# Enhancing Spin Polarization Using Ultrafast Angular Streaking

G. S. J. Armstrong  Centre for Theoretical Atomic, Molecular and Optical
Physics, School of Mathematics and Physics, Queen’s University Belfast,
University Road, Belfast, BT7 1NN, Northern Ireland, UK
<EMAIL_ADDRESS>D. D. A. Clarke  School of Physics and CRANN
Institute, Trinity College Dublin, Dublin 2, Ireland  J. Benda  Institute of
Theoretical Physics, Faculty of Mathematics and Physics, Charles University, V
Holešovičkách 2, 180 00 Prague 8, Czech Republic  J. Wragg  Centre for
Theoretical Atomic, Molecular and Optical Physics, School of Mathematics and
Physics, Queen’s University Belfast,
University Road, Belfast, BT7 1NN, Northern Ireland, UK  A. C. Brown  Centre
for Theoretical Atomic, Molecular and Optical Physics, School of Mathematics
and Physics, Queen’s University Belfast,
University Road, Belfast, BT7 1NN, Northern Ireland, UK  H. W. van der Hart
Centre for Theoretical Atomic, Molecular and Optical Physics, School of
Mathematics and Physics, Queen’s University Belfast,
University Road, Belfast, BT7 1NN, Northern Ireland, UK

###### Abstract

Through solution of the multielectron, semi-relativistic, time-dependent
Schrödinger equation, we show that angular streaking produces strongly spin-
polarized electrons in a noble gas. The degree of spin polarization increases
with the Keldysh parameter, so that angular streaking — ordinarily applied to
investigate tunneling — may be repurposed to generate strongly spin-polarized
electron bunches. Additionally, we explore modifications of the angular
streaking scheme that also enhance spin polarization.

Photoelectron spin alignment represents an exploitable degree of freedom for
probing both the structure and dynamics of matter. Spin-polarized electron
bunches facilitate the imaging of magnetic domains in surfaces Altman  _et
al._ (1991); N. Rougemaille and A.K. Schmid (2010) and thin films Wu  _et al._
(2013); Chen  _et al._ (2015); Prieto  _et al._ (2016), as well as the study
of magnetization profiles in magnetic nanostructures Ding  _et al._ (2005);
Zdyb  _et al._ (2006); Niu  _et al._ (2017). Polarized electron beams are also
instrumental in fundamental experiments addressing electron scattering and
fluorescence emission from atoms and molecules Uhrig  _et al._ (1994); Prinz
_et al._ (1995); Green  _et al._ (2004); Maseberg and Gay (2009), and even
play a vital role in tests of the Standard Model with high-energy, electron-
positron colliders Vauth and List (2016). Harnessing the spins of field-
ionized electrons for real-time imaging and spectroscopy demands reliable and
efficient experimental strategies for generating electron bunches with well-
defined spin polarization. In this letter, we demonstrate numerically that the
contemporary technique of attosecond angular streaking Eckle  _et al._ (2008a,
b) (or the “attoclock”), traditionally used to investigate electron tunneling,
can be repurposed to produce strongly spin-polarized electrons.

The production of spin-polarized electrons through the laser-driven ionization
of atoms and molecules has been examined for over 50 years Fano (1969);
Lambropoulos (1973); Huang  _et al._ (1979); Dixit  _et al._ (1981); Nakajima
and Lambropoulos (2002). Their realization through the multiphoton ionization
of noble gases by circularly polarized pulses has constituted an especially
significant focus of experimental Hartung  _et al._ (2016); Liu  _et al._
(2018a); Trabert  _et al._ (2018) and theoretical Barth and Smirnova (2013,
2014) efforts alike. In many of these studies, spin polarization was obtained
in long circular pulses from the natural energy separation of corotating and
counter-rotating ($p_{\pm 1}$) electrons, which tend to have opposing spin
orientations Trabert  _et al._ (2018). In particular, low-energy photoelectron
emission can be strongly spin-polarized, due to the suppression of emission of
corotating electrons. In atoms with significant spin-orbit splittings, such as
Xe, this effect is made more apparent: the energy separation between the
ejected electrons enables resolution of the final ionic ground-state level
following ionization Liu  _et al._ (2018a); Trabert  _et al._ (2018). Diatomic
molecules such as NO have also been identified as promising candidates for
spin-polarized electron production Liu and Barth (2016); Liu  _et al._ (2017),
enabled by the opposing spin orientations of the valence $\pi_{\pm}$ orbitals,
as well as their spectral separation. Strong-field schemes for producing spin-
polarized electrons using orthogonal two-color pulses Han  _et al._ (2019) and
bicircular pulses Milošević (2016); Ayuso  _et al._ (2017); Milošević (2018)
have also been proposed.

Short-pulse schemes such as angular streaking Eckle  _et al._ (2008a, b) may
provide an alternative means of tuning the spin polarization. The attoclock
employs a few-cycle, near-circularly-polarized laser pulse of infrared
wavelength to modify the binding potential of the target to form a rotating
barrier, through which an electron may tunnel. The brevity of the pulse serves
to localize the ejected-electron wavepacket within an angular interval,
allowing a dominant emission direction to be defined. If emission is directed
away from the major axis of the laser polarization ellipse, this could imply
that electron ejection lags behind the laser field, by a time needed for
tunneling. Considerable controversy exists regarding this interpretation, with
conflicting findings of tunneling being instantaneous Eckle  _et al._ (2008a,
b); Pfeiffer  _et al._ (2011); Torlina  _et al._ (2015); Bray  _et al._
(2018); Sainadh  _et al._ (2019) and non-instantaneous Boge  _et al._ (2013);
Landsman  _et al._ (2014); Camus  _et al._ (2017). A detailed review on this
subject is given in Ref. Kheifets (2020).

Regardless of the attoclock’s ability to provide a tunneling time,
calculations have shown that in an attoclock scheme, $p_{\pm 1}$ electrons
ejected from noble gases emerge at different angles to the major axis of the
laser polarization, and are thereby spatially separated to some extent Kaushal
_et al._ (2015); Liu  _et al._ (2018b). These electrons are also separated in
energy Barth and Smirnova (2011) and favor opposite spin orientations Trabert
_et al._ (2018), such that the differential in both energy and emission angle
may render the attoclock scheme promising for the realization of spin-
polarized electron bunches.

Figure 1: Schematic diagram of ionization pathways for Kr. Upper panel: arrows
indicate spin orientation, and their thickness indicates the relative yield.
Lower panel: The $p_{\pm 1}$ electrons ejected in the attoclock scheme emerge
with different momentum vectors, and their differing spin preferences enables
spin polarization. Values of the spin polarization $\chi$ are indicated. See
text for further details.

Figure 1 illustrates this separation of co- and counter-rotating electrons in
the angular streaking of the Kr atom. Note that in this work, we define the
electric field such that the $p_{-1}$ electron is counter-rotating, and the
$p_{+1}$ electron corotating. An elliptically-polarized, few-cycle pulse
ejects valence $4p_{\pm 1}$ electrons, leaving the residual Kr+ ion in either
the ${}^{2}P_{3/2}$ ground state, or the ${}^{2}P_{1/2}$ excited state, with a
fine-structure splitting of about 0.67 eV. As shown in Fig. 1, the
${}^{2}P_{3/2}$ ionic state is fourfold degenerate, coupling to a $p_{-1}$
electron that prefers to be spin down, and a $p_{+1}$ electron that prefers to
be spin up Trabert  _et al._ (2018). The ${}^{2}P_{1/2}$ excited state is
twofold degenerate: the $p_{+1}(p_{-1})$ electron must be spin down(up). This
degeneracy imposes limits on the highest possible degree of spin polarization:
if the ion is left in the ${}^{2}P_{1/2}$ excited state, spin polarization
(given by the $\chi$ values indicated in Fig. 1) can reach 100%, whereas if it
is left in the ${}^{2}P_{3/2}$ ground state, the spin polarization can attain
a maximum magnitude of 50%. Counter-rotating electrons dominate the yield in
strong-field ionization, and hence the strongest spin polarization to be
expected is –50%. As shown in the lower panel of Fig.1, the spatial separation
of $p_{\pm 1}$ electrons enabled by the angular streaking scheme Kaushal  _et
al._ (2015) means that strong spin polarization may be obtained in certain
spatial directions.

We explore spin polarization driven by angular streaking of Kr using the ab
initio, $R$-matrix with time-dependence (RMT) method Nikolopoulos  _et al._
(2008); Moore  _et al._ (2011); Clarke  _et al._ (2018); Brown  _et al._
(2020). RMT provides a solution of the time-dependent Schrödinger equation in
full dimensionality for single ionization of multielectron atomic and
molecular systems. The atomic structure input for the RMT calculations is
obtained from both non-relativistic and semi-relativistic calculations using
the rmatrx i codes Berrington  _et al._ (1995). Our atomic structure model of
the Kr atom is described in Ref. Wragg  _et al._ (2019). We use bound orbitals
obtained from Hartree-Fock calculations Clementi and Roetti (1974), and allow
the ejection of both $4s$ and $4p$ electrons.

The time-varying laser field is treated classically and within the electric
dipole approximation. The $\hat{z}$-axis is along the laser propagation
direction, so that the field is constrained to the $xy$-plane. Thus, we have

$\bm{\mathcal{E}}(t)=\frac{{\cal
E}_{0}}{\sqrt{1+\epsilon^{2}}}\sin^{4}\left(\frac{\omega
t}{2N_{c}}\right)\left[\cos\omega t\;\hat{\bf x}+\epsilon\sin\omega
t\;\hat{\bf y}\right],$ (1)

where ${\cal E}_{0}$ is the peak electric field strength, $\omega$ is the
laser frequency, $N_{c}$ is the number of laser cycles, and $\epsilon$ is the
ellipticity. The field retains this profile for time $t\in[0,2\pi
N_{c}/\omega]$, and is zero otherwise. The pulses considered in this work ramp
on over four cycles, followed by an equal number of cycles of ramp-off, so
that $N_{c}=8$ in all cases. We propagate the wave function using an Arnoldi
propagator of order 8, for a total of 60 fs. The ejected electron is described
up to a distance of $3730\ a_{0}$ from the nucleus, where $a_{0}$ is the Bohr
radius.

Following time propagation, we obtain the photoelectron momentum distribution
in the laser polarization plane. The contributions of co- and counter-rotating
electrons, as well as both spin-up and spin-down electrons, are resolved by
decoupling the spin and orbital angular momenta of the ejected electron from
those of the residual Kr+ ion. The ejected electron wavefunction is
transformed to momentum space using a Fourier transform.

We first investigate the laser-driven response of Kr in a conventional angular
streaking scheme. Fig. 2 shows the polarization-plane, photoelectron momentum
distributions for Kr, ionized by an 8-cycle, $2\times 10^{13}$ W/cm2,
elliptically polarized ($\epsilon=0.87$), 780-nm pulse, separated into
contributions from co- ($p_{+1}$) and counter-rotating ($p_{-1}$) electrons
(Figs. 2a and 2b respectively). The elliptically polarized laser pulse
confines the dominant ionization yield to an angular range, in contrast to the
approximately angle-independent yield induced by circular pulses. The
momentum-integrated distribution in Fig. 2c demonstrates a clear angular
separation of around 10∘ between co- and counter-rotating electrons. Such
separation is typically attributed to differing deflections caused by the
Coulomb potential Kaushal  _et al._ (2015); Liu  _et al._ (2018b). Counter-
rotating electrons provide close to 90% of the total yield.

(a)  Corotating ($p_{+1}$)

(b)  Counter-rotating ($p_{-1}$)

(c)

Figure 2: Momentum distribution for (a) corotating ($p_{+1}$) and (b) counter-
rotating ($p_{-1}$) electrons ionized from Kr by an 8-cycle, 780-nm, 2$\times
10^{13}$ W/cm2 pulse of ellipticity $\epsilon=0.87$. The radial labels
indicate momentum values (in a.u.). Note the different orders of magnitude in
the scales of (a) and (b). (c) The respective momentum-integrated
distributions.

Intuitively, the strongest spin polarization should be attained if the yield
of corotating electrons can be minimized. This is naturally achieved at low
photoelectron energies Liu  _et al._ (2018a); Trabert  _et al._ (2018); Barth
and Smirnova (2013). However, angular streaking provides an additional source
for this minimization, by confining the dominant photoelectron yield to a
limited angular range, thereby creating the minima seen in Fig. 2c. Spin
polarization should then reach an optimal value close to $\phi=0,180$∘, where
the corotating yield is minimal.

(a)  Spin up

(b)  Spin down

(c)

Figure 3: Momentum distribution for (a) spin up and (b) spin down electrons
ionized from Kr by an 8-cycle, 780-nm, 2$\times 10^{13}$ W/cm2 pulse of
ellipticity $\epsilon=0.87$. Note the different orders of magnitude in the
color scales. (c) Momentum-integrated spin polarization, its dependence on the
Kr+ $J$ value, and its single-threshold ($J=3/2$) estimate.

To investigate this, we calculate the spin-resolved momentum distributions for
photoelectrons ejected from Kr in the same scheme as in Fig. 2. Figs. 3a and
3b show the polarization-plane momentum distributions for spin-up and spin-
down electrons. The energy- and angle-dependence of both distributions show a
clear resemblance to the distribution in Fig. 2b, which is not surprising,
given that counter-rotating electrons dominate the overall yield. Their
dominance also explains the observation that spin down electrons are strongly
favored.

To assess the angular variation of spin polarization in greater detail, we
integrate the spin-resolved momentum distributions over momentum, and define
the angular distributions of spin-up and spin-down electrons as ${\cal
P}_{\uparrow}(\phi)$ and ${\cal P}_{\downarrow}(\phi)$ respectively. From
these quantities, we obtain an angle($\phi$)-dependent spin polarization given
by $\chi(\phi)=[{\cal P}_{\uparrow}(\phi)-{\cal P}_{\downarrow}(\phi)]/[{\cal
P}_{\uparrow}(\phi)+{\cal P}_{\downarrow}(\phi)].$ Fig. 3c shows this quantity
as a function of azimuthal angle (solid black curve). A significant spin
polarization of around $-0.4$ is attained close to $\phi=0,180$∘, as expected
(see Fig. 2c). Away from these angles, corotating electrons make stronger,
though still small, contributions. Since corotating electrons prefer to be
spin up in this case, their contribution weakens the total spin polarization.
Indeed, the weakest spin polarization observed here, of around $-0.28$, is
attained close to $\phi=100,280$∘, where the corotating electron yield is
maximal. At other angles, the spin polarization varies, since it comprises
angle-dependent combinations of co- and counter-rotating electron yields. The
angle dependence of the spin polarization is related to the angle-dependent
ratio of counter-rotating to corotating electron yields, $R(\phi)$.
Considering only the ionic ground state, and assuming the spin weights
indicated in Fig. 1, a simple, single-threshold estimate for the spin
polarization can be obtained via
$\chi\approx-\frac{1}{2}[R(\phi)-1]/[R(\phi)+1]$. We plot this estimate in
Fig. 3c, alongside the spin polarization calculated using the RMT method.
Thus, an angular separation of co- and counter-rotating electrons which
produces a large value of $R(\phi)$ ought to yield strong spin polarization.

The calculated spin polarization is significantly weaker than this estimate,
due to contributions from the $J=1/2$ ionic state which are strongly spin up.
As shown in Fig. 3c, when we include only the ground ionic state in our
analysis, the calculated spin polarization agrees well with the simple
estimate. Under these conditions, the ionic excited state contributes around
8% of the total yield.

Of course, in a strong-field scheme some degree of spin polarization is likely
to be observed, due to the natural imbalance between co- and counter-rotating
electron yields. This is demonstrated in recent experiments Trabert  _et al._
(2018); Liu  _et al._ (2018a), which use long circular pulses to obtain an
energy-dependent spin polarization. To investigate the particular
effectiveness of the angular streaking scheme, we show in Fig. 3c the spin
polarization calculated using a 16-cycle, circularly polarized pulse
($\epsilon=1$). Relative to this scheme, angular streaking clearly enhances
the spin polarization, yielding a minimum spin polarization which is around
20% lower than the angle-independent value of around –0.33 obtained using a
long circular pulse, equivalent to around 40% progress towards the limit of
–0.5. The pronounced angular minimum in the corotating electron yield created
by the attoclock facilitates its advantage over the long-pulse scheme,
conferring a higher degree of spin polarization and dominant emission in a
well defined angular interval.

Although a strong spin polarization is obtained under these conditions, the
total photoelectron yield is rather low. Although the yield may be enhanced by
increasing the peak laser intensity, this would be detrimental to the spin
polarization, since corotating electron contributions would increase. This
effect is made clear in Fig. 4, where we show the calculated mean value of the
energy-integrated spin polarization as a function of peak laser intensity. The
vertical bars indicate the range of values taken by the spin polarization over
all azimuthal angles. Over this range of intensities, the average spin
polarization reduces in magnitude by around 15%, as do the extreme (most and
least negative) values.

Figure 4: Average energy-integrated spin polarization as a function of laser
intensity. Vertical bars indicate the range of spin polarization values
obtained over the full range in azimuthal angle $\phi$.

The need for a relatively low peak intensity for the purpose of spin
polarization contrasts with the typical attoclock requirement of relatively
high peak intensity to access the tunneling regime (note the Keldysh parameter
values in Fig. 4). This indicates that the attoclock scheme may be applied to
good effect in the multiphoton regime to generate strong spin polarization. We
note also that since a low intensity is advantageous for spin polarization,
focal-volume averaging is likely to enhance its degree. Therefore, we expect
that strong spin polarization should be detectable in low-intensity
experiments at a wavelength of 780 nm.

Figure 5: Angular distributions for (a) corotating and counter-rotating
electrons, (b) spin-up and spin-down electrons, and (c) spin polarization of
electrons ionized from Kr by an 8-cycle, 390-nm, 2$\times 10^{13}$ W/cm2 pulse
of ellipticity $\epsilon=0.87$.

Of course, it would be useful if strong spin polarization could be obtained by
aligning the minimum in the corotating electron signal with a strong counter-
rotating electron signal. Such a separation may be possible at shorter
wavelengths, where the reduction in ejected-electron angular momentum ought to
increase the angle between corotating and counter-rotating electrons.

To do this, we choose a wavelength of 390 nm, and retain an ellipticity of
0.87, and a peak laser intensity of $2\times 10^{13}$ W/cm2 (as in Fig. 3).
This wavelength may be viable in experiment, merely requiring frequency-
doubling within the typical attoclock setup. Fig. 5(a) shows the calculated
angular distributions for corotating and counter-rotating electrons. The
angular separation of 39∘ is significantly larger than that observed in Fig. 2
at 780 nm, and significant counter-rotating electron yield is observed at
angles where the yield of corotating electrons is minimal. The photoelectron
yield is also considerably larger than that obtained at 780 nm. Fig. 5(b)
shows the calculated angular distributions for spin-up and spin-down
electrons. Spin-down electrons contribute almost twice the yield of spin-up
electrons. The angle-dependent spin polarization, shown in Fig. 5(c), reaches
a value of $-0.3$ at around 200∘, close to the minimum in the corotating-
electron signal. Although this is a smaller degree of spin polarization than
that obtained in the 780-nm case at the same peak intensity, its angular
location lies in a region of relatively high photoelectron yield, and strong
values span a broader angular range. The main reason for the lower degree of
spin polarization is that the minimum in the corotating electron yield is not
as pronounced as that obtained at 780 nm (see Fig. 2).

In Fig. 5(c), we compare the spin polarization obtained using the angular
streaking scheme with the result for a 16-cycle, circular pulse
($\epsilon=1$). As at 780 nm (see Fig. 3c), we find a significant enhancement
of the spin polarization in the angular streaking scheme. In this case the
enhancement is around 30%, with the 16-cycle pulse yielding a spin
polarization of around –0.21 at all angles.

Figure 6:  (a) Angular distribution of spin-up and spin-down electrons ionized
from Kr by an 8-cycle, 248-nm, $2\times 10^{13}$ W/cm2 pulse of ellipticity
$\epsilon=0.87$. (b) Average energy-integrated spin polarization as a function
of laser wavelength at an intensity of $2\times 10^{13}$ W/cm2. Vertical bars
indicate the range of spin polarization values obtained over the full range in
azimuthal angle $\phi$.

As was the case at 780 nm, we find that spin polarization is reduced in
magnitude as peak laser intensity is increased. We also find that further
shortening of the wavelength is detrimental to spin polarization, since the
minimum in the corotating electron yield is not as deep as that observed at
780 nm. Fig. 6(a) shows that if the laser wavelength is reduced to 248 nm
(with the laser intensity fixed at $2\times 10^{13}$ W/cm2), spin-up and spin-
down electrons contribute on a similar level, in stark contrast to the
dominance of spin-down electrons seen in Fig. 3 and Fig. 5(b). Consequently,
the energy-integrated spin polarization approaches zero as laser wavelength is
decreased to 248 nm, as shown in Fig. 6(b). As the wavelength is increased to
585 nm and beyond, the spin polarization appears to reach a plateau. This
indicates that even with longer wavelengths, it may be difficult to achieve a
spin polarization significantly larger than that obtained at 780 nm.
Therefore, the parameters chosen in this work appear to be close to optimal
for the generation of spin-polarized electrons within a single-pulse angular
streaking scheme.

Our findings are not limited to atomic krypton. The angular streaking scheme
should enhance spin polarization for any atomic or molecular system in which
spin polarization can be generated by long, circularly polarized pulses. The
streaking scheme introduces angular separation between co-rotating and
counter-rotating electrons, and thus enhances the spatial separation of
interfering pathways. In addition, the scheme has the advantage that it is
robust over focal-averaging. As the laser intensity decreases, the degree of
spin polarization increases. Hence, the angular streaking scheme should be
applicable to a wide range of systems.

In conclusion, we have investigated the use of angular streaking as a
generator of spin-polarized electrons. We have shown that the angular
streaking scheme can induce spin polarization by minimizing the corotating
electron yield over an angular range, thereby creating a region of strongly
spin-polarized electrons. Additionally, we have shown that the angular
streaking scheme may be used for this purpose when implemented in the low-
intensity, multiphoton regime, far from the tunneling conditions under which
the scheme is usually employed. We suggest this as a new application of
angular streaking, from which the useful product of spin-polarized electrons
may be obtained.

The data presented in this article may be accessed at Ref. pur (2020). The RMT
code is part of the UK-AMOR suite, and can be obtained for free at Ref. rep
(2019). This work benefited from computational support by CoSeC, the
Computational Science Centre for Research Communities, through CCPQ. The
authors acknowledge funding from the UK Engineering and Physical Sciences
Research Council (EPSRC) under grants EP/P022146/1, EP/P013953/1,
EP/R029342/1, and EP/T019530/1. This work relied on the ARCHER UK National
Supercomputing Service (www.archer.ac.uk), for which access was obtained via
the UK-AMOR consortium funded by EPSRC.

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