# Phenomenology of a Rydberg impurity in an ideal Bose Einstein condensate

Aileen A. T. Durst<EMAIL_ADDRESS>Matthew T. Eiles<EMAIL_ADDRESS>Max
Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187
Dresden, Germany

(September 3, 2024)

###### Abstract

We investigate the absorption spectrum of a Rydberg impurity immersed in and
interacting with an ideal Bose-Einstein condensate. Here, the impurity-bath
interaction can greatly exceed the mean interparticle distance; this
discrepancy in length scales challenges the assumptions underlying the
universal aspects of impurity atoms in dilute bosonic environments. Our
analysis finds three distinct parameter regimes, each characterized by a
unique spectral response. In the low-density regime, we find that the Rydberg
impurity is dressed by the surrounding bath similarly to the known Bose
polaron. Transitioning to intermediate densities, the impurity response, given
by sharp quasiparticle peaks, fragments into an intricate pattern bearing the
hallmarks of a diverse molecular structure. Finally, at high density, a
universal Gaussian response emerges as the statistical nature of the bath
dominates its quantum dynamics. We complement this analysis with a study of an
ionic impurity, which behaves equivalently. Our exploration offers insights
into the interplay between interaction range, density, and many-body behavior
in impurity systems.

The dynamics of strongly-correlated quantum mixtures pose a significant
challenge to theoretical description, even at the level of a single impurity
immersed in a non-interacting bath. The apparent simplicity of the Hamiltonian
of such a mixture,

$\displaystyle\hat{H}$
$\displaystyle=\sum_{\boldsymbol{k}}\frac{\boldsymbol{k}^{2}}{2\mu}\hat{b}_{\boldsymbol{k}}^{\dagger}\hat{b}_{\boldsymbol{k}}+\sum_{\boldsymbol{k},\boldsymbol{q}}V(\boldsymbol{q})\hat{b}_{\boldsymbol{k}+\boldsymbol{q}}^{\dagger}\hat{b}_{\boldsymbol{k}},$
(1)

belies the rich complexity of the phenomena emergent in this many-particle
system [1, 2, 3]. In Equation 1, which is written in a frame centered on the
zero-momentum impurity, $\hat{b}_{\boldsymbol{k}}^{\dagger}$ and
$\hat{b}_{\boldsymbol{k}}$ denote the bath creation and annihilation
operators, $V(\boldsymbol{q})$ is the interspecies interaction, and $\mu$ is
the reduced mass of the impurity and a bath atom [4]. $\hat{H}$ is commonly
used to describe dilute gases with a mean interparticle distance
$\rho^{-1/3}$, where $\rho$ is the density, greatly exceeds the range of the
potential $V(\boldsymbol{r})$. This justifies its replacement by a zero-range
pseudopotential proportional to the bath-impurity scattering length $a$ [5].
Within this approximation, the physics of the system becomes universal,
depending only on the scattering length, the dimensionality of the system, and
the density and quantum statistics of the bath [6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16]. Measurements of repulsive and attractive polaron quasiparticles
and weakly bound molecules in ultracold gases have provided strong evidence
for this universal behavior [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 16].

However, this universality is not expected to apply when the interaction range
is comparable to the typical interparticle distance. Such is the case for a
Rydberg impurity, where the highly excited Rydberg electron with principal
quantum number $n$ mediates the impurity-bath interaction by scattering off of
the bath particles. The $s$-wave electron-atom scattering length $a_{s}$ (see
Figure 1) determines the overall strength of this interaction [27, 28, 29].
For a Rydberg $\ket{nS}$ state this leads to the isotropic potential

$\displaystyle V_{\mathrm{Ryd}}(r)=2\pi
a_{s}\absolutevalue{\psi_{n00}(r)}^{2},$ (2)

which, unlike a zero-range potential, can in principle support several bound
states [30, 31, 32, 33, 34]. The appearance of the Rydberg wave function,
$\psi_{n00}(r)$, causes the range $R_{0}$ and depth $V_{0}$ of this highly
oscillatory potential to vary as $n^{2}$ and $n^{-6}$, respectively [34].
Recently, the ultracold toolbox has been expanded to include other impurity
systems with finite-ranged interactions, such as dipolar atoms [35, 36, 37]
and ion-atom mixtures [38, 39, 40, 41]. These break the zero-range
universality in a similar fashion, and raise the question of whether or not a
unified description of a finite-ranged impurity in a quantum environment
exists.

In this article, we approach this question through an exploration of the
behavior of a Rydberg impurity interacting with an ideal Bose-Einstein
Condensate (BEC) at zero temperature. Previous studies [42, 43, 44, 45] have
treated such an impurity in isolation from the zero-range Bose polaron due to
the large discrepancy in length and energy scales. However, we show that these
different impurities share the same underlying physics determined by the
universal parameter $a\rho^{1/3}$. Even though the spectral response becomes
more complicated in finite-ranged impurity systems, each component can be
understood and generally described. To further support these findings, we also
consider an ionic impurity. Together, this leads to an extension of the
universal description of Bose polarons to a broader class of interactions.
Further, we attempt to unify the disparate interpretations provided by the
many approaches developed for such impurity problems, which include the field-
theoretical quasiparticle methods describing polaron physics, the few-body
picture of molecular physics, and semiclassical methods originating in the
theory of pressure broadening.

Figure 1: Two scattering lengths characterize the interaction of a Rydberg
impurity with its environment. Atoms probing the interior of the Rydberg atom
collide with the highly excited electron directly; these interactions are
characterized by the atom-electron scattering length $a_{\mathrm{s}}$. In
contrast, distant atoms interact with the Rydberg atom as a single entity, and
the atom-impurity scattering length $a_{\mathrm{Ryd}}$ is characteristic of
this interaction.

To investigate and characterize the universal aspects of the Rydberg impurity,
we performed a detailed numerical study of the absorption spectrum
$A(\omega)$. This is obtained from the Fourier transform of the auto-
correlation function

$S(t)=\langle
e^{i\hat{H}_{0}t}e^{-i\hat{H}t}\rangle=\left(\sum_{\alpha}e^{i(\epsilon_{0}-\omega_{\alpha})t}\absolutevalue{\bra{0}\ket{\alpha}}^{2}\right)^{N},$
(3)

where the expectation value is taken with respect to the non-interacting BEC
state (the ground state of $\hat{H}_{0}$) [44]. Since the ideal BEC is in a
pure product state, the final expression of the many-body response only
requires the eigenstates $(\ket{0})$, $\ket{\alpha}$ and energies
$(\epsilon_{0})$, $\omega_{\alpha}$ of the (non)-interacting two-body
Hamiltonian of the Rydberg atom and a single boson. We solve for the two-body
physics by employing the eigenchannel R-matrix method [46], which yields the
energy-dependent logarithmic derivative of the scattering wave function
$\ket{\alpha}$ for $r>R_{0}$. This allows us to efficiently calculate
thousands of box-continuum states, molecular bound states, and the zero-energy
scattering length $a_{\mathrm{Ryd}}$. Figure 1 shows
$a_{\mathrm{Ryd}}(a_{s})$, which carries information about both the
interaction range $R_{0}\sim 2n^{2}$ and the dependence of two-body bound
states on $V_{0}$. We study a Rydberg impurity with $n=50$ as an illustrative
and generic example, and compute $A(\omega)$ as a function of $a_{s}$
111Further details about the interaction potential and the numerical
parameters for the calculation can be found in the Supplementary Material at:
URL to be inserted by publisher.. In this way we adjust $V_{0}$ independent of
$R_{0}$ 222While it is not possible to tune the electron-atom scattering
length in experiment, it is a convenient theoretical tool due to this
decoupling of $V_{0}$ and $R_{0}$. To change $a_{\mathrm{Ryd}}$
experimentally, $n$ can be varied to explore the different interaction regimes
studied here..

Figure 2: $A(\omega)$ of a $50S$ Rydberg impurity in a BEC with
$\rho=10^{12}\,$cm-3. The mean-field energy shifts $E_{\mathrm{zr}}$ (teal)
and $E_{\mathrm{Ryd}}$ (green), as well as the bare and dressed dimer energies
$E_{\mathrm{b}}$ (solid white) and $E_{\mathrm{b}}+E_{\mathrm{zr}}$ (dashed
white) are overlaid.

Figure 2 shows $A(\omega)$ when $R_{0}\rho^{1/3}\ll 1$. In this regime, we
recover all of the features known from the limit of a zero-range impurity [49,
3, 12]. When $a_{\mathrm{s}}>-0.06a_{0}$, $V_{\mathrm{Ryd}}$ does not support
a two-body bound state, and $A(\omega)$ exhibits a single peak at negative
energy indicating the formation of an attractive polaron. The mean-field
energy of the zero-range approximation of the Rydberg potential,
$E_{\mathrm{zr}}=2\pi a_{\mathrm{Ryd}}\rho/\mu$, describes the position of
this feature. However, this description fails dramatically near a scattering
resonance. Instead, the mean-field description of the full Rydberg potential,
$E_{\mathrm{Ryd}}=\rho\int V_{\mathrm{Ryd}}(r)\mathrm{d}^{3}r={2\pi
a_{s}\rho}/{m_{e}}$, [44, 28, 50] follows the center of spectral weight
smoothly across unitarity, even as the spectral feature diffuses and cannot be
associated with a well-defined quasiparticle [51, 3]. Using the Born
approximation for the Rydberg scattering length,
$a_{\mathrm{Ryd}}^{\mathrm{B}}=a_{s}\mu/m_{e}$, one can rewrite
$E_{\mathrm{Ryd}}=2\pi a_{\mathrm{Ryd}}^{\mathrm{B}}\rho/\mu$ to have the same
structure as $E_{\mathrm{zr}}$.

To the red of the resonance, $E_{\mathrm{zr}}$ again describes the brightest
spectral feature, which is located at positive energy and therefore identified
as a repulsive polaron. Below this feature lies a series of negative energy
peaks associated with the molecular bound state, which can be multiply
occupied to form dimers, trimers, and the like. These interact with the
residual bath through the same Rydberg interaction as the bare atom, and as a
result they are dressed by bath excitations in the same fashion. Each
ultralong-range Rydberg molecule therefore possesses some quasiparticle
character inherited from the repulsive polaron and forms a "molaron" with a
binding energy shifted from that of the bare dimer, $E_{\mathrm{b}}$, to
$E_{\mathrm{b}}+E_{\mathrm{zr}}$ [52, 53, 54, 55]. Myriad experiments have
observed this vibrational spectrum, confirming its basic structure but not yet
providing conclusive evidence for the many-body shift $E_{\mathrm{zr}}$ [56,
57, 58, 43]. This is not surprising, since theoretical uncertainties [59, 60,
61, 62, 63] would have hidden this small shift.

Figure 3: $A(\omega)$ of a $50S$ Rydberg impurity in a BEC with (a)
$\rho=10^{13}\,$cm-3, (b) $\rho=5\cdot 10^{13}\,$cm-3, and (c) $\rho=5\cdot
10^{14}\,$cm-3. In (a), the mean-field energy $E_{\mathrm{zr}}$ (teal) is
shown alongside the mean-field energies of various molaron states. In panels
(b) and (c) cuts of $A(\omega)$ at fixed $a_{s}$ (white curves) show the
lineshape more clearly. $E_{\mathrm{Ryd}}$ is shown in green.

Figure 3(a) displays $A(\omega)$ over a broader range of $a_{s}$ and at ten
times the density of Figure 2, which causes the molaron peaks to accumulate
appreciable spectral weight as $R_{0}\rho^{1/3}\sim 1$. This reveals an
intriguing internal structure due to the appearance of additional two-body
bound states as the potential deepens. This structure suggests a nomenclature,
the $[n_{0},\dots,n_{i},\dots,n_{M}]_{M}$-molaron, where $M$ is the total
number of two-body bound states supported by the potential and $n_{i}$ is the
bosonic occupation number of the $i$th bound state, with $i=0$ representing
the bare atom. The peak position of the $[0_{0}]_{M}$-molaron, i.e. the
polaron, coincides with $E_{zr}$. Exemplary molaron peaks are labeled in
Figure 3(a). At every scattering resonance each quasiparticle peak undergoes
the same fragmentation seen in Figure 2. For example, at the second scattering
resonance, each of the $[n_{i}]_{1}$ states broadens and eventually splits
into a multitude of states $[n_{i},m_{i+1}]_{2}$.

An important consequence of the large extent of the Rydberg potential is that
$a_{\mathrm{Ryd}}$ is large and positive except close to a resonance, where it
dips below zero. As a result, the attractive polaron exists only in a very
limited parameter space. At the transition from a repulsive to an attractive
polaron, $a_{\mathrm{Ryd}}$ vanishes at a Ramsaeur-Townsend zero [64, 65].
Despite its non-zero interaction potential, the Rydberg atom effectively does
not interact with the bath – the scattering phase shift vanishes. Here, the
molaron features sharpen, losing quasiparticle weight as they more closely
resemble bare molecules.

At higher density (Figure 3b) the spectral weight shifts entirely into molaron
states. Their absorption peaks blur together and $A(\omega)$ takes on a
Gaussian profile with mean values $E_{\mathrm{Ryd}}$ whose emergence is
explained by the central limit theorem [42, 44, 66]. However, the regression
to a Gaussian distribution does not occur at the same density for each
$a_{s}$: the distinct molaron peaks, still resolvable in the vicinity of the
Ramsauer-Townsend zeros in Figure 3(b), merge to form a Gaussian spectral
profile only at higher density (panel (c)).

Figure 4: Effective quasiparticle weight of the Rydberg impurity, calculated
by averaging $S(t)$ over late times. The solid white curve shows
$a_{\mathrm{Ryd}}\rho^{1/3}=1$, which demarcates the two phases: quasiparticle
(molaron or polaron limit with $Z=1$) and semiclassical / statistical ($Z=0$).
The dashed white line shows $R_{0}\rho^{1/3}=1$, the semiclassical transition.

This progression from an individual Lorentzian at zero density through
asymmetric lineshapes and polymer formation to a symmetric Gaussian
distribution at high density is largely consistent with the semi-classical
theory of pressure broadening [67, 68, 69, 50, 70, 71, 72, 73]. This theory
predicts the occurrence of "satellite" peaks at integer multiples of the
extrema of the interaction potential, an obvious parallel to the molaron
structure, as well as their blending together into a Gaussian response as the
number of particles within the range of the potential, $R_{0}^{3}\rho$,
increases [66, 69]. Our calculation shows that a more accurate condition takes
into account the zero-energy scattering length, and thus generalizes the above
condition to $\rho^{-1/3}\ll a_{\mathrm{Ryd}}$.

This more accurate condition is depicted in Figure 4, which shows a
qualitative estimate of the quasiparticle weight given by a temporal average
of $S(t)$ in the long time limit. The curve $a_{\mathrm{Ryd}}\rho^{1/3}=1$
indicates the transition between a spectrum dominated by quasiparticle
features (where $S(t)\gg 0$) to a purely statistical state characterized by
the Gaussian lineshape with no quasiparticle weight. In the extreme limit
$a_{\mathrm{Ryd}}\to 0^{+/-}$, the above condition is never satisfied and
$A(\omega)$ will show distinct peaks no matter the density. This explains the
regions with large quasiparticle weight even at high densities seen in Figure
4 where the impurity has a small but non-zero quasiparticle character due to
its vastly reduced coupling to the dense environment.

In contrast, the region of approximately zero quasiparticle weight extends to
very low densities when $a_{\mathrm{Ryd}}$ diverges. This corresponds to the
parameter regime where the loss of quasiparticle characteristics and a
broadening of the well-defined polaron peak into a diffuse continuum is known
from zero-range impurities as well [6, 52, 74]. Consequently, the spectral
broadening of the attractive polaron peak near resonance in the zero-range
polaron and the emergence of the Gaussian feature in Rydberg polaron studies
share the same underlying physical origin.

This collection of phenomena is not limited to a Rydberg atom, but is shared
by all impurity systems: the quantitative differences between the absorption
spectra of various impurities are a matter of degree rather than of kind. This
can be seen by comparing the spectrum of a neutral impurity [52, 49] or an
ionic impurity [38] with the present results.

Figure 5: $A(\omega)$ of an ionic impurity at $\rho=2\,R_{\mathrm{Ion}}^{-3}$.
The green line shows the mean-field energy shift $E_{\mathrm{Ion}}$. The inset
highlights the fragmentation of molaron states.

In the latter case, the interaction is often taken to be a regularized
polarization potential [75, 38]

$\displaystyle V_{\mathrm{Ion}}(r)$
$\displaystyle=-\frac{\alpha}{(r^{2}+b^{2})^{2}}\frac{r^{2}-c^{2}}{r^{2}+c^{2}},$
(4)

with a characteristic length scale $R_{\mathrm{Ion}}=\sqrt{2\mu\alpha}$. We
calculated $A(\omega)$ for $c=0.0023\,R_{\mathrm{Ion}}$, varying $b$ to adjust
the potential depth, as shown in Figure 5. We observe polaron features and
fragmentation of the molaron states as they cross a scattering resonance,
familiar from Figure 3(a). At the density of $\rho=2\,R_{\mathrm{Ion}}^{-3}$,
the molaron states possess significant spectral weight and hint at the
emerging Gaussian lineshape centered around the mean-field energy shift
$E_{\mathrm{Ion}}=\rho\int V_{\mathrm{Ion}}(r)\mathrm{d}^{3}r$, which becomes
completely dominant at higher densities as in Figure 3(c) [76]. As with the
Rydberg impurity, by writing $E_{\mathrm{Ion}}=2\pi
a_{\mathrm{Ion}}^{B}\rho/\mu$ we see that this feature is simply characterized
by the Born approximation for the scattering length, the reduced mass and the
bath density.

With the insights provided by these two impurity systems, we have shown that
the universal parameter $a\rho^{1/3}$ determines the qualitative behavior of
their absorption spectra. This unites the phenomena of finite-ranged
impurities with those known from the well-studied zero-range impurity (the
"Bose polaron"). Even deeply bound molecular states, whose energies depend on
the details of the two-body interaction, respond identically to the influence
of the bath. At sufficiently high densities, these details become irrelevant
and the system response is universal, depending only on its scattering length
in the Born approximation within a mean-field description.

The huge length scales of the Rydberg atom are especially appropriate for the
approximations made in $\hat{H}$: the neglect of boson-boson interactions and
kinetic energy correlation. Rydberg impurities therefore may serve as a more
suitable platform for studying molarons and refining their theoretical
description than ground-state atoms, even though the bath-induced density
shift of the molecule peaks, typically on the scale $\leavevmode\nobreak\ 4\pi
n^{2}\rho/\mu$, is small relative to the binding energies. These subtle many-
body energy shifts, including the shift of the bare atomic line (the repulsive
Rydberg polaron), could be isolated by a careful study of the density-
dependence and asymmetry of the absorption peaks, particularly for light
atoms. Additionally, fermionic environments, which have been partially
explored for Rydberg and ionic impurities, provide another avenue for future
investigation [45, 77].

###### Acknowledgements.

We are grateful for many enlightening discussions with P. Giannakeas, A.
Eisfeld, and S. Wüster.

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##  Appendix A Supplementary Information

In the following, we provide additional details and parameters for the
calculations presented in the main text.

###  A.1 Loschmidt Echo

The Loschmidt-Echo $S(t)$, also known as the autocorrelation function, is the
time-dependent overlap of the bare BEC state and the interacting state
following the sudden introduction of the impurity at $t=0$. Beginning with the
BEC wave function
$\ket{\psi_{\mathrm{BEC}}}=1/\sqrt{N!}b_{0}^{\dagger^{N}}\ket{\mathrm{vac}}$,
where

$\displaystyle N=\frac{\rho_{0}\cdot 2L^{3}}{\pi}$ (5)

is the number of $s$-wave BEC particles, $S(t)$ is given by

$\displaystyle S(t)$
$\displaystyle=\bra{\psi_{\mathrm{BEC}}}e^{i\hat{H}_{0}t}e^{-i\hat{H}t}\ket{\psi_{\mathrm{BEC}}}$
(6)
$\displaystyle=\left(\sum_{\alpha}e^{i(\epsilon_{0}-\omega_{\alpha})t}\absolutevalue{\bra{0}\ket{\alpha}}^{2}\right)^{N}.$
(7)

Here the state $\ket{0}$ is the ground state of the non-interacting two-body
Hamiltonian

$h_{0}(r)=-\frac{\nabla_{r}^{2}}{2\mu}$ (8)

with eigenenergy $\epsilon_{0}$, while $\ket{\alpha}$ are the eigenstates of
the interacting two-body Hamiltonian

$h(r)=-\frac{\nabla^{2}_{r}}{2\mu}+V_{\mathrm{Ryd}}(r)$ (9)

with eigenenergies $\omega_{\alpha}$. Calculating $S(t$) ultimately involves
determining the interacting eigenstates $\ket{\alpha}$ and their corresponding
eigenenergies $\omega_{\alpha}$. A more complete derivation can be found in
[45]. We perform the time evolution of $S(t)$ up to
$t_{\mathrm{max}}=1000\,\mu$s in order to compute $A(\omega)$ from directly
integrating the Fourier transform of $S$. To have a numerically well-defined
Fourier transformation we multiply $S(t)$ with a exponential decay
$\exp[-t/(0.4\cdot t_{\mathrm{max}})]$. This decay time is chosen to be large
in the present study to avoid obscuring any interesting results by numerical
broadening of the spectral features. Tests for shorter decay times, which
model the finite Rydberg lifetime, did not show significant deviation from the
calculations performed here.

###  A.2 Rydberg potential

For an $s$-state Rydberg electron the interaction potential
$V_{\mathrm{Ryd}}(r)$ is isotropic and spherically symmetric. To simplify the
theoretical analysis, we neglect the effect of a finite quantum defect and use
the electronic wave function $\psi_{nlm}(r)$ of the hydrogen atom. Further, we
assume that the electron-atom scattering length $a_{s}$, which sets the
strength of the overall interaction, is energy-independent. These assumptions
yield the two-body interaction potential,

$\displaystyle V_{\mathrm{Ryd}}(r)=2\pi
a_{s}\absolutevalue{\psi_{n00}(r)}^{2},$ (10)

which captures the features of more sophisticated calculations to a semi-
quantitative degree. In the main text our results are computed for an
electronic $n=50$ state of a mass-balanced system with the mass of 84Sr,
$\mu=83.91342/2\,[\mathrm{au}]$, in a box with radial extent of
$L=550n^{2}a_{0}$.

Within the Born approximation, the zero-energy scattering length is given by
$a_{\mathrm{Ryd}}^{\mathrm{B}}=\frac{m_{e}}{\mu}a_{s}$, from which we obtain
the density shift $E_{\mathrm{Ryd}}=\frac{2\pi a_{s}}{m_{e}}$. This gives the
position of the Gaussian feature in the high density regime.

###  A.3 Ion potential

The interaction between an ion and a neutral atom has a long-range tail
$\propto-\alpha/(2r^{4})$, with $\alpha$ the polarizability of the neutral
atom. To avoid problems at short inter-particle distances we include a short-
range regularization, which gives us the ionic interaction potential

$\displaystyle V(r)$
$\displaystyle=-\frac{\alpha/2}{(r^{2}+b^{2})^{2}}\frac{r^{2}-c^{2}}{r^{2}+c^{2}}$
(11)

with characteristic range $R_{\mathrm{Ion}}=\sqrt{2\mu\alpha}$. We fix the
free parameters to be $c=0.0023\,R_{\mathrm{Ion}}$, $\alpha=320$, and
$\mu=86.9092/2\,[\mathrm{au}]$, corresponding to the polarizability and mass
of 87Rb atoms.

Within the Born approximation, the zero-energy scattering length is

$\displaystyle a_{\mathrm{Ion}}^{\mathrm{B}}$
$\displaystyle=-\frac{R_{\mathrm{ion}}^{2}\pi}{4b}\frac{(b^{2}+2bc-c^{2})}{(b+c)^{2}}.$
(12)

From this, or alternatively from the integral $\rho\int
V_{\mathrm{Ion}}(r)\mathrm{d}^{3}r$, we obtain the density shift
$E_{\mathrm{Ion}}=\frac{2\pi}{\mu}\rho a_{\mathrm{Ion}}^{\mathrm{B}}$ giving
the position of the Gaussian feature in the high density regime.

###  A.4 Spectrum of $h$

In the following, we describe our general approach to calculating the spectrum
of a given two-body Hamiltonian $h$ with a generic interaction potential which
can be truncated at some finite distance. When we give details, such as the
number of basis states and dimensions of the quantization volume, they are
specific to the calculation of the Rydberg impurity. The ionic impurity can
require, due to the slower decay of its interaction, a larger basis size and
matching of interacting and free wave fucntions at a larger radius to achieve
convergence.

We separate the bound state calculation from the continuum state calculation
in order to save computational effort, since we want to avoid diagonalizing a
huge matrix to obtain many hundreds of continuum states.

In our calculations for the continuum of single-particle states
$\ket{\alpha}$, we employ the eigenchannel R-matrix approach. This method
obviates the need to solve the Schrödinger equation numerically over the
entire quantization volume. We partition the space around the impurity into
two distinct regions: one (roughly for $0<r<3n^{2}$) where the Rydberg
potential differs significantly from zero, and another $3n^{2}<r<L$ where the
interaction potential is negligible and the wave function is known
analytically. At the boundary of the interaction volume, we compute the log-
derivative of the wave function at a specific energy $E$,

$-b_{\beta}(E)=\frac{\partial\ln r\psi_{\beta}}{\partial r}.$ (13)

We compute $b_{\beta}$ using the streamlined eigenchannel approach detailed in
Ref. [46], using 500 B-spline functions of order 12 to span the range from the
inner boundary $r_{0}=200a_{0}$ to $r_{1}=3n^{2}$. By now matching the
analytical log-derivative of the free particle solutions outside of the range
of the interaction potential with those inside, we compute the energy-
dependent phase shift $\delta(E)$ and the scattering length $a_{\mathrm{imp}}$
of the potential as follows:

$\displaystyle\tan(\delta(E))$
$\displaystyle=\frac{b(E)j_{0}(kr_{1})r_{1}+\partial_{r}j_{0}(kr_{1})}{b(E)y_{0}(kr_{1})r_{1}+\partial_{r}y_{0}(kr_{1})}$
(14) $\displaystyle a_{\mathrm{imp}}$
$\displaystyle=-\mathrm{lim}_{k\rightarrow 0}\tan(\delta(k))/k.$ (15)

In a second step we discretize the continuum by imposing the hard wall
boundary condition at $r=L$. To achieve this, we do an energy search for all
wave functions that have zero amplitude at the box boundary. In total we use
about 10000 interacting states $\ket{\alpha}$ up to a energy cut-off
($E_{\mathrm{max}}=300\,$[MHz]) to represent the continuum. Especially close
to a resonance the continuum couples strongly and a good numerical
representation of the continuum becomes particularly important. The calculated
overlaps of the low-lying box-continuum states with the free BEC wave function
is shown in Figure 6, for two different interaction strengths. In both cases
we see an exponential decay with energy, however for a value close to
resonance $a_{s}=-0.315a_{0}$ (blue line) the overlaps tend to be one or two
orders of magnitude larger than they are for an interaction strength far from
resonance $a_{s}=-0.2a_{0}$ (red line). This underscores the importance of
considering a comprehensive continuum description, especially in proximity to
a resonance.

Figure 6: The energy-dependent Frank-Condon overlaps of the continuum states
of $h$ with the ground state of the non-interaction system $h_{0}$.

To calculate the bound states, we use a basis of around 20000 B-splines
spanning the entire box and standard diagonalization routines designed for
sparse matrices. This step is especially important in order to accurately
obtain bound states close to threshold which decay very slowly at large $r$.

###  A.5 Ion in the high density limit

Here we show the absorption spectrum of an ionic impurity for the same
parameters in the text, except at a density ten times greater. Here, the
Gaussian lineshape is again clear, and the peak position follows
$E_{\mathrm{Ion}}$. Some deviation from the smooth Gaussian can be seen near
to one of the Ramsauer Townsend zeros.

Figure 7: $A(\omega)$ of an ionic impurity at
$\rho=30\,R_{\mathrm{Ion}}^{-3}$. The green line shows the mean-field energy
shift $E_{\mathrm{Ion}}$.


# Generation of arbitrarily polarized GeV lepton beams via nonlinear Breit-
Wheeler process

Kun Xue  MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of
Condensed Matter, School of Physics, Xi’an Jiaotong University, Xi’an 710049,
China  Ren-Tong Guo  MOE Key Laboratory for Nonequilibrium Synthesis and
Modulation of Condensed Matter, School of Physics, Xi’an Jiaotong University,
Xi’an 710049, China  Feng Wan  MOE Key Laboratory for Nonequilibrium Synthesis
and Modulation of Condensed Matter, School of Physics, Xi’an Jiaotong
University, Xi’an 710049, China  Rashid Shaisultanov  Max-Planck-Institut für
Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  Helmholtz-Zentrum
Dresden-Rossendorf, Bautzner Landstraße 400, 01328 Dresden, Germany  Yue-Yue
Chen  Department of Physics, Shanghai Normal University, Shanghai 200234,
China  Zhong-Feng Xu  MOE Key Laboratory for Nonequilibrium Synthesis and
Modulation of Condensed Matter, School of Physics, Xi’an Jiaotong University,
Xi’an 710049, China  Xue-Guang Ren  MOE Key Laboratory for Nonequilibrium
Synthesis and Modulation of Condensed Matter, School of Physics, Xi’an
Jiaotong University, Xi’an 710049, China  Karen Z. Hatsagortsyan
<EMAIL_ADDRESS>Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany  Christoph H. Keitel  Max-Planck-
Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  Jian-
Xing Li<EMAIL_ADDRESS>MOE Key Laboratory for Nonequilibrium Synthesis
and Modulation of Condensed Matter, School of Physics, Xi’an Jiaotong
University, Xi’an 710049, China

###### Abstract

Generation of arbitrarily spin-polarized lepton (here refer in particular to
electron and positron) beams has been investigated in the single-shot
interaction of high-energy polarized $\gamma$ photons with an ultraintense
asymmetric laser pulse via nonlinear Breit-Wheeler (BW) pair production. We
develop a fully spin-resolved semi-classical Monte Carlo method to describe
the pair creation and polarization in the local constant field approximation.
In nonlinear BW process the polarization of created pairs is simultaneously
determined by the polarization of parent $\gamma$ photons, the polarization
and asymmetry of scattering laser field, due to the spin angular momentum
transfer and the asymmetric spin-dependent pair production probabilities,
respectively. In considered all-optical method, dense GeV lepton beams with
average polarization degree up to about $80\%$ (adjustable between the
transverse and longitudinal components) can be obtained with currently
achievable laser facilities, which could be used as injectors of the polarized
$e^{+}e^{-}$ collider to search for new physics beyond the Standard Model.

Ultrarelativistic spin-polarized lepton (here refer in particular to electron
and positron) beams have many important applications in particle and high-
energy physics Wardle  _et al._ (1998); Žutić  _et al._ (2004); The BESIII
Collaboration. (2019), especially in $e^{+}e^{-}$ collider, such as
International Linear Collider (ILC) Moortgat-Pick  _et al._ (2008); Bear  _et
al._ (2013), Compact Linear Collider (CLIC) Usun Simitcioglu  _et al._ (2018);
Ari  _et al._ (2016) and Circular Electron Positron Collider (CEPC) Duan  _et
al._ (2019); Nikitin (2020). In those experiments, the longitudinal
polarization of leptons can change interaction cross section and consequently
provides high sensitivities Moortgat-Pick  _et al._ (2008) through, e.g.,
suppressing background from $WW$ boson and single $Z$ boson production via
$WW$ fusion Moortgat-Pick  _et al._ (2008), enhancing different triple gauge
couplings in $WW$ pair production Moortgat-Pick  _et al._ (2008); Diehl  _et
al._ (2003) and improving top vector coupling in top quark production
Chakraborty  _et al._ (2003); while, the transverse polarization can cause
asymmetric azimuthal distribution of final-state particles Bartl  _et al._
(2007) and then brings a way to study new physics beyond the Standard Model
(BSM) Herczeg (2003); Ananthanarayan and Rindani (2004a, 2005, 2018) through,
e.g., measuring relative phases among helicity amplitudes in $WW$ pair
production Fleischer  _et al._ (1994), probing mixture of scalar-electron
states Hikasa (1986) and searching for graviton in extra dimensions Rizzo
(2003). Commonly, longitudinal and transverse polarizations are studied
separately since those corresponding effects are independent to each other
Moortgat-Pick  _et al._ (2008); Bartl  _et al._ (2007). However, it deserves
to point out that the arbitrarily spin-polarized (ASP) lepton beams (having
both longitudinal and transverse polarization components) also attract broad
interests, because they can introduce three mutually orthogonal axes (required
by fully reconstructing the density matrix of a spin-1/2 particle), modify
effective BSM vertices Dass and Ross (1977); Burgess and Robinson (1991);
Ananthanarayan and Rindani (2004a), and thus play an unique role in future BSM
experiments in $e^{+}e^{-}$ colliders, e.g., rendering special spin structure
functions as being observable in vector and axial-vector type BSM interactions
Ananthanarayan and Rindani (2018), producing polarized top quark pairs as a
probe of new physics Harlander  _et al._ (1997); Godbole  _et al._ (2006) and
diagnosing spin and chirality structures of new particles in antler-topology
processes Choi  _et al._ (2015).

In conventional methods, the transversely polarized lepton beams can be
directly obtained in a storage ring via Sokolov-Ternov effect Sokolov and
Ternov (1964, 1968); Baier and Katkov (1967); Baier (1972); Derbenev and
Kondratenko (1973), which demands a long polarization time since large-size
static magnetic fields are relatively weak ($\sim$Tesla), and the
longitudinally polarized ones can be created via high-energy circularly
polarized (CP) $\gamma$ photons interacting with high-$Z$ target Omori  _et
al._ (2006); Alexander  _et al._ (2008); Abbott  _et al._ (2016) (Bethe-
Heitler $e^{+}e^{-}$ pair production process Heitler (1954)), in which the low
luminosity of $\gamma$ photons requires a large amount of repetitions to yield
a dense positron beam for further applications Variola (2014). The transverse
and longitudinal polarizations can be transformed to each other by a spin
rotator, which demands quasi-monoenergetic beams with a risk of beam intensity
reduction Buon and Steffen (1986); Moffeit  _et al._ (2005).

Modern ultrashort ultraintense laser pulses Yoon  _et al._ (2019); Danson  _et
al._ (2019); Gales  _et al._ (2018) enable alternative efficient methods to
generate dense polarized lepton beams in femtoseconds via nonlinear quantum
electrodynamics (QED) processes Ritus (1985), e.g. nonlinear Compton
scattering Goldman (1964); Nikishov and Ritus (1964); Brown and Kibble (1964);
CAI  and Breit-Wheeler (BW) $e^{+}e^{-}$ pair production Reiss (1962); Ivanov
_et al._ (2005); Seipt and King (2020); Wistisen (2020); Wan  _et al._ (2020).
As reported, the leptons can be greatly transversely polarized in a standing-
wave Del Sorbo  _et al._ (2017, 2018); Seipt  _et al._ (2018), elliptically
polarized (EP) Li  _et al._ (2019), or bichromatic laser Seipt  _et al._
(2019); Song  _et al._ (2019); Chen  _et al._ (2019) but not in a
monochromatic symmetric laser Kotkin  _et al._ (2003); Ivanov  _et al._
(2004); Karlovets (2011). And, longitudinally polarized positrons can be
produced by CP $\gamma$ photons through the helicity transfer Li  _et al._
(2020a) (similar to the Bethe-Heitler process). Moreover, two-step schemes
have also been proposed: low-energy polarized leptons are first generated by
polarized photocathodes Pierce  _et al._ (1980); Kuwahara  _et al._ (2012);
Zitzewitz  _et al._ (1979), polarized atoms Barth and Smirnova (2013) or
molecular photodissociation Rakitzis  _et al._ (2003); Sofikitis  _et al._
(2017, 2018), and then accelerated to ultrarelativistic energies via laser-
Wen  _et al._ (2019); Wu  _et al._ (2019a) or beam-driven Wu  _et al._
(2019b); Nie  _et al._ (2021) plasma wakefield (conventional accelerators work
as well). All those proposals provide leptons either greatly transversely or
longitudinally polarized, however, how to generate above mentioned unique ASP
lepton beams is still a great challenge.

$\hat{y}$$\hat{x}$$\hat{z}$$\gamma_{p}$$e^{-}$$e^{+}$

polarized $\gamma_{p}$

asymmetric laser

(a1)(a2)(a3)$\xi_{3}$$\xi_{1}$$\xi_{2}$Poincaré
sphere(b1)(b2)(b3)$\hat{y}$$\hat{x}$$\hat{z}$$\overline{{\bm{S}}}_{+}$Polarization
states of $e^{+}$

Magnet

Figure 1:  Scenario of generation of ASP lepton beams via nonlinear BW
process. A LP asymmetric laser pulse (propagating along $-\hat{z}$ direction
and polarizing along $\hat{x}$ axis) head-on collides with polarized $\gamma$
photons ($\gamma_{p}$) to create ASP electron and positron beams. (a1)-(a3)
indicate LP, CP and EP $\gamma$ photons, respectively, and (b1)-(b3) show
average polarizations of created positrons $\overline{\bm{S}}_{+}$
corresponding to (a1)-(a3), respectively. The red-solid arrow and ellipsoid
indicate the direction and amplitude of $\overline{\bm{S}}_{+}$, respectively.
The red-dashed arrows in (b1) ($\parallel\hat{{y}}$) and (b2)
($\parallel\hat{{z}}$) indicate particular cases of neglecting the
polarization of $\gamma$ photons and employing symmetric laser field,
respectively.

In this Letter, the generation of ASP GeV lepton beams has been investigated
in the interaction of polarized $\gamma$ photons with a counter-propogating
ultraintense linearly polarized (LP) asymmetric laser pulse [see the
interaction scenario in Fig. 1]. We develop a fully spin-resolved semi-
classical Monte Carlo algorithm to describe photon-polarization-dependent pair
production and polarization in nonlinear BW process. We find that the
polarization of created pairs is simultaneously determined by the polarization
of parent $\gamma$ photons, the polarization and asymmetry of scattering laser
field, due to the spin angular momentum transfer and the asymmetric spin-
dependent pair production probabilities, respectively [see details in Fig. 2
and Eq. (4)]. As employing unpolarized $\gamma$ photons or ignoring the
polarization of parent $\gamma$ photons the pair polarization will rely on the
laser polarization and asymmetry [see Fig. 1(b1)], and as employing a
symmetric laser field the transverse polarization of the total beam will be
suppressed (since the polarization directions in adjacent half laser cycles
are opposite) and only the longitudinal polarization can be retained [see Fig.
1(b2)]. Our simulations show that dense GeV lepton beams with adjustable
polarization degree up to about 80% can be obtained with currently achievable
laser facilities to the benefit of many unique applications [see details in
Fig. 3].

(a)$\hat{\bf P}_{1}$(${\bm{E}}$)$\hat{\bf P}_{2}$(-${\bm{B}}$)$\hat{\bf
P}$$\theta_{\alpha}$$\theta_{1}$$\theta_{2}$${\bm{S}}_{+}$$2\theta_{\alpha}$${\bm{D}}_{mag.}$${\overline{\bm{P}}}_{+}$${\bm{D}}_{pol.}^{(LP)}$(c)$\hat{y}$$\hat{x}$$\theta_{\alpha}$$\overline{\bm{S}}_{+}$$\hat{\bf
P}$(b)${\bm{D}}_{mag.}$${\overline{\bm{P}}}_{+}$${\bm{D}}_{pol.}^{(LP)}$${\bm{D}}_{pol.}^{(CP)}$${\bm{S}}_{+}$$\hat{\bf
P}_{1}$(${\bm{E}}$)$\hat{\bf P}_{2}$(-${\bm{B}}$)$\hat{{\bf
v}}_{+}$(d)$\hat{y}$$\hat{x}$$\hat{z}$$\overline{\bm{S}}_{+}$$\varphi$$\theta_{\alpha}$

Figure 2:  (a) and (b): Polarization of sample positron ${\bm{S}}_{+}$ created
by LP and EP $\gamma$ photons, respectively. In our interaction scheme [see
Fig. 1] we employ $\hat{\bf P}_{1}=\hat{\bm{E}}\approx\hat{{\bf a}}_{+}$ and
$\hat{\bf P}_{2}=-\hat{\bm{B}}\approx\hat{{\bm{b}}}_{+}$.
${\overline{\bm{P}}}_{+}$ indicates the direction of the instantaneous SQA
with two transverse components ${\bm{D}}_{mag}$, ${\bm{D}}_{pol}^{(LP)}$ and
one longitudinal component $D_{pol}^{(CP)}$ [see Eq.(3)]. For LP $\gamma$
photon in (a), $D_{pol}^{(CP)}=0$ and $\theta_{\alpha}$ indicates the
polarization angle to $\hat{\bf P}_{1}$. $\theta_{1}$ and $\theta_{2}$ are the
angles of $\hat{\bf P}$ to ${\bm{D}}_{pol.}^{(LP)}$ and ${\bm{D}}_{mag.}$,
respectively. (c) and (d): Average polarization of positrons
$\overline{\bm{S}}_{+}$ created by LP and EP $\gamma$ photons, respectively.
The red arrow and circle (ellipsoid) indicate the direction and amplitude of
$\overline{\bm{S}}_{+}$ [see Eq. (4)], respectively.

In strong laser field, a rich pair yield via nonlinear BW process requires the
nonlinear QED parameter
$\chi_{\gamma}\equiv|e|\sqrt{-(F_{\mu\nu}k_{\gamma}^{\nu})^{2}}/m^{3}\gtrsim
1$ Ritus (1985); Baier  _et al._ (1998), and the created pairs may further
emit photons via nonlinear Compton scattering, which is unnegligible as
another nonlinear QED parameter
$\chi_{e}\equiv|e|\sqrt{-(F_{\mu\nu}p^{\nu})^{2}}/m^{3}\gtrsim 1$ Ritus
(1985). Here, $e$ and $m$ are the electron charge and mass, respectively,
$k_{\gamma}$ and $p$ the 4-momenta of $\gamma$ photon and positron (electron),
respectively, and $F_{\mu\nu}$ the field tensor. Relativistic units with
$c=\hbar=1$ are used throughout. The photon polarization can be characterized
by the unit vector $\hat{\bf P}={\rm cos}(\theta_{\alpha})\hat{\bf P}_{1}+{\rm
sin}(\theta_{\alpha})\hat{\bf P}_{2}\cdot e^{i\theta_{\beta}}$, and
corresponding Stokes parameters are $(\xi_{1},\xi_{2},\xi_{3})=[{\rm
sin}(2\theta_{\alpha}){\rm cos}(\theta_{\beta})$, ${\rm
sin}(2\theta_{\alpha}){\rm sin}(\theta_{\beta})$, ${\rm
cos}(2\theta_{\alpha})]$ McMaster (1961); Boyarkin (2011). Here $\hat{{\bf
P}}_{1}$ and $\hat{{\bf P}}_{2}$ are two orthogonal basis vectors,
$\theta_{\alpha}$ the polarization angle, $\theta_{\beta}$ the absolute phase,
$\xi_{1}$ and $\xi_{3}$ describe the linear polarizations, and $\xi_{2}$
circular polarization. The fully spin-resolved pair production probability
$W_{pair}$ has been calculated via the QED operator method of Baier-Katkov-
Fadin Baier  _et al._ (1973) in the local constant field approximation Ritus
(1985); Baier  _et al._ (1998); Di Piazza  _et al._ (2018); Ilderton (2019);
Di Piazza  _et al._ (2019); Seipt and King (2020) (valid at the invariant
field parameter $a_{0}=|e|E_{0}/m\omega\gg 1$, where $E_{0}$ and $\omega_{0}$
are the laser field amplitude and frequency, respectively); see the complex
expression of $W_{pair}$ in sup .

Let’s first summarize our methods of numerical simulation and analytical
estimation. Note that in nonlinear BW process the polarization of electrons is
similar with that of positrons, thus for simplicity we only discuss the case
of positrons below. To study the positron polarization ${\bm{S}}_{+}$, we
first sum over the electron polarization ${\bm{S}}_{-}$ in $W_{pair}$, and the
probability relying on ${\bm{S}}_{+}$ is simplified as:

$\displaystyle\frac{{\rm d}^{2}W_{pair}^{+}}{{\rm d}\varepsilon_{+}{\rm
d}t}=W_{0}(C+{\bm{S}}_{+}\cdot{\bm{D}}),$ (1)

where $W_{0}=\alpha m^{2}/\left(\sqrt{3}\pi\varepsilon_{\gamma}^{2}\right)$,
$C={\rm
IntK}_{\frac{1}{3}}(\rho)+\frac{\varepsilon_{-}^{2}+\varepsilon_{+}^{2}}{\varepsilon_{-}\varepsilon_{+}}{\rm
K}_{\frac{2}{3}}(\rho)-\xi_{3}{\rm K}_{\frac{2}{3}}(\rho)$,
${\bm{D}}=\left(\xi_{3}\frac{\varepsilon_{\gamma}}{\varepsilon_{-}}-\frac{\varepsilon_{\gamma}}{\varepsilon_{+}}\right){\rm
K}_{\frac{1}{3}}(\rho)\hat{{\bm{b}}}_{+}-\xi_{1}\frac{\varepsilon_{\gamma}}{\varepsilon_{-}}{\rm
K}_{\frac{1}{3}}(\rho)\hat{{\bf
a}}_{+}+\xi_{2}\bigg{[}\frac{\varepsilon_{+}^{2}-\varepsilon_{-}^{2}}{\varepsilon_{-}\varepsilon_{+}}{\rm
K}_{\frac{2}{3}}(\rho)+\frac{\varepsilon_{\gamma}}{\varepsilon_{+}}{\rm
IntK}_{\frac{1}{3}}(\rho)\bigg{]}\hat{\bf v}_{+}$, $\hat{\bm{b}}_{+}=\hat{\bf
v}_{+}\times\hat{{\bf a}}_{+}$
$\approx-\hat{\bm{k}}_{\gamma}\times\hat{\bm{E}}=-\hat{\bm{B}}$ is an unit
vector anti-parallel to the magnetic field ${\bm{B}}$ (with direction vector
$\hat{\bm{B}}$) in the rest frame of positron, ${\bm{E}}$ the electric field
with direction vector $\hat{\bm{E}}$, $\hat{\bf v}_{+}$ and $\hat{\bf a}_{+}$
the unit vectors of the positron velocity and acceleration, respectively,
$\rho=2\varepsilon_{\gamma}^{2}/\left(3\chi_{\gamma}\varepsilon_{-}\varepsilon_{+}\right)$,
${\rm IntK}_{\frac{1}{3}}(\rho)\equiv\int_{\rho}^{\infty}{\rm d}z{\rm
K}_{\frac{1}{3}}(z)$, ${\rm K}_{n}$ the $n$-order modified Bessel function of
the second kind, $\alpha$ the fine structure constant, $\varepsilon_{\gamma}$,
$\varepsilon_{-}$ and $\varepsilon_{+}$ the energies of parent $\gamma$
photon, created electron and positron, respectively, with
$\varepsilon_{\gamma}=\varepsilon_{+}+\varepsilon_{-}$. When a $\gamma$ photon
decays into a pair, the positron spin state is collapsed into one of its basis
states defined by the instantaneous spin quantization axis (SQA) along the
energy-resolved average polarization vector Baier  _et al._ (1973):

$\displaystyle{\rm SQA}\parallel{\overline{\bm{P}}_{+}}={\bm{D}}/C,$ (2)

which can be rewritten as

$\displaystyle{\overline{\bm{P}}}_{+}$ $\displaystyle=$
$\displaystyle\left[{\bm{D}}_{mag.}+{\bm{D}}_{pol.}^{(LP)}+{\bm{D}}_{pol.}^{(CP)}\right]/C$
(3) $\displaystyle=$
$\displaystyle\left[|{\bm{D}}_{mag.}|\hat{\bm{D}}_{mag.}+|{\bm{D}}_{pol.}^{(LP)}|\hat{\bm{D}}_{pol.}^{(LP)}+|{\bm{D}}_{pol.}^{(CP)}|\hat{\bm{D}}_{pol.}^{(CP)}\right]/C,$

where $\hat{\bm{D}}_{mag.}=-\hat{{\bf b}}_{+}\approx\hat{\bm{B}}$,
$\hat{\bm{D}}_{pol.}^{(LP)}=\xi_{3}\hat{{\bm{b}}}_{+}-\xi_{1}\hat{{\bf
a}}_{+}$ and $\hat{\bm{D}}_{pol.}^{(CP)}=\xi_{2}\hat{{\bf v}}_{+}$ rely on the
magnetic field $\hat{\bm{B}}$, linear polarizations $\xi_{1}$ and $\xi_{3}$,
and circular polarization $\xi_{2}$, respectively, with corresponding factors
$|{\bm{D}}_{mag.}|=\frac{\varepsilon_{\gamma}}{\varepsilon_{+}}{\rm
K}_{\frac{1}{3}}(\rho)$,
$|{\bm{D}}_{pol.}^{(LP)}|=\frac{\varepsilon_{\gamma}}{\varepsilon_{-}}{\rm
K}_{\frac{1}{3}}(\rho)$ and
$|{\bm{D}}_{pol.}^{(CP)}|=\frac{\varepsilon_{+}^{2}-\varepsilon_{-}^{2}}{\varepsilon_{-}\varepsilon_{+}}{\rm
K}_{\frac{2}{3}}(\rho)+\frac{\varepsilon_{\gamma}}{\varepsilon_{+}}{\rm
IntK}_{\frac{1}{3}}(\rho)$, respectively. As the photon polarization is
ignored, ${\rm SQA}\parallel\hat{\bm{D}}_{mag.}\parallel\hat{\bm{B}}$, i.e.,
the positrons are polarized along the magnetic field ${\bm{B}}$ Chen  _et al._
(2019) [see Fig. 1(b1)]. The polarization geometries of positrons created by
LP and EP $\gamma$ photons are illustrated in Figs. 2(a) and (b),
respectively. For LP $\gamma$ photon, $\theta_{\beta}=0$,
$\hat{\bm{D}}_{pol.}^{(CP)}=0$, $(\xi_{1},\xi_{2},\xi_{3})=[{\rm
sin}(2\theta_{\alpha})$, 0, ${\rm cos}(2\theta_{\alpha})]$, and
$\hat{\bm{D}}_{pol.}^{(LP)}=\xi_{3}\hat{{\bf b}}_{+}-\xi_{1}\hat{{\bf
a}}_{+}={\rm cos}(2\theta_{\alpha})\hat{\bf P}_{2}-{\rm
sin}(2\theta_{\alpha})\hat{\bf P}_{1}$. For general EP $\gamma$ photon with
$\xi_{2}\neq 0$, the longitudinal polarization component must be taken into
account.

For a positron beam, the average transverse polarizations
$\overline{\bm{S}}_{T}$ in adjacent half laser cycles are opposite and cancel
each other out due to the laser field oscillation, and consequently,
$\overline{\bm{S}}_{T}$ is proportional to the asymmetry of the laser field,
which can be characterized by the relative deviation between the pair
production probabilities in positive and negative half cycles ${\cal
A}_{field}=(W_{pair}^{+,pos.}-W_{pair}^{+,neg.})/(W_{pair}^{+,pos.}+W_{pair}^{+,neg.})$.
Thus, one can estimate

$\displaystyle\overline{{\bm{S}}}_{+}=\left\\{{{\cal
A}_{field}\cdot\left[\overline{\bm{D}}_{mag.}+\overline{\bm{D}}_{pol.}^{(LP)}\right]+\overline{\bm{D}}_{pol.}^{(CP)}}\right\\}/{\overline{C}},$
(4)

where
$\overline{\bm{D}}_{mag.}=\int_{0}^{\varepsilon_{\gamma}}{\bm{D}}_{mag.}{\rm
d}\varepsilon_{+}$,
$\overline{\bm{D}}_{pol.}^{(LP)}=\int_{0}^{\varepsilon_{\gamma}}{\bm{D}}_{pol.}^{(LP)}{\rm
d}\varepsilon_{+}$,
$\overline{\bm{D}}_{pol.}^{(CP)}=\int_{0}^{\varepsilon_{\gamma}}{\bm{D}}_{pol.}^{(CP)}{\rm
d}\varepsilon_{+}$ and $\overline{C}=\int_{0}^{\varepsilon_{\gamma}}C{\rm
d}\varepsilon_{+}$. For LP $\gamma$ photons,
$|\overline{\bm{D}}_{mag.}|=|\overline{\bm{D}}_{pol.}^{(LP)}|$ with
${\theta}_{1}={\theta}_{2}$ results in
$\overline{\bm{S}}_{+}\parallel-\hat{\bf P}$ [see Fig. 2(c)]; for more general
EP ones, as $\theta_{\beta}$ increases, $\overline{\bm{S}}_{+}$ will rotate
anti-clockwise by an azimuth angle $\varphi$, which can be calculated by Eq.
(4) [see Fig. 2(d)]. $\overline{\bm{S}}_{T}={\cal A}_{field}{{\cal A}}_{pol.}$
is dominated by ${\cal A}_{field}$ with ${\cal
A}_{pol.}=\left[\overline{\bm{D}}_{mag.}+\overline{\bm{D}}_{pol.}^{(LP)}\right]/\overline{C}$
[see Fig. 4(a)], while the average longitudinal polarization
$\overline{{\bm{S}}}_{L}$ is solely determined by
$\overline{\bm{D}}_{pol.}^{(CP)}/\overline{C}\propto\xi_{2}$ as expected. In
symmetric laser field with ${\cal A}_{field}=0$, $\overline{\bm{S}}_{T}$ is
very little and only $\overline{{\bm{S}}}_{L}$ can be obtained by employing
longitudinally polarized $\gamma$ photons ($\xi_{2}\neq 0$) Li  _et al._
(2020a) [see Fig. 1(b2)]. The momentum and spin dynamics of the pairs
propagating through the laser field are calculated following a Monte Carlo
algorithm Wan  _et al._ (2020), including the radiative depolarization effects
and spin procession Thomas (1926, 1927); Bargmann  _et al._ (1959). See more
details of our simulation method in sup .

(c)$\theta_{y}$(mrad)

$\theta_{x}$ (mrad)

$\overline{\bm{S}}_{T}$(d)$\theta_{y}$(mrad)

$\theta_{x}$ (mrad)

$\overline{\bm{S}}_{L}$(e)$\theta_{y}$(mrad)

$\theta_{x}$ (mrad)

log10(d$N_{+}$/d$\theta_{x}$)(f)$\varepsilon_{+}$(GeV)

$\overline{\bm{S}}_{T}$$|$$\overline{\bm{S}}_{L}$

log10(d$N_{+}$/d$\varepsilon_{+}$)

(a)$\theta_{\alpha}$

$\theta_{\beta}$

$63.3\%$(b)$\theta_{\alpha}$

$\theta_{\beta}$

$53.5\%$

Figure 3:  (a) and (b): $\overline{\bm{S}}_{T}$ and $\overline{\bm{S}}_{L}$ of
positrons with respect to $\theta_{\alpha}$ and $\theta_{\beta}$,
respectively, analytically estimated by Eq. (4) with ${\cal
A}_{field}=0.8378$. The black points correspond to
($\theta_{\alpha}=50^{\circ}$, $\theta_{\beta}=70^{\circ}$). (c)-(f) are the
numerical simulation results with ($\theta_{\alpha}=50^{\circ}$,
$\theta_{\beta}=70^{\circ}$). (c)-(e): $\overline{\bm{S}}_{T}$,
$\overline{\bm{S}}_{L}$ and the angle-resolved positron density ${\rm
log}_{10}({\rm d}^{2}N_{+}/{\rm d}\theta_{x}{\rm d}\theta_{y})$ (rad-2) with
respect to the deflection angles $\theta_{x}$ = arctan($p_{+,x}/p_{+,z}$) and
$\theta_{y}$ = arctan($p_{+,y}/p_{+,z}$), respectively. The black curves
indicate $\overline{\bm{S}}_{T}$, $\overline{\bm{S}}_{L}$ and ${\rm
log}_{10}({\rm d}N_{+}/{\rm d}\theta_{x})$ (mrad-1) summing over $\theta_{y}$
vs $\theta_{x}$, respectively. (f) $\overline{\bm{S}}_{T}$ (red),
$\overline{\bm{S}}_{L}$ (blue) and the energy-resolved positron density ${\rm
log}_{10}({\rm d}N_{+}/{\rm d}\varepsilon_{+})$ (GeV-1) (green) vs
$\varepsilon_{+}$. Other parameters are given in the text.

Then, we illustrate typical results of created ASP positron beams in Fig 3.
The employed laser and $\gamma$ photon parameters are as follows. A tightly
focused LP bichromatic Gaussian laser pulse Salamin and Keitel (2002); sup  (a
frequency-chirped laser pulse Galow  _et al._ (2011) can work similarly)
propagates along $-\hat{z}$ direction and polarizes along $\hat{x}$ axis [see
Fig. 1], with a phase difference $\Delta\phi=\pi/2$ to obtain the maximal
field asymmetry, peak intensity $I_{0}\approx 1.11\times 10^{22}$ W/cm2
(corresponding invariant field parameters $a_{1}=60$ and $a_{2}=15$),
wavelengths $\lambda_{1}=2\lambda_{2}=1\mu$m, pulse durations
$\tau_{1}=\tau_{2}=15T_{1}$ with periods $T_{1}=2T_{2}$, and focal radii
$w_{1}=w_{2}=5\mu$m. This kind of laser pulse is currently feasible in
petawatt laser facilities Yoon  _et al._ (2019); Danson  _et al._ (2019);
Gales  _et al._ (2018). While, a cylindrical polarized $\gamma$ photon beam
propagates along $\hat{z}$ direction, with an initial energy
$\varepsilon_{\gamma}=1.8$GeV, energy spread
$\Delta\varepsilon_{\gamma}/\varepsilon_{\gamma}=0.06$, angular divergence
$\Delta\theta_{\gamma}=0.3$mrad, beam radius $w_{\gamma}=1\mu$m, beam length
$L_{\gamma}=5\mu$m, photon number $N_{\gamma}=10^{6}$ and density
$n_{\gamma}\approx 6.37\times 10^{16}$cm-3 having a transversely Gaussian and
longitudinally uniform distribution. Such a $\gamma$ photon beam may be
generated via synchrotron radiation Alexander  _et al._ (2008); Baynham  _et
al._ (2011), bremsstrahlung Abbott et al. (2016), linear Omori  _et al._
(2006) or nonlinear Compton scattering King and Tang (2020); Tang  _et al._
(2020); Li  _et al._ (2020b). The pair production is remarkable at these
parameters since $\overline{\chi}_{\gamma}\approx 0.96$.

(b)$\theta_{\alpha}$

$\overline{\bm{S}}_{T}$

$N_{+}/N_{\gamma}$

$\times 10^{-2}$ MCNo Rad.Exc. Pol.(a)$a_{1}$ $\overline{\bm{S}}_{T}$ ${\cal
A}_{field}$ ${\cal A}_{pol.}$$\overline{\bm{S}}_{T}$(c)$\varepsilon_{+}$(GeV)

${\bm{S}}_{T}$

log${}_{10}($d$N_{+}/$d$\varepsilon_{+})$

$\theta_{\alpha}=90^{\circ}$(d)$\varepsilon_{+}$(GeV)

${\bm{S}}_{T}$

log${}_{10}($d$N_{+}/$d$\varepsilon_{+})$

$\theta_{\alpha}=0^{\circ}$

Figure 4:  (a) ${\cal A}_{pol.}$ (black-dash-dotted), ${\cal A}_{field}$
(blue-dashed) and $\overline{\bm{S}}_{T}$ (red-solid) analytically calculated
by Eq. (4) vs $a_{1}$ ($a_{2}=a_{1}/4$), and the green points
$\overline{\bm{S}}_{T}$ are our numerical results.
$\theta_{\alpha}=90^{\circ}$. At $a_{1}$ = 60, 80 and 120, respectively, the
corresponding $\overline{\chi}_{\gamma}\approx$ 0.96, 1.20 and 1.63,
respectively. (b) $\overline{\bm{S}}_{T}$ and the yield ratio of positrons
$N_{+}/N_{\gamma}$ (green-solid) vs $\theta_{\alpha}$. The red-solid, blue-
dash-dotted and black-dotted curves indicate the results of
$\overline{\bm{S}}_{T}$ calculated by our Monte Carlo method, artificially
ignoring the radiative depolarization effects of the positrons propagating
through the laser field, and excluding the polarization effects of parent
$\gamma$ photons in nonlinear BW process, respectively. (c) and (d):
${\bm{S}}_{T}$ (red-solid) and
log${}_{10}($d$N_{+}/$d$\varepsilon_{+})$(GeV-1) (blue) vs $\varepsilon_{+}$
at $\theta_{\alpha}=90^{\circ}$ and $\theta_{\alpha}=0^{\circ}$, respectively.
Here $\theta_{\beta}=0^{\circ}$, and other parameters are the same as those in
Fig. 3.

According to Eq. (4), the polarizations of created positron beam
($\overline{\bm{S}}_{T}$ and $\overline{\bm{S}}_{L}$) can be controlled by
adjusting the polarization of parent $\gamma$ photons ($\theta_{\alpha}$ and
$\theta_{\beta}$) [see Figs. 3(a) and (b)], which indicates the spin angular
momentum transfer from parent $\gamma$ photons to created pairs.
$\overline{\bm{S}}_{T}\propto\sqrt{\xi_{1}^{2}+\xi_{3}^{2}}=\sqrt{{\rm
sin}^{2}(2\theta_{\alpha}){\rm cos}^{2}(\theta_{\beta})+{\rm
cos}^{2}(2\theta_{\alpha})}$ is mainly determined by $\theta_{\alpha}$ and can
reach above 80%, while $\overline{\bm{S}}_{L}\propto\xi_{2}={\rm
sin}(2\theta_{\alpha}){\rm sin}(\theta_{\beta})$ periodically varies with
respect to $\theta_{\alpha}$ and $\theta_{\beta}$ and its amplitude can
achieve about 60%. For a specific case with $\theta_{\alpha}=50^{\circ}$ and
$\theta_{\beta}=70^{\circ}$, the analytical estimations are
$\overline{\bm{S}}_{T}\approx 63.3\%$ and $\overline{\bm{S}}_{L}\approx
53.5\%$ [see the black-star points in Figs. 3(a) and (b)], and corresponding
numerical results are $\overline{\bm{S}}_{T}\approx 62.1\%$ and
$\overline{\bm{S}}_{L}\approx 50.3\%$ sup . The little deviations are derived
from that in analytical estimations we employ a constant ${\cal A}_{field}$,
which actually has spatial and temporal profiles in our numerical simulations.
As the created positrons propagate through the laser field, the average
polarizations slightly decrease to $\overline{\bm{S}}_{T}\approx 59.4\%$ and
$\overline{\bm{S}}_{L}\approx 44.8\%$ [see Figs. 3(c) and (d)] due to the
quantum radiative depolarization effects Li  _et al._ (2019) [see sup  and
Fig. 4(b)]. $\overline{\bm{S}}_{T}\propto{\cal A}_{field}$ is asymmetric in
angular distribution due to asymmetric ${\cal A}_{field}$, which doesn’t
affect the symmetry of $\overline{\bm{S}}_{L}\propto\xi_{2}$. The yield rate
of positrons $N_{+}/N_{\gamma}\approx 0.1$ is much higher than that of the
common method employing Bethe-Heitler pair producation ($\sim
10^{-3}-10^{-4}$) Omori  _et al._ (2006); Alexander  _et al._ (2008); Abbott
_et al._ (2016), since $N_{+}\sim\alpha a_{0}$ is rather large in ultraintense
laser field Ritus (1985); Baier  _et al._ (1973); see Fig. 3(e). The flux of
the positron beam is approximately $3.0\times 10^{19}$/s with duration
$\tau_{+}\approx\tau_{\gamma}$ (due to the relativistic effect). The emittance
$\epsilon\approx w_{+}\theta_{div.}\sim 10^{-2}$ mm$\cdot$mrad fulfills the
experimental requirements of the beam injectors Artru  _et al._ (2008), with
radius $w_{+}\approx w_{\gamma}=1\mu$m and angular divergence
$\theta_{div.}\sim 30$mrad [FWHM in Fig. 3(e)]. Due to the stochastic effects
of the pair production and further radiation, the energy of the positron beam
spreads with a density peak at $\varepsilon_{+}\approx 0.3$GeV [see Fig.
3(f)]. With the increase of $\varepsilon_{+}$, $\overline{\bm{S}}_{T}$
($\overline{\bm{S}}_{L}$) monotonically decreases (increases) from above 90%
(0%) to about $50\%$ (above $85\%$), since the pair polarization is mainly
determined by the polarization of the laser (parent $\gamma$ photons) at low
(high) $\varepsilon_{+}$ (see sup ). For $\varepsilon_{+}=$ 0.4GeV, 0.8GeV and
1.2GeV, respectively, corresponding $\overline{\bm{S}}_{T}$
($\overline{\bm{S}}_{L}$) is about 58.7%, 52.6% and 49.8% (42.6%, 63.4% and
78.8%), respectively, and brilliance about 4.6$\times$1020, 1.4$\times$1020
and 0.5$\times$1020 positrons/(s mm2 mrad2 0.1% BW), respectively, with
angular divergence (FWHM) of about 24.9 mrad2, 15.9 mrad2 and 13.0 mrad2,
respectively.

We underline that as the ultra-relativistic laser intensity $a_{0}(\sim
a_{1})\propto\chi_{\gamma}$ continuously increases, $W_{pair}^{+,pos.}$ and
$W_{pair}^{+,neg.}$ are both gradually approaching the saturation threshold,
and thus $\overline{\bm{S}}_{T}\propto{\cal A}_{field}$ decreases continuously
[see Fig. 4(a)]. As employing unpolarized $\gamma$ photons or artificially
ignoring the polarization effects of $\gamma$ photons in nonlinear BW process,
for given parameters $\overline{\bm{S}}_{T}$ is only about 50.8%, however,
employing polarized $\gamma$ photons $\overline{\bm{S}}_{T}$ can achieve up to
76.2% with a peak yield rate $N_{+}/N_{\gamma}\approx 0.11$ at
$\theta_{\alpha}=90^{\circ}$. And in a broad range of
$45^{\circ}\lesssim\theta_{\alpha}\lesssim 135^{\circ}$ one can obtain
$\overline{\bm{S}}_{T}\gtrsim 60\%$ with $N_{+}/N_{\gamma}\gtrsim 0.08$ [see
Fig. 4(b)]. The energy-resolved $\overline{\bm{S}}_{T}$ and densities at
$\theta_{\alpha}=90^{\circ}$ and $0^{\circ}$ are represented in Figs. 4(c) and
(d), respectively. It’s interesting that at $\theta_{\alpha}=0^{\circ}$ even
though $\overline{\bm{S}}_{T}$ is very little, highly polarized positron beams
can still be obtained by energy choosing by magnets.

For experimental feasibility, the impact of the laser and $\gamma$ photon
parameters (e.g., the laser intensity, pulse duration, colliding angle, and
the angular spreading, energy and energy spreading of the $\gamma$ photon
beam) on the positron polarization is investigated, and the results keep
uniform (see sup ). Moreover, for a more general case: an electron beam head-
on collides with a LP bichromatic laser pulse to generate positrons, the
positrons are only weakly transversely polarized, since the polarization of
intermediate $\gamma$ photons is nearly parallel to that of the laser field
sup .

In conclusion, we reveal the fully spin-resolved pair polarization mechanism
in nonlinear BW process, which can be observed by the average polarization
vector in an asymmetric (e.g. well known bichromatic and frequency-chirped)
laser field. And we put forward an efficient method to generate dense ASP GeV
lepton beams with the polarization degree up to about $80\%$ with currently
achieveable petawatt laser facilities Yoon  _et al._ (2019); Danson  _et al._
(2019); Gales  _et al._ (2018), which have unique applications for high-energy
physics and particle physics, in particular, as injectors of the polarized
$e^{+}e^{-}$ colliders for searching for BSM new physics Moortgat-Pick  _et
al._ (2008); Ananthanarayan and Rindani (2004b, 2005, 2018); Herczeg (2003);
Dass and Ross (1977); Burgess and Robinson (1991); Ananthanarayan and Rindani
(2004a); Harlander  _et al._ (1997); Godbole  _et al._ (2006); Rizzo (2003);
Choi  _et al._ (2015).

Acknowledgement: This work is supported by the National Natural Science
Foundation of China (Grants Nos. 12022506, 11874295, 11875219 and 11905169),
and the National Key R&D Program of China (Grant No. 2018YFA0404801).

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