{e}^{\prime}_{i}=e^{\prime}_{i}-k^{\prime}(e^{\prime}_{i}\cdot
k)/(k^{\prime}\cdot k)$ and relation RitusGroup

$\displaystyle\alpha^{\prime}A_{1}=(\ell-\beta)A_{0}+4\beta A_{2}.$ (14)

As a result, the ”partial” cross sections $d^{2}\sigma_{1,2}$, averaged and
summed over the initial and final electron spin projections, respectively,
read

$\displaystyle{d^{2}\sigma_{1}}$ $\displaystyle=$
$\displaystyle\frac{2\alpha^{2}d\varphi d\omega^{\prime}}{\xi\chi
m^{2}~{}N_{0}}\sum\limits_{\ell=\ell_{\rm
min}}^{\infty}\,\frac{1}{|\vec{q}-l\omega|}\,[\xi^{2}~{}A_{1}^{2}\sin^{2}\varphi$
(15) $\displaystyle+$
$\displaystyle\xi^{2}\frac{u^{2}}{4(1+u)}\left(A_{1}^{2}-A_{0}A_{2}\right)]~{},$
$\displaystyle{d^{2}\sigma_{2}}$ $\displaystyle=$
$\displaystyle\frac{2\alpha^{2}d\varphi d\omega^{\prime}}{\xi\chi
m^{2}~{}N_{0}}\sum\limits_{\ell=\ell_{\rm
min}}^{\infty}\,\frac{1}{|\vec{q}-l\omega|}\,[-A_{0}^{2}-\xi^{2}\,A_{1}^{2}\sin^{2}\varphi$
(16) $\displaystyle+$
$\displaystyle\xi^{2}\left(1+\frac{u^{2}}{4(1+u)}\left(A_{1}^{2}-A_{0}A_{2}\right)\right)]~{},$

The sum $d^{2}\sigma\equiv d^{2}\sigma_{1}+d^{2}\sigma_{2}$ (for monochromatic
pulse) is the known expression for the unpolarized cross section

$\displaystyle{d^{2}\sigma}$ $\displaystyle=$
$\displaystyle\frac{4\alpha^{2}d\varphi d\omega^{\prime}}{\xi
m^{2}\chi}\sum\limits_{\ell=\ell_{\rm
min}}^{\infty}\,\frac{1}{|\vec{q}-l\omega|}\,[-A_{0}^{2}$ (17)
$\displaystyle+$
$\displaystyle\xi^{2}\left(1+\frac{u^{2}}{2(1+u)}\left(A_{1}^{2}-A_{0}A_{2}\right)\right)]~{},$

where $\ell_{\rm min}=u\xi(1+\xi^{2}/2)/2\chi$.

The difference in $d^{2}\sigma_{1,2}$ leads to asymmetry (cf. Eq. (4))

$\displaystyle{\cal
A}(\varphi,\omega^{\prime})=\frac{d^{2}\sigma_{1}-d^{2}\sigma_{2}}{d^{2}\sigma}~{}.$
(18)

The sign of ${\cal A}$, plus or minus, is related to the direction of
$\gamma^{\prime}$-polarization relative to the axes $\mathbf{e}^{\prime}_{1}$
or $\mathbf{e}^{\prime}_{2}$, respectively. The degree of polarization is
determined according to Eq. (5).

Since the asymmetry is defined for any pair of $\varphi$ and
$\omega^{\prime}$, the average asymmetry $\langle{\cal A}\rangle_{\varphi}$
can be introduced

$\displaystyle\langle{\cal
A}(\omega^{\prime})\rangle_{\varphi}=\frac{1}{2\pi}\int\limits_{0}^{2\pi}d\varphi\,{\cal
A}(\varphi,\omega^{\prime})~{}.$ (19)

It has a physical meaning and can be studied experimentally. We emphasize that
asymmetry $\cal A$ is equal to the $\gamma^{\prime}$ spin variable
$\xi_{3}^{f}$.

Finally note, that $\cos\varphi$ changes sign upon transformation
$\varphi\to\varphi+\pi$. Using the symmetry property of the basis functions
(8) $A_{m}(l,\alpha^{\prime},\beta)$ with $\alpha^{\prime}=z\cos\varphi$,
$A_{m}(l,-\alpha^{\prime},\beta)=(-1)^{l+m}A_{m}(l,\alpha^{\prime},\beta)$)
Ritus-79 , one can find that the bi-linear combinations $A_{0}^{2}$,
$A_{1}^{2}$, $A_{0}A_{2}$, and $A_{1}^{2}\sin^{2}\varphi$ are even functions
of $\cos\varphi$, which leads to the symmetry relation

$\displaystyle d^{2}\sigma_{i}(\varphi)=d^{2}\sigma_{i}(\varphi+\pi)~{}.$ (20)

##  III Finite field strength (${\mathbf{\xi}}\leq 1$)

###  III.1 Low field strength with ${\mathbf{\xi}}\ll 1$

For methodological purposes, we consider here a monochromatic pulse with a
weak field strength with $\xi\ll 1$, where the lowest harmonic with $l=1$
dominates. The functions $A_{m}(l=1,\alpha^{\prime},\beta)$ for small values
of the parameters $\alpha^{\prime}\sim\xi$, $\beta\sim\xi^{2}$ read

$\displaystyle
A_{0}\simeq\frac{\alpha^{\prime}}{2}(1+\frac{\beta}{2}),\,A_{1}\simeq\frac{1}{2}-\frac{{\alpha^{\prime}}^{2}}{16}-\frac{\beta}{4},\,A_{2}\simeq\frac{\alpha^{\prime}}{8}(1+\beta).$

In the frame with $\vec{p}=0$ the total cross section is

$\displaystyle\frac{d\sigma}{d\Omega}=\sigma_{0}\left(\frac{\omega^{\prime}}{\omega}\right)^{2}\left(\frac{\omega^{\prime}}{\omega}+\frac{\omega}{\omega^{\prime}}-2\sin^{2}\theta\cos^{2}\varphi\right)~{},$
(21)

where $\sigma_{0}=\alpha^{2}/2m^{2}$ and the identity
$d\omega^{\prime}/|\vec{q}-\ell\omega|={\omega^{\prime}}^{2}d\cos\theta/\ell
k\cdot p$ is used. Averaging over $\varphi$ leads to the well-known Klein-
Nishina cross section.

The partial cross sections read

$\displaystyle\frac{d\sigma_{1}}{d\Omega}=\sigma_{0}\left(\frac{\omega^{\prime}}{\omega}\right)^{2}\left(\frac{1}{2}\left(\frac{\omega^{\prime}}{\omega}+\frac{\omega}{\omega^{\prime}}\right)+2\sin^{2}\varphi-1\right)~{},$
(22) $\displaystyle\frac{d\sigma_{2}}{d\Omega}$ $\displaystyle=$
$\displaystyle\sigma_{0}\left(\frac{\omega^{\prime}}{\omega}\right)^{2}\left(\frac{1}{2}\left(\frac{\omega^{\prime}}{\omega}+\frac{\omega}{\omega^{\prime}}\right)-2\sin^{2}\varphi+1\right.$
(23) $\displaystyle-$
$\displaystyle\left.2\sin^{2}\theta\cos^{2}\varphi\right)~{}.$

Equations (22), (23) may be written in a compact form known from textbooks
Akhiezer_Berestetsky ; LL4 ; Greiner .

$\displaystyle\frac{d\sigma_{i}}{d\Omega}=\frac{1}{2}\sigma_{0}\left(\frac{\omega^{\prime}}{\omega}\right)^{2}\left(\frac{\omega^{\prime}}{\omega}+\frac{\omega}{\omega^{\prime}}-2+4(\mathbf{e}_{0}\mathbf{e}^{\prime}_{i})^{2}\right)~{},$
(24)

where $\mathbf{e}_{0}=\mathbf{x}$ and $\mathbf{e}^{\prime}_{i}$ are the
polarization vectors of $\gamma$ and $\gamma^{\prime}$, respectively.

Equations (22), (23) lead to the asymmetry

$\displaystyle{\cal
A}(\varphi,\omega^{\prime})=2\frac{2\sin^{2}\varphi+\sin^{2}\theta\cos^{2}\varphi-1}{\frac{\omega^{\prime}}{\omega}+\frac{\omega}{\omega^{\prime}}-2\sin^{2}\theta\cos^{2}\varphi}.$
(25)

In coplanar geometry with $\varphi=0$, the asymmetry is proportional to
$-\cos^{2}\theta$, i.e. negative. This means that the polarization of the
recoil photon is directed along the $\mathbf{e}^{\prime}_{2}$ axis, or along
the laser beam polarization. In the frame with $\varphi=\pi/2$, the asymmetry
is positive, it is directed along the $\mathbf{e}_{1}$ axis, and is again
parallel to the laser beam polarization. In the general case, the asymmetry of
${\cal A}$ for separate $\ell>1$ depends on the value of $\varphi$ and the
parity of the harmonics.

For example, the asymmetry ${\cal A}(\varphi,\omega^{\prime})$ for individual
harmonics $\ell=1...4$ at $\xi=0.1$ for $\varphi=0$ and $\pi/2$ as a function
of $\omega^{\prime}/E_{e}$ is shown in Fig. 1 in the top and bottom panels,
respectively. This analysis (for separate harmonics) is made mainly for
methodological purposes and comparison with Ref. Serbo2004 .

For $\varphi=0$ (coplanar frame), the asymmetry is negative for all harmonics.
For $\ell=1$ it has a bump-like behavior with a bump position
$\omega^{\prime}/E_{e}\approx 0.17$. For $\ell>1$, the number of bumps
increases and the position of the first bump shifts towards larger values of
$\omega^{\prime}$.

Figure 1: (Color online) The asymmetry ${\cal A}(\varphi,\omega^{\prime})$ for
monochromatic background as a function of $\omega^{\prime}$ for $\ell=1...4$
at $\varphi=0$ and ${\pi}/{2}$ is shown in the top and bottom panels,
respectively. The electron energy $E_{e}=16.5$ GeV, laser beam frequency
$\omega=1.55$ eV, and $\xi=0.1$.

In the case of $\varphi={\pi}/{2}$, the asymmetry is positive and large for
odd $\ell$ and negative for even $\ell$ (for separate harmonics!). This result
is consistent with Serbo2004  (cf. Fig. 8).

###  III.2 Monochromatic background with $\xi\leq 1$

Consider now the result for the coherent sum of all harmonics in (15), (16)
for the monochromatic plane wave background (mPW). Here and below, the
electron energy $E_{e}=16.5$ GeV and the laser beam frequency $\omega=1.55$ eV
are chosen according to the LUXE kinematics LUXE_exp . The unpolarized cross
section $d\sigma/d\omega^{\prime}$ (17) as a function of $\omega^{\prime}$ is
exhibited in Fig. 2.

Figure 2: (Color online) The differential unpolarized cross sections
$d\sigma/d\omega^{\prime}$ (17) as functions of $\omega^{\prime}$ for
different $\xi$ for monochromatic plane wave background.

The result is consistent with other similar researches (see, for example,
relevant references in TitovPEPAN ). At small $\xi^{2}\ll 1$, the cross
sections have a step-like behavior (cf. bottom curve in Fig. 2), where each
new step corresponds to the inclusion of a new harmonic as
$\omega^{\prime}\to\omega^{\prime}_{\rm max}(\ell)$ with

$\displaystyle\omega^{\prime}_{\rm max}(\ell)=\frac{\ell k\cdot
q}{q_{0}+|\vec{q}|\cos\theta+\ell\omega(1-\cos\theta)}~{}.$ (26)

The height of the steps is proportional to $\xi^{2}$. It is explained by the
properties of the basis functions $A_{m}$ [4], and the presence of the factor
$\xi^{2}$ in (15) and (16). As $\xi$ increases, the relative height of the
steps decreases and at $\xi=1$ becomes insignificant (see the top curve in
Fig. 2) cf. Titov2014 ; impact .

Figure 3: (Color online) The left panels show the partial cross sections
$d\sigma_{1,2}/d\omega^{\prime}$ as functions of $\omega^{\prime}$. The right
panels show the differential cross sections $d^{2}\sigma_{1,2}/d\varphi
d\omega^{\prime}$ as functions of the azimuthal angle $\varphi$ at fixed
$\omega^{\prime}=0.2\,E_{e}$. Background is a monochromatic plane wave.

The partial cross sections $d\sigma_{1,2}/d\omega^{\prime}$ as functions of
$\omega^{\prime}$ for different $\xi$ are shown in the left panels of Fig. 3.
In the region $\omega^{\prime}<\omega^{\prime}_{1}=\omega^{\prime}_{\rm
max}(\ell=1)$, $d\sigma_{2}/d\omega^{\prime}$ has a deep minimum.
The distributions of $d^{2}\sigma_{i}/d\varphi d\omega^{\prime}$ over
azimuthal angle at fixed $\omega^{\prime}=0.2\,E_{e}$, close to the position
of the minimum in $d\sigma_{2}/d\omega^{\prime}$, are shown in the right
panels of Fig. 3. Here we assume the symmetry property (20). At $\varphi=0$
($\pi$), $d^{2}\sigma_{2}\gg d^{2}\sigma_{1}$, whereas at $\varphi=\pi/2$,
$d^{2}\sigma_{1}\gg d^{2}\sigma_{2}$, which results in negative and positive
asymmetries ${\cal A}(\varphi)$, for these two cases, respectively.

Figure 4: (Color online) The left panel: The asymmetries as functions of
$\varphi$ at $\omega^{\prime}=0.2\,E_{e}$ for different $\xi$. The right
panel: The average asymmetries $\langle{\cal A}\rangle_{\varphi}$ as functions
of $\omega^{\prime}$. Background is a monochromatic plane wave.

The asimuthal angle dependence of ${\cal A}(\varphi,\omega^{\prime})$ at fixed
$\omega^{\prime}=0.2E_{e}$ for different $\xi$ is exhibited in the left panel
of Fig. 4. It is negative with a large absolute value at $\varphi=0$ ($\pi$)
and large and positive at $\varphi\simeq\pi/2$. In both cases, polarization of
$\gamma^{\prime}$ is along the laser beam polarization with a degree of
polarization ${\cal P}_{12}\simeq 0.9$.

The averaged asymmetry as a function of $\omega^{\prime}$ is exhibited in the
right panel of Fig. 4. At small $\omega^{\prime}$ it is positive with a bump
around $\omega^{\prime}/E_{e}\sim 0.2$, then change a sign to negative and
vanishes at $\omega^{\prime}\to E_{e}$.

###  III.3 The finite pulse

In this subsection, we study the effect of a finite pulse, where the linearly
polarized background field is determined by the e.m. four-potential
$A(\phi)=(0,{\mathbf{A}}(\phi))$ with

$\displaystyle\mathbf{A}(\phi)=f(\phi)\left[\mathbf{a}\cos(\phi)\right]~{},$
(27)

where $\phi=k\cdot x$ is the invariant phase, and
${\mathbf{a}}={\mathbf{x}}m\xi/e$. For the sake of definiteness, the envelope
function $f(\phi)$ is chosen as a hyperbolic secant:
$f(\phi)=1/[\cosh\phi/\Delta]$. The dimensionless quantity $\Delta$ is related
to the pulse duration $2\Delta=2\pi N$, where $N$ has the meaning of the
number of cycles in the laser pulse. It is related to the time duration of the
pulse $\tau_{N}=2N/\omega$ (for the dependence of some observables on the
envelope shape, see, for example TitovPEPAN ; Titov2014 ).

Figure 5: (Color online) The differential unpolarized cross sections (35) for
different $\xi$. The left and right panels correspond to the number of
oscillations in the pulse $N=2$ and 5, respectively. The corresponding result
for monochromatic background is exhibited by the thin solid curves.

Figure 6: (Color online) The left panels: The differential partial cross
sections $d\sigma_{i}/d\omega^{\prime}$ for a number of oscillations in a
pulse $N=2$ and different $\xi$. The right panels: The differential cross
sections $d^{2}\sigma_{i}/d\varphi d\omega^{\prime}$ as functions of the
azimuthal angle $\varphi$ at fixed $\omega^{\prime}=0.2\,E_{e}$.

Figure 7: (Color online) The same as in Fig. 6 but for $N=5$.

Figure 8: (Color online) The asymmetries as functions of the azimuthal angle
$\varphi$ for different $\xi$, at $\omega^{\prime}=0.2\,E_{e}$. The left and
right panels correspond to the number of oscillations in the pulse $N=2$ and
5, respectively.

Figure 9: (Color online) The average asymmetries $\langle{\cal
A}\rangle_{\varphi}$ as functions of $\omega^{\prime}$ for different $\xi$.
The left and right panels correspond to the number of oscillations in the
pulse $N=2$ and 5, respectively.

In our consideration, the $S$-matrix element is determined in the same way as
in the monochromatic case, using the exact Volkov solution for electron wave
functions with e.m. four-potential (27). The transition matrix element is
calculated using the Fourier integral Titov2020 ; Titov2014 . As a result, an
auxiliary continuous variable $\ell$ appears. The product $\omega\ell$ has the
meaning of the fraction of the laser beam energy involved in the non-linear
Compton process. Together with the replacement of summation over discrete $l$
by the integral over continuous $d\ell$ in the transition matrix, there is one
more difference: now the pre-exponential factor and the exponential factor in
electron wave functions containing linear and quadratic field terms of
$A(\phi)$ form new basis functions ${\widetilde{A}_{m}}$. The structure of the
matrix element remains similar to the monochromatic case (7). The new basis
functions ${\widetilde{A}_{m}}$ read impact

$\displaystyle\widetilde{A}_{m}(\ell)=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}d\phi\,f^{m}(\phi)\cos^{m}(\phi)\,{\rm
e}^{i\ell\phi-i{\cal P}(\phi)}~{},$ (28)

where

$\displaystyle{\cal P}(\phi)=\tilde{\alpha}(\phi)-\tilde{\beta}(\phi)~{},$
(29)
$\displaystyle\tilde{\alpha}(\phi)=\alpha^{\prime}\,h_{1}(\phi),\qquad\tilde{\beta}(\phi)=4\beta\,h_{2}(\phi)$
(30)

with

$\displaystyle
h_{n}(\phi)=\int\limits_{-\infty}^{\phi}d\phi^{\prime}f^{n}(\phi^{\prime})\cos^{n}\phi^{\prime}~{},$
(31)

where $\alpha^{\prime}$ and $\beta$ are determined in (9) with
$z={2\xi\ell}\sqrt{{u/u_{\ell}}(1-u/u_{\ell})}$ and $u_{\ell}=2\ell\chi/\xi$.
The function $\widetilde{A}_{0}(\ell)$ is regularized using the prescription
of Boca-2009  which leads to the identity

$\displaystyle{\alpha^{\prime}}\widetilde{A}_{1}(\ell)=\ell\widetilde{A}_{0}(\ell)+4\beta\widetilde{A}_{2}(\ell)~{}.$
(32)

The differential cross sections $d^{2}\sigma_{i}$ have the form of equations
(15,16) with the replacement of $\sum\limits_{\ell}$ by
$\int\limits_{\ell_{\rm min}}^{\infty}d\ell$, where the lower limit of the
integral is $\ell_{\rm min}=um^{2}/2k\cdot p$ and the replacement
$A_{m}\to\widetilde{A}_{m}$

$\displaystyle{d^{2}\sigma_{1}}$ $\displaystyle=$
$\displaystyle\frac{2\alpha^{2}d\varphi d\omega^{\prime}}{\xi\chi
m^{2}E_{e}~{}N_{0}}\int\limits_{\ell_{\rm
min}}^{\infty}~{}d\ell\,[\xi^{2}~{}|\widetilde{A}_{1}|^{2}\sin^{2}\varphi$
(33) $\displaystyle+$
$\displaystyle\xi^{2}\frac{u^{2}}{4(1+u)}\left(|\widetilde{A}_{1}|^{2}-{\rm
Re}\widetilde{A}_{0}\widetilde{A}^{*}_{2}\right)]~{},$
$\displaystyle{d^{2}\sigma_{2}}$ $\displaystyle=$
$\displaystyle\frac{2\alpha^{2}d\varphi d\omega^{\prime}}{\xi\chi
m^{2}E_{e}~{}N_{0}}\int\limits_{\ell_{\rm
min}}^{\infty}~{}d\ell\,[-|\widetilde{A}_{0}|^{2}-\xi^{2}\,|\widetilde{A}_{1}|^{2}\sin^{2}\varphi$
(34) $\displaystyle+$
$\displaystyle\xi^{2}\left(1+\frac{u^{2}}{4(1+u)}\left(|\widetilde{A}_{1}|^{2}-{\rm
Re}\widetilde{A}_{0}\widetilde{A}^{*}_{2}\right)\right)]~{},$

where replacement $|\vec{q}-l\omega|\simeq|\vec{p}|\simeq E_{e}$ is used. The
unpolarized cross section is the sum

$\displaystyle{d^{2}\sigma}={d^{2}\sigma_{1}}+{d^{2}\sigma_{2}}~{}.$ (35)

Utilization of basis functions $\widetilde{A}_{1,2}$ in (28) lead to a
multidimensional integral for $d\sigma_{1,2}/d\omega^{\prime}$ which needs
large computing resources. We calculate them as follows. For given wave
envelope functions $f(\phi)$, we first calculate two arrays of auxiliary
functions $h_{1,2}(\phi)$ (31) in the range $\phi_{\rm
min}\leq\phi\leq\phi_{\rm max}$, where $\phi_{\rm max,min}=\pm\,30\,\Delta$
with $10^{5}$ points ($\Delta=\pi N$ and $N$ is the number of oscillations in
a pulse). In fact, $\phi_{\rm max,min}$ are the limits of integration in (28).
Then, using the FORTRAN interpolation software PARINT, we extract the phase
function ${\cal P}(\phi)$ (29) and determine the complex integrand in (28) at
any point in the integration domains in (33), (34). This technique reduces the
dimension of the integrals for $d\sigma_{i}$ and the time of their
calculation. The function $\widetilde{A}_{0}$ is determined by Eq.(32).

The flux parameter $N_{0}$ in (33, 34) is related to the average square of the
e.m. strength and expressed thought the envelope function as integral
$1/2\pi\int_{-\infty}^{\infty}d\phi(f^{2}(\phi)+{f^{\prime}}^{2}(\phi))\cos^{2}\phi$
and for chosen $f(\phi)$ is equal $N_{0}={\Delta}/{2\pi}(1+1/{3\Delta^{2}})$.

Below we will limit ourselves to the number of oscillations in a pulse $N=2$
and 5 and field strength $\xi=0.1$, 0.5 and 1. The unpolarized cross sections
for a finite pulse are exhibited in Fig. 5. This result is close to our
previous works (see, for example TitovPEPAN ; impact ). For completeness and
easy references we also exhibited in Fig. 5 result for monochromatic
background shown by solid thin curves. Again, one can see a step-like behavior
of the cross section at small $\xi$. The result for N=5 is closer to the
monochromatic background and, for N=10 they are almost identical impact .

The partial cross sections $d\sigma_{i}/d\omega^{\prime}$ for N=2 and 5 are
shown in the left panels of Figs. 6 and 7, respectively. In general, the
results for $N=2$ and 5 are close to each other. It can be seen that the
result for the finite pulse background is qualitatively close to the result
for the monochromatic PW background shown in Figs 2 and 3: now the cross
sections are slightly smoother functions of $\omega^{\prime}$ .

The same is true for the azimuthal angle distributions of differential cross
sections shown in the right panels of Figs. 6 and 7, and asymmetries shown in
Fig. 8. The average asymmetries shown in Fig. 9 resemble the result of the
monochromatic PW calculation, cf. Fig. 4 (left). The symmetry property (20) is
also valid for a finite pulse, which is confirmed by direct numerical
calculation.

To summarize this part, we can conclude that in general, our result for a
finite pulse is similar to the case of the monochromatic plane wave
background, i.e. in coplanar geometry with $\varphi=0$ and in a frame with
$\varphi=\pi/2$ the recoil photon $\gamma^{\prime}$ is polarized almost
parallel to the polarization of the laser beam with a degree polarization
${\cal P}$ close to unity.

##  IV Ultra-large pulse intensity, ${\mathbf{\xi}}\gg 1$

For large values of $\xi\gg 1$ the main contribution to probabilities of non-
linear processes comes from the central part of the wave envelope and the
diagonal terms in the amplitude squared with $\ell=\ell^{\prime}$ TitovPRA2013
. Therefore, as in TK2020 , for further analysis we use an extension of the
formalism developed by Nikishov and Ritus RitusGroup , with explicit
consideration of the $\gamma^{\prime}$ polarization. Strictly speaking, the
validity of this approximation for the chosen variables depends on the value
of the recoil photon energy $\omega^{\prime}$. We will discuss this issue
later.

The partial and unpolarized differential sections $d^{2}\sigma_{i}$,
$d^{2}\sigma$ have the following form:

$\displaystyle{d^{2}\sigma_{1}}$ $\displaystyle=$
$\displaystyle\frac{4\alpha^{2}d\varphi
d\omega^{\prime}}{m^{2}\xi\chi\,E_{e}}\int\limits_{\ell_{\rm
min}}^{\infty}d\ell\,[\xi^{2}~{}\widehat{A}_{1}^{2}\sin^{2}\varphi$ (36)
$\displaystyle+$
$\displaystyle\xi^{2}\frac{u^{2}}{4(1+u)}\left(\widehat{A}_{1}^{2}-\widehat{A_{0}}\widehat{A}_{2}\right)]~{},$
$\displaystyle{d^{2}\sigma_{2}}$ $\displaystyle=$
$\displaystyle\frac{4\alpha^{2}d\varphi
d\omega^{\prime}}{m^{2}\xi\chi\,E_{e}}\int\limits_{\ell_{\rm
min}}^{\infty}d\ell\,[-\widehat{A}_{0}^{2}-\xi^{2}\,\widehat{A}_{1}^{2}\sin^{2}\varphi$
(37) $\displaystyle+$
$\displaystyle\xi^{2}\left(1+\frac{u^{2}}{4(1+u)}\left(\widehat{A}_{1}^{2}-\widehat{A}_{0}\widehat{A}_{2}\right)\right)]~{}.$
$\displaystyle{d^{2}\sigma}$ $\displaystyle=$
$\displaystyle{d^{2}\sigma_{1}}+{d^{2}\sigma_{2}}~{},$ (38)

where $\ell_{\rm min}=u\xi(1+\xi^{2}/2)/2\chi$.

The bi-linear combinations of $\widehat{A}_{k}$ are expressed through the Airy
functions $\Phi$ and their derivatives $\Phi^{\prime}$

$\displaystyle\widehat{A}_{0}^{2}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{2\pi^{2}}\Phi^{2}(y)~{},\qquad
g^{2}=\frac{4}{\xi^{2}\sin^{2}\psi}\frac{\sigma}{y},$
$\displaystyle\widehat{A}_{1}^{2}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{2\pi^{2}}(\rho^{2}\Phi^{2}(y)+\frac{\sigma}{\xi^{2}y}{\Phi^{\prime}}^{2}(y))~{},$
$\displaystyle\widehat{A}_{0}\widehat{A}_{2}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{2\pi^{2}}(\rho^{2}-\frac{\sigma}{\xi^{2}})\Phi^{2}(y)~{},$
(39)

where the argument of the Airy function $y$ is related to the auxiliary
variables $\rho=\cos\psi$, $\tau$ and $\sigma$ as

$\displaystyle y$ $\displaystyle=$
$\displaystyle\left(\frac{u}{2\chi\sin\psi}\right)^{2/3}\sigma~{},\quad\sigma=1+\tau^{2}$
$\displaystyle\rho^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\xi^{2}}(1+\frac{\xi^{2}}{2})(\frac{u_{\ell}}{u}-1)\cos^{2}\varphi$
$\displaystyle\tau^{2}$ $\displaystyle=$
$\displaystyle(1+\frac{\xi^{2}}{2})(\frac{u_{\ell}}{u}-1)\sin^{2}\varphi$ (40)

with $u_{\ell}=2\ell\chi/\xi(1+\xi^{2}/2)$.

The validity of our approach at $\xi\gg 1$ is determined by condition
$\ell_{\rm min}\gg 1$. Assuming that for $\omega^{\prime}\ll E_{e}$,
$u\simeq\omega^{\prime}/E_{e}$ we have

$\displaystyle\omega^{\prime}/E_{e}\gg\omega^{\prime}_{\rm
min}/E_{e}=4\chi/\xi^{3}~{}.$ (41)

For $E_{e}=16.5$ GeV and $\omega=1.55$ eV, $\omega^{\prime}_{\rm
min}/E_{e}\simeq 0.78/\xi^{2}$. Indeed, this $\omega^{\prime}_{\rm min}/E_{e}$
is a small value, but nevertheless it should be remembered that it is finite.

Figure 10: (Color online) The left panel: The unpolarized differential cross
sections $d\sigma/d\omega^{\prime}$ as functions of $\omega^{\prime}$ for
$\xi=5,\ 20$ and 50 shown by the blue dot-dashed, red dashed, and black solid
curves, respectively. The right panel: The partial cross sections
$d\sigma_{i}/d\omega^{\prime}$ as functions of $\omega^{\prime}$ for $\xi=5$
and 50.

The unpolarized cross sections $d\sigma/d\omega^{\prime}$ integrated over the
azimuthal angle $\varphi$ for $\xi=5,$ 20 and 50 are shown in the left panel
of Fig. 10. Here and below we consider $\omega^{\prime}$ in the range
$\omega^{\prime}_{\rm min}<\omega^{\prime}<E_{e}$. The cross sections decrease
monotonically with increasing $\omega^{\prime}$, disappearing in the limit
$\omega^{\prime}\to E_{e}$.

Figure 11: (Color online) The left and right panels correspond to the
azimuthal angle distributions of the partial differential cross sections
$d^{2}\sigma_{i}/d\omega^{\prime}$ for $\omega^{\prime}/E_{e}=0.2$ and 0.6,
respectively, for $\xi=5$ and 50.

The partial distributions $d\sigma_{1,2}/d\omega^{\prime}$ integrated over the
azimuthal angle $\varphi$ for $\xi=5$ and 50 are shown in the right panel of
Fig. 10. One can see that $d\sigma_{2}>d\sigma_{1}$. However, it is impossible
to determine the alignment of $\gamma^{\prime}$-polarization only from the
predominance of $\sigma_{2}$, averaged over the azimuthal angle.

For this purpose, Fig. 11 shows the distributions of partial cross sections as
functions of the azimuthal angle for $\omega^{\prime}/E_{e}=0.2$ and 0.6
exhibited in the left and right panels, respectively. Note that the structure
of $d^{2}\sigma_{i}$ confirms the symmetry property (20).

The corresponding asymmetries are shown in Fig. 12.

Figure 12: (Color online) The asymmetries ${\cal A}(\varphi,\omega^{\prime})$
as functions of the azimuthal angle $\varphi$ at $\omega^{\prime}/E_{e}=0.2$
and 0.6 exhibited in the left and right panels, respectively.

In the vicinity of $\varphi=0\,(\pi)$, $d^{2}\sigma_{2}\gg d^{2}\sigma_{1}$
$({\cal A}\sim-1)$, which leads to alignment of the $\gamma^{\prime}$
polarization along the laser beam polarization. The degree of polarization
${\cal P}$ is close to unity. These areas provide the main contribution to the
integral cross sections. On the other hand, in vicinity of $\varphi=\pi/2$,
$d^{2}\sigma_{1}$ dominates (${\cal A}>0$). This again leads to the alignment
of $\gamma^{\prime}$ polarization along the laser beam polarization.

Figure 13: (Color online) The asymmetries ${\cal A}(0,\omega^{\prime})$,
${\cal A}(\pi/2,\omega^{\prime})$, and the average asymmetries $\langle{\cal
A}\rangle_{\varphi}$ at small and large values of $\xi$, shown in the left and
right panels, respectively, at $\omega^{\prime}/E_{e}=0.2$.

For convenience, the asymmetries ${\cal A}(0,\omega^{\prime})$, ${\cal
A}(\pi/2,\omega^{\prime})$, and $\langle{\cal A}\rangle_{\varphi}$ at
$\omega^{\prime}/E_{e}=0.2$ in a wide region of $\xi$ are shown in Fig. 13.
The left and right panels correspond to the intervals $0.01\leq\xi\leq 1$ and
$5\leq\xi\leq 100$, respectively. The asymmetry in coplanar geometry ${\cal
A}(\varphi=0,\omega^{\prime})$ is negative with a large absolute value
regardless of the field strength. When $\varphi=\pi/2$, the asymmetry ${\cal
A}(\varphi=\pi/2,\omega^{\prime})$ is positive, while the average asymmetry
varies from large to small values, when $\xi$ varies from small to large
values $\xi$. At $\xi\gg 1$, the average asymmetry is close to zero, which is
a consequence of the azimuthal angle distributions exhibited in Fig. 12.

##  V Summary

In summary, we have analyzed polarization of the recoil photon
$\gamma^{\prime}$ in non-linear Compton scattering. The polarization axes
${\mathbf{e}^{\prime}_{1,2}}$ are chosen according to (12). Polarization of
$\gamma^{\prime}$ is described by the asymmetry ${\cal{A}}$ which is equal to
the intrinsic spin variable $\xi^{f}_{3}$ of $\gamma^{\prime}$. The initial
photon polarization is chosen along the ${\mathbf{x}}$ axis.

We have analyzed the $\gamma^{\prime}$ polarization in the LUXE experiment
with the electron energy $E_{e}=16.5$ GeV and optical laser frequency of 1.55
eV in a wide range of $\xi$.

The result of our study shows that for small $\xi<1$ and
$\omega^{\prime}/E_{e}\simeq 0.2$ the asymmetry
${\cal{A}}(\varphi,\omega^{\prime})$ is large and negative or positive for
$\varphi=0$ or $\varphi=\pi/2$, respectively, which indicates that the recoil
photon is polarized along the $\mathbf{e}^{\prime}_{2}$ or
$\mathbf{e}^{\prime}_{1}$ axes, respectively. In both cases, $\gamma^{\prime}$
is polarized along the laser beam polarization and the degree of polarization
${\cal P}$ is close to unity. The average asymmetry $\langle
A\rangle_{\varphi}$ has a bump-like behavior with a bump position
$\omega^{\prime}/E_{e}\approx 0.2$. The finite pulse does not modify
qualitatively the result for the monochromatic plane wave background.

Concerning our model for the finite pulse, it should be noted that despite the
transparency and validity of our approach, it involves the calculation of
multiple integrals with rapidly oscillating functions, which requires a large
amount of computational resources, especially for a large number of
oscillations $N$ in the pulse and large $\xi$. This causes inconvenience, say,
in using our approach as an input for kinetic codes. Nevertheless, it can be
used as a test for approximate approaches such as the local-constant-field
approximation (LCFA) Ritus-79 ; Piazza2018  or ”locally monotonic”
approximation (LMA) Heinzl2020 ; for LCFA and LMA, see review AM_Review  and
references therein. On the other hand, our results for a finite pulse coincide
qualitatively with the results of the monochromatic PW background, so the
latter can be used in qualitative estimations.

At ultra-high laser beam intensity with $\xi\gg 1$ the asymmetry
${\cal{A}}(\varphi,\omega^{\prime})$ is negative (positive) at $\varphi=0$
($\pi/2$). In both cases, $\gamma^{\prime}$ is polarized along the laser beam
polarization. In case of $\varphi=0\,(\pi)$ the degree of polarization is
close to unity. The average asymmetry is close to zero.

Our calculation was performed for polarization axes in the form (11), (12). A
different choice of axes (see for example Seipt:2020diz ) would lead to a
different dependence of partial cross sections on the azimuthal angle, which
can be the subject of experimental study of this issue. However, the main
conclusions of Seipt:2020diz  relative to the directions and degree of
$\gamma^{\prime}$ polarization are close to ours.

In spite of the fact that the partial cross sections in III B, III C and IV
are expressed in a similar form, they are different because correspond to the
different initial conditions: monochromatic background, background with the
finite pulse, and large field intensity with $\xi\gg 1$, respectively. The
difference appears in a different structure of the basis functions, which
leads to different methods of calculating multidimensional integrals for cross
sections. We choose such presentation as a transparent and convenient for
those who want to repeat our results.

Finally note that our result assumes the preferred configuration of experiment
on $e^{+}e^{-}$ pair production in the two-step electron-laser interaction
with linearly polarized laser beams (see, for example, Fig. 3 in LUXE_exp ).
The first and the second steps are the nlCo and nlBW processes, respectively.
The mutual polarization of these two laser beams must be perpendicular since
this configuration leads to enhanced yield of $e^{+}e^{-}$ pair production
RitusGroup ; TK2020 .

The yield of $e^{+}e^{-}$ pairs in electron-laser interaction based on the
nlCo$\otimes$nlBW-type folding model was estimated at a probabilistic level in
TAK2021 . The polarization of recoil photons was not taken into account.
Accounting for the $\gamma^{\prime}$ polarization can be a further development
of such studies.

##  VI Acknowledgments

I am grateful to B. Kämpfer for our fruitful previous collaboration on
studying different topics of strong-field QED and to O. V. Teryaev for
discussions of various aspects of spin physics.

Data Availability Statement: No Data associated in the manuscript.

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