# Exploring the Origin of Solar Energetic Electrons I: Constraining the
Properties of the Acceleration Region Plasma Environment

Ross Pallister Department of Mathematics, Physics & Electrical Engineering,
Northumbria University,
Newcastle upon Tyne, UK
NE1 8ST  Natasha L. S. Jeffrey Department of Mathematics, Physics & Electrical
Engineering, Northumbria University,
Newcastle upon Tyne, UK
NE1 8ST

###### Abstract

Solar flare electron acceleration is an efficient process, but its properties
(mechanism, location) are not well constrained. Via hard X-ray (HXR) emission,
we routinely observe energetic electrons at the Sun, and sometimes we detect
energetic electrons in interplanetary space. We examine if the plasma
properties of an acceleration region (size, temperature, density) can be
constrained from in-situ observations, helping to locate the acceleration
region in the corona, and infer the relationship between electrons observed
in-situ and at the Sun. We model the transport of energetic electrons,
accounting for collisional and non-collisional effects, from the corona into
the heliosphere (to 1.0 AU). In the corona, electrons are transported through
a hot, over-dense region. We test if the properties of this region can be
extracted from electron spectra (fluence and peak flux) at different
heliospheric locations. We find that cold, dense coronal regions significantly
reduce the energy at which we see the peak flux and fluence for distributions
measured out to 1.0 AU, the degree of which correlates with the temperature
and density of plasma in the region. Where instrument energy resolution is
insufficient to differentiate the corresponding peak values, the spectral
ratio of [7-10) to [4-7) keV can be more readily identified and demonstrates
the same relationship. If flare electrons detected in-situ are produced in,
and/or transported through, hot, over-dense regions close to HXR-emitting
electrons, then this plasma signature should be present in their lower-energy
spectra (1–20 keV), observable at varying heliospheric distances with missions
such as Solar Orbiter.

tablenum SIXtablenum

##  1 Introduction

The Sun is an efficient particle accelerator capable of producing kilo-, Mega-
and Giga-electronvolt (keV, MeV, GeV) energetic particles (e.g., Holman et
al., 2011; Vilmer et al., 2011; Klein & Dalla, 2017), in transient events such
as solar flares and coronal mass ejections (CME) via magnetic reconnection
(e.g., Parker, 1957; Sweet, 1958; Priest & Forbes, 2000) and/or shocks (e.g.,
Forbes et al., 2006). In solar flares, a large fraction ($\sim 10-50$%) of the
released magnetic energy is converted into energetic particles, including
energetic electrons (e.g., Emslie et al., 2012). However, the processes that
accelerate and transport energetic electrons are widely debated with the exact
configuration(s) of the acceleration mechanism(s), environment and location(s)
still undetermined.

In this standard model, hard X-ray (HXR) producing energetic electrons (e.g.,
Kontar et al., 2011) are transported along newly formed magnetic field lines
in the corona, precipitating into the dense layers of the lower atmosphere and
losing energy. So-called flare-produced solar energetic electrons (SEEs) can
also be detected in the heliosphere (e.g., Lin, 1985), either via in-situ
measurements or by their radio emissions (c.f., Pick & Vilmer, 2008), but the
connection between these distinct electron populations, and indeed their
connecting magnetic topology, is still poorly understood. Moreover, multi-
messenger diagnostics, whether remote sensing or in-situ, can be complicated
by various particle propagation and emission effects such as e.g., Coulomb
collisions (e.g., Jeffrey et al., 2014), X-ray albedo (e.g., Bai & Ramaty,
1978), radio wave scattering (e.g., Kontar et al., 2017b), turbulence
(parallel and cross-field diffusion) (e.g., Kontar et al., 2014) and field
line meandering (e.g., Laitinen et al., 2016). HXR-emitting electrons are
produced close to or within hot and dense flaring loops, possibly via
turbulence (Kontar et al., 2017a; Stores et al., 2021), and we see the
signature of that plasma in their low-energy spectra ($\leq 30$ keV). Non-
thermal electrons accelerated out of a hot, dense plasma and/or non-thermal
electrons moving through a hot, dense region and undergoing (partial-)
thermalization at lower-energies ($\leq 50$ keV) (e.g., Jeffrey et al., 2019;
Kontar et al., 2015), will retain or imprint the properties of that plasma
environment.

Most studies to date explore the connection between different flare electron
populations by studying the properties of the higher energy power law (usually
above $40$ keV). After the launch of the Ramaty High Energy Solar
Spectroscopic Imager (RHESSI; Lin et al. (2002)), several studies examined
flares with both HXR-producing electrons and SEEs detected at 1.0 AU. Krucker
et al. (2007) compared HXR spectra with WIND/3DP (Lin et al., 1995) electron
spectra at $\sim 1.0$ AU. For so-called ‘prompt’ events, where the SEE release
time appears to coincide with the flare HXR burst, Krucker et al. (2007) found
a clear correlation for both power-law spectral indices and total number of
electrons, which is consistent with a single process accelerating both
electron populations. Under the assumption that both HXR-producing and
escaping electron populations are accelerated by a similar mechanism within
similar plasma conditions, we expect near identical spectra. However, in
Krucker et al. (2007), the peak flux spectrum was harder then the inferred
cold-thick-target electron spectra at the Sun possibility suggesting the
effects of coronal or heliospheric transport processes. A near-identical study
using flare data from solar cycle 24, Dresing et al. (2021), also found a
strong correlation of about 0.8 between remote HXR-producing and in-situ
spectral indices. Dresing et al. (2021) observed an increased correlation for
events with ‘significant anisotropy’, suggesting that transport effects reduce
the signature of the acceleration region properties in the data. In Wang et
al. (2021), sixteen SEE producing flares were examined; they determined that
the spectral index of HXR-producing electrons was no less than the observed
high-energy spectral index of SEEs (above a spectral break energy), showing a
positive correlation with the high-energy spectral index of SEEs. Further, the
spectral analysis (extending down to $\sim 5$ keV) suggested that the source
of SEEs was high in the corona at a heliocentric distance of $\geq 1.3$ solar
radii. In contrast, other studies looking at Type III radio bursts (e.g., Reid
et al., 2011) offer conflicting results, suggesting that outward propagating
electrons are accelerated between heights of $40-60$ Mm ($\approx 1.07$ solar
radii), much closer to the flare location at the Sun.

Only with a better understanding of the acceleration environment and transport
processes therein can we explore the connection between flare-accelerated
electron populations. To this end, we aim to constrain the properties of an
acceleration region (e.g., its size, temperature, density, turbulence
profile); properties not currently constrained well by remote flare
observation or in-situ detection alone. In this initial study (Paper I) we use
solely SEE transport modelling in the corona and heliosphere alone to
investigate the possibility of extracting acceleration region plasma
properties from in-situ data, constraining the likely location of their
acceleration, and hence relation with HXR-producing electrons. The study
mainly concentrates on the electron spectral observations at heliospheric
locations of $0.4-1.0$ AU, that can now be explored with both Parker Solar
Probe (PSP; Fox et al. (2016)) and Solar Orbiter (SolO; Müller et al. (2020)).
Later studies will combine these constraints with those derived from coronal
HXR-centric studies to explore the possibility of a unified acceleration
region parameter range for prompt flare events. We examine if more attention
should be given to the understudied electron spectral range of $1-30$ keV, and
if signatures of hot and dense plasma can be extracted from in-situ datasets.
Most studies concentrate on studying and comparing the properties of
interplanetary electrons with their HXR-emitting counterparts above $20-30$
keV only. However, the analysis of HXR-producing electrons shows that the
lower portion of the spectrum $\leq 20$ keV is directly related to the
presence and properties of the surrounding coronal plasma (e.g., Kontar et
al., 2015). We suggest that any hot, over-dense plasma signature will help to
locate the origin of such particles and in particular their relationship to
HXR-emitting electrons. For this task, we assume that electron spectra (peak
flux and fluence) in the range of $1-20$ keV are appropriate for this purpose.

In Section 2 we present the model of electron transport for both the coronal
region and heliospheric components, and the coronal plasma properties used in
the former. In Section 3 we present the results of the study, including
different diagnostics that can be used to estimate the acceleration plasma
environment from a partially-thermalized electron distribution. In Section 4,
we summarize and discuss the main results.

##  2 Coronal and heliospheric transport model

We have developed an electron acceleration and transport model for the inner
corona and heliosphere, starting at $1$ solar radius $R_{\sun}$. It is assumed
that the extent of the simulated coronal region beyond the solar surface is
always negligible compared to 1.0 AU (maximum 40 Mm vs approx. 150,000 Mm, or
0.026%).

The model is composed of two discrete domains: a hot and over-dense (flare)
coronal ‘acceleration’ region of length $L$ with given uniform electron
temperature $T$ and density $n_{e}$, and a sparser cold plasma ($T$ $\approx
0$) representing the wider heliosphere for $z>L$ (see Figure 1). Contrary to
the name, the current study neglects acceleration mechanisms and will instead
inject a single non-thermal electron distribution into a region dominated by
collisional effects, henceforth referred to as the ‘collisional’ region111The
acceleration of the electrons out of the background thermal plasma will be
discussed in Paper II.. The extent and plasma properties of this collisional
region are varied within reasonable limits to investigate the effect each
variable has on the electron population ejected from this region. Following
different solar flare observations of (above-the-) loop-top sources (e.g.,
Caspi et al., 2014; Jeffrey et al., 2015; French et al., 2020), we choose a
sensible range of parameters for the coronal region: temperature $T$ ranging
between $10-30$ MK, electron number density $n_{e}$ ranging between
$10^{9}-10^{10}$ cm-3 and region size $L$ ranging between $10-40$ Mm ($\sim
14^{\prime\prime}-55^{\prime\prime}$).

Figure 1: Simplified cartoon indicating one possible magnetic topology (such
as emerging flux) that could lead to the production of both flare HXR-
producing, and in-situ detected electrons in close proximity, in hot and over-
dense regions close to the flare energy release site. In Paper I, we
concentrate on studying the properties of those electrons accelerated during
the flare but observed in-situ, by modelling the transport of an injected
electron population through a hot and over-dense coronal region related to the
flare (black-lined rectangle) of various temperature $T$, number density
$n_{e}$ and size $L$, and then out into the heliosphere until 1.0 AU.

The effects acting on electrons in the following transport equations are
broadly separable into two types: collisional and non-collisional (and the
latter into focusing and diffusion). In the heliospheric environment,
adiabatic focusing and pitch-angle diffusion act in opposite to align or
disperse particles (respectively) with regards to the magnetic field, the
former being more significant where the magnetic field is strong and the
latter where the mean free path is short. In the coronal region, collisional
energy losses may dominate in the dense environment, whereas collisions will
be negligible in the sparser heliospheric domain where non-collisional effects
have previously been demonstrated to take precedence (due to low heliospheric
densities described by empirical models and observations e.g., Newkirk Jr,
1967; Saito et al., 1977; Leblanc et al., 1998; Fludra et al., 1999; McCauley
et al., 2018).

Forces due to gravity and electric fields are neglected. In this preliminary
study, we neglect the effects of Langmuir wave turbulence and Landau damping
that can produce a clear spectral break at $\sim 40$ keV or lower (Kontar &
Reid (2009); Section 4 briefly discusses how wave-particle interactions may
change the diagnostics outlined in Section 3). The effects of cross field
particle diffusion and field line meandering which are vital for large spread
events (Laitinen et al., 2016) are neglected here.Particle (re-)acceleration
in the heliosphere, such as that induced by shocks associated with co-rotating
interaction regions (CIRs) (as described by Heber et al., 1999; Allen et al.,
2021; Zhao et al., 2019) and the heliospheric current sheet (e.g., Zharkova &
Khabarova, 2012; Khabarova et al., 2015) are also ignored in the present
study.

###  2.1 Governing Fokker-Planck Equation

To describe the transport of a chosen electron distribution function
$f(t,z,v,\mu)$ in time $t$, along a guiding magnetic field $z$, speed $v$ and
cosine of the pitch-angle ($\beta$) to the guiding magnetic field
$\mu=\cos\beta$, we use the following Fokker-Planck equation (e.g., Lifshitz &
Pitaevskii, 1981; Karney, 1986),

$\begin{split}\frac{\partial f}{\partial t}+\mu v\frac{\partial f}{\partial
z}=&\underbrace{-\frac{v(1-\mu^{2})}{2L_{z}}\frac{\partial
f}{\partial\mu}}_{\text{adiabatic focusing}}\\\
&\underbrace{+\frac{\partial}{\partial\mu}\left[D_{\mu\mu}\frac{\partial
f}{\partial\mu}\right]}_{\text{non-collisional pitch-angle scattering}}\\\
&\underbrace{+\frac{\Gamma}{2v^{2}}\left[\frac{\partial}{\partial
v}\left(2vG(u)\frac{\partial f}{\partial
v}+4u^{2}G(u)f\right)\right]}_{\text{collisional energy losses}}\\\
&\underbrace{+\frac{\Gamma}{2v^{3}}\left[\frac{\partial}{\partial\mu}\left((1-\mu^{2})\biggl{[}{\rm
erf}(u)-G(u)\biggr{]}\,\frac{\partial
f}{\partial\mu}\right)\right]}_{\text{collisional pitch-angle scattering}}\\\
\end{split}$ (1)

The first two terms on the right hand side of Equation 1 model a simple
heliospheric environment with the presence of only adiabatic focusing and
(parallel) pitch-angle scattering following e.g., Roelof (1969); Dröge &
Kartavykh (2009); Agueda & Vainio (2013). Both terms are present within the
defined collisional coronal ‘acceleration’ region $z<L$ and the wider
heliosphere. $D_{\mu\mu}$ is the pitch-angle diffusion coefficient which
quantifies the diffusion of an electron subject to its pitch-angle, velocity
and local mean free path $\lambda$. The diffusion coefficient and its $\mu$
derivative are given by:

$D_{\mu\mu}=K(1-\mu^{2})(|\mu|^{q-1}+h)$ (2) $\frac{\partial
D_{\mu\mu}}{\partial\mu}=K\mu\bigg{[}(q-1)(1-\mu^{2})|\mu|^{q-3}-2(|\mu|^{q-1}+h)\bigg{]}$
(3)

where

$K=\frac{3v}{2\lambda(4-q)(2-q)}$ (4)

and $q=5/3$ is the spectral index of magnetic field fluctuations, taken as a
Kolmogorov spectrum and constant $h=0.01$, which accounts for unmodelled
scattering affects that are otherwise dominated at $|\mu|>0$ in the
heliosphere. Following Alcock (2018), the mean free path $\lambda$ is
calculated as in Equation 5,

$\lambda=\lambda_{\earth}\Bigg{(}\frac{z}{z_{\earth}}\Bigg{)}^{\kappa}\Bigg{(}\frac{p}{p_{\text{min}}}\Bigg{)}^{2\xi}$
(5)

dependent on the mean free path at 1.0 AU, $\lambda_{\earth}=0.3$ AU, the
ratio of electron position $z$ relative to 1.0 AU ($=z_{\earth}$) and the
ratio of current and minimum electron momenta $p$ (derived from the minimum
allowed kinetic energy in the heliosphere). $\kappa$ and $\xi$ are parameters
that quantify the degree to which the electron momentum and radial distance
from the Sun affect the mean free path. For this study they are given as
$\kappa=0.5$ and $\xi=-0.2$, and not explored further. The magnetic field at
any point along $z$ is given by Equation 6 (Dulk & McLean, 1978),

$B=\frac{1}{2}\Bigg{(}\frac{z}{R_{\sun}}-1\Bigg{)}^{-\frac{3}{2}}$ (6)

and $L_{z}$ in Equation 1 is ratio of the local magnetic field to its spatial
gradient.

$L_{z}=\frac{B(z)}{\left(-dB/dz\right)}$ (7)

The final two terms in Equation 1 describe collisional energy losses and
pitch-angle scattering respectively (e.g., Jeffrey et al., 2014; Kontar et
al., 2015) where $\Gamma=4\pi e^{4}$ln$\Lambda n_{e}/m_{e}^{2}$, for electron
charge $e$ [statC], Coulomb logarithm ln$\Lambda$ and electron mass $m_{e}$
[g]. The error function erf$(u)$ and the Chandrasekhar function $G(u)$ are
given by

$\text{erf}(u)\equiv(2/\sqrt{\pi})\int_{0}^{u}\text{exp}(-u^{2})du$ (8)

and

$G(u)\equiv\frac{\text{erf}(u)-u\text{erf}^{\prime}(u)}{2u^{2}}$ (9)

where $u$ is the dimensionless velocity $u=v/(\sqrt{2}v_{th})$ and
$v_{th}=\sqrt{k_{B}T_{e}/m_{e}}$. The error function and $G(u)$ control the
lower-energy ($E\approx k_{B}T_{e}$) electron collisional interactions
ensuring that they become indistinguishable from the background thermal
plasma.

###  2.2 Conversion to Stochastic Differential Equations (SDEs)

The Fokker-Planck equation (Equation 1) can be rewritten as a Kolmogorov
forward equation (Kolmogorov, 1931) and then converted to a set of time-
dependent stochastic differential equations (e.g., Gardiner (1986); Strauss &
Effenberger (2017)) that describe the evolution of $z$, $E$, and $\mu$ in Itô
calculus. The electron transport equations are solved numerically by a series
of first-order Euler expressions, returning the evolution of electron velocity
$v$, pitch-angle $\mu$ and position $z$ along the guiding field:

$\begin{split}v_{i+1}&=v_{i}-\frac{\Gamma}{v_{i}^{2}}\left(\text{erf}(u_{i})-2u_{i}\text{erf}^{\prime}(u_{i})+G(u_{i})\right)\,\Delta
t\\\ &+\sqrt{\frac{2\Gamma G(u_{i})}{v_{i}}\Delta t}\;W_{v}(t)\end{split}$
(10) $\begin{split}\mu_{i+1}&=\mu_{i}\\\
&-\frac{\Gamma\mu_{i}(\text{erf}(u_{i})-G(u_{i}))}{v^{3}_{i}}\,\Delta
t+\Bigg{(}\frac{dD_{\mu\mu}}{d\mu}+\frac{v_{i}(1-\mu^{2}_{i})}{2L_{z}}\Bigg{)}\,\Delta
t\\\
&+\sqrt{\left(2D_{\mu\mu}+\frac{\Gamma(1-\mu_{i}^{2})(\text{erf}(u_{i})-G(u_{i}))}{v^{3}_{i}}\right)\,\Delta
t}\;W_{\mu}(t)\end{split}$ (11)

$z_{i+1}=z_{i}+\mu_{i}v_{i}\Delta t$ (12)

$W$ is a value randomly chosen each time step from a Brownian normal
distribution with a mean of 0 and variance of 1. This provides the stochastic
element to the electron transport. In the coronal region, we set the time step
$\Delta t$ at a constant value of $10^{-2}$ s; in order for collisional
effects to have significance, there must be multiple simulation steps within
the relatively small coronal region. With initial kinetic energies of order 10
keV, beamed electrons are expected to exit the largest modelled coronal region
(40 Mm) within a few tenths of a second. Setting $\Delta t$ = $10^{-2}$ s
allows even the most energetic electrons to have multiple simulation steps
within the coronal region.

As described in Lemons et al. (2009) and Jeffrey et al. (2014), at velocities
less than

$v\leq\bigg{(}\Gamma\frac{8}{3}\sqrt{\frac{m_{e}}{2\pi k_{B}T_{e}}}\Delta
t\bigg{)}^{1/2}$ (13)

the analytical equation

$v\simeq\bigg{(}v_{0}^{2}+\Gamma\frac{8}{3}\sqrt{\frac{m_{e}}{2\pi
k_{B}T_{e}}}\Delta t\bigg{)}^{1/2}$ (14)

can be used to determine $v$ to remove any divergence at low $v$ in Equations
1 and 10. For such velocities, $\mu$ can be drawn randomly from an isotropic
distribution between -1 and +1.

Outside of the coronal collisional region, we use a time step $\Delta t$ =
$1.0$ s and stop updating the velocity terms, as the mean free path in the
simulated heliosphere is large enough that further collisional effects are
assumed to be negligible enough to ignore. Any electrons with kinetic energies
below 1 keV are taken to be thermal and have their pitch-angle values frozen,
rather than randomised every step, saving computation time otherwise spent
propagating electrons below energies of interest.

###  2.3 Initial, plasma and boundary conditions

At a singular point called $z=R_{\sun}$, $10^{6}$ electrons are initialised
with a beamed pitch-angle $\mu_{0}=0.99$, where positive values correspond to
a direction outwards from the Sun and $\mu_{0}=\pm 1.0$ is perpendicular to
the solar surface. Since electrons are not accelerated from a thermal
population in this initial study, we create a power-law velocity (energy)
distribution between the energies of 5 keV and 100 keV, matching the electron
energies detected via remote-sensing HXR observations at the Sun, with a power
index in energy $E$ of $\delta=3$ ($E^{-\delta}$).

The electrons are initialised at the lowest point of a coronal region,
simulated for a length of time sufficient for all particles to be ejected into
the heliosphere. Heliospheric electrons are then simulated for an additional
period of $t=50,000$s, after which the majority of the initial distribution
have reached positions $z>1.0$ AU. This procedure is performed for all
combination of coronal region properties.

As mentioned, we test the following coronal ‘flaring’ conditions: temperature
$T$ equal to either 10 MK, 20 MK or 30 MK, electron number density $n_{e}$
equal to $1\times 10^{9}$ cm-3, $5\times 10^{9}$ cm-3 or $1\times 10^{10}$
cm-3, and region length $L$ equal to 10 Mm, 20 Mm, 30 Mm or 40 Mm.

The kinetic energies and pitch-angles of the population for each set of
coronal region properties are examined at various points in space and time. In
particular, we will examine the time-integrated electron fluence (spectra) at
one location in space, and the flux peak spectra, where the energy spectra are
created using the peak flux in each energy bin or channel, at the following
heliospheric locations of 0.01 AU, 0.4 AU and 1.0 AU. It is assumed that
electrons are transported out to empty space, such that no planetary
magnetospheres are present in the simulation.

As the transport equations do not inherently prevent an unphysical pitch-angle
value ($|\mu|>1$), care must be taken when modelling particles close to this
boundary. In this case they are reset in the same position with a uniformly
random pitch angle $|\mu|=[0.89,1.00)$. Particles cannot be transported below
$R_{\sun}$ and are perfectly reflected at this boundary. The collisional
region is not present in the heliospheric component of the simulation, and so
any ejected electrons that would re-enter those ranges of distance from the
Sun are similarly reflected. This does not represent a physical effect, but
rather a way to prevent extended thermalisation of particles that may be
trapped close to the boundary without affecting statistics. We find that in
all cases the coronal ejecta population remains relatively anisotropic, such
that electrons that do re-encounter the boundary represent a very small
proportion of the heliospheric energy spectra.

##  3 Results

Figure 2: Example ejected electron fluence (blue) versus energy (left) and
pitch-angle (right). The injected distribution (power law, beamed) is denoted
by the black curve in the left plot and by the dashed black line in the right.
The injected distribution travels through a ‘flare’ region with the following
plasma properties: length $L=30$ Mm, temperature $T=20$ MK and number density
$n_{e}=5\times 10^{9}$ cm-3. A thermal distribution corresponding to a plasma
temperature of $T=20$ MK is also shown (grey dotted, left)

###  3.1 Collisional region plasma properties

The ejected electron energy and pitch-angle fluence distributions of an
example parameter set are shown in Figure 2. The ejected fluence corresponds
to time-integrated distributions measured at one spatial location (similar to
spacecraft data), here namely the furthest boundary from the Sun of the
simulated collisional region of length $L$ ($z=R_{\sun}+L$). The ejecta
fluence distribution (blue) is compared with the injected beamed power law
population (black) and a thermal distribution corresponding to the ambient
plasma temperature (grey dotted) of $T=20$ MK. Prior to ejection, the
population steadily evolves towards a thermal shape, though the high-energy
tail ($>$10 keV) remains relatively unchanged from injection, as expected in
lower density ($n_{e}=5\times 10^{9}$ cm-3) plasma. These electrons are
ejected quickly and do not undergo enough collisions to begin to significantly
thermalise above 10 keV. The population as a whole remains relatively
anisotropic out to the point of ejection, though some pitch-angle scattering
is evident if largely insignificant222Here, there is an obvious bias towards
high positive pitch-angles since we inject a beamed distribution, and
measurement is taken at the first crossing of a given boundary. However, the
aim here is to study the effects of different plasma properties on a given
distribution, not exhaustively examine different pitch-angle injections.. We
examine these properties in the respective fluence distributions of all
parameter sets in order to determine how they are affected by changing the
region size, plasma temperature and density.

Figure 3: Collisional-boundary electron energy (upper) and pitch-angle (lower)
fluence distributions independently varying each parameter between their
minimum (blue) and maximum (pink) values, while fixing the other parameters at
intermediate values. Dotted lines represent the thermal distributions
associated with the given temperature values. The fluence peaks at higher
energies for smaller, hotter and sparser regions, which also correspond to
less isotropic ejecta distributions. As expected, the electron spectra above
20 keV remains relatively unchanged regardless of region parameters.

In Figure 3 these distributions are compared for the minimum and maximum
tested values of $L$, $T$ and $n_{e}$ in turn, fixing the other two parameters
at intermediate values. In all parameter sets, the fluence (energy)
distribution remains more or less unchanged above 20 keV, as these electrons
are so energetic as to be practically unaffected by the plasma prior to
ejection for the properties we test for. As such the focus of this analysis
will be on energies below this approximate value, in particular in the 1-20
keV range, where spectral variations due to hot, over-dense plasma
environments should show, if present.

Changing the physical size of the region shows the smallest change in the
energy and pitch-angle distributions. The smaller region has a fluence peak at
a slightly higher energy and fewer electrons ejecting at lower energies.
Smaller regions also appear to have somewhat less isotropic ejecta. As the
electrons are travelling across a shorter distance (and so a shorter time)
before ejection, the population has had less time to thermalise than for a
larger region, therefore resulting in the distributions retaining more of the
features of the non-thermal injected population.

Between plasma temperatures of 10 MK (blue) and 30 MK (pink), a higher $T$
shows a higher energy for the fluence peak. In similarly sized regions or
equivalent density, hotter ambient temperatures result in the population being
thermalised to higher energies before ejection. Such temperatures also appear
to make the population less isotropic than for colder regions (since the
collisional times are increased in a hotter plasma).

In higher density regions, the population evolves more towards a thermal
distribution before ejection than for a lower density, becoming more isotropic
in the process. This is due to the greater number of collisions an average
electron will undergo in this time, making the process of thermalisation more
efficient. Conversely, the population in the lower density region has fewer
average collisions and retains more of the injected distribution’s shape.

Figure 4: The temperature dependence of the ejected electron fluence
distributions is compared for minimum (left) and maximum (right) simulated
densities, and $L=30$ Mm. Distribution shape is very similar in all low-
density regions, but the spectral differences due to temperature can be
clearly seen in higher density cases.

The effects of plasma temperature and density are not entirely separable, as
both play a role in the electron collisional time and by extension the
efficiency of thermalisation. In Figure 4 we compare how varying the electron
temperature for high and low densities affects the resulting fluence spectra
at the collisional region boundary.

In the low-density case, there are too few collisions for the distribution to
thermalise above the lowest measured energies before ejection. The fluence
peaks have approximately the same heights and correspond to the same band of
kinetic energy. At higher densities however we see a much more significant
divergence between the fluence distributions. Low temperatures greatly
suppress the fluence peak and decelerate the injected distribution to energies
of order $\sim 1$ keV. For higher temperatures, the fluence peaks have
comparable heights to the low density region but show more electrons being
thermalised between $1-10$ keV energies, corresponding to the thermal energy
of the ambient plasma. It can be seen that ejecta from low-density regions
have little temperature dependence making it more difficult to constrain the
region properties based on ejecta distributions than it would be for a higher
density. Differences in lower ($10$ MK) temperature and higher temperature
($\geq 20$ MK) can be clearly distinguished in high density plasma ($10^{10}$
cm-3). However, for any temperature plasma, there are clear spectral
differences between the low and high density cases.

In Figure 5, for the ejected fluence spectra at the collisional region
boundary, we plot the energy at which the fluence spectra peaks for each
studied plasma region varying in $L$, $T$ and $n_{e}$. We determine if such
properties are useful to help diagnose the plasma properties of the flare
‘acceleration’ region. At higher densities the energy at which the fluence
peaks drops significantly at all temperatures, though to a lesser degree in
hotter regions. Higher temperatures themselves increase this peak energy in
regions of equal density, with seemingly diminishing returns at $L\geq 20$ Mm
and $n_{e}\geq 5\times 10^{9}$ cm-3.

For each $L$, we see the general pattern of energy of the fluence peak
increasing with hotter plasma temperature and decreasing with increasing
plasma number density (with changes due to temperature more evident in higher
density regions). In this example, a peak fluence sitting at an energy of
$\sim 5$ keV could correspond to $T=20$ MK (high flare temperature), with $L$
ranging between $20-40$ Mm and $n$ between $5\times 10^{9}-1\times 10^{10}$
cm-3 (i.e., higher density). We will now extend the use of these diagnostics
into the heliosphere where such spectra can be observed.

Figure 5: Peak (in energy) of ejected electron fluence for every simulated
parameter set, with each window corresponding to a different region size. The
parameter set labels (x-axis) are formatted in order of region temperature and
electron density. Increasing the electron density of the region significantly
reduces the energy at which the fluence peaks for most region sizes, with a
greater degree of deceleration for colder plasmas.

###  3.2 Extending transport to the heliosphere

The distributions for energy, pitch angle, and ejection time generated by the
coronal component are taken as injection profiles and iterated out to 1.0 AU.
In order to draw an approximate equivalence with the energy resolution of
modern spacecraft in the heliosphere (and archived data), all data discussed
after this point will be displayed with 1 and 3 keV binning unless otherwise
stated.

Figure 6: Fluence (upper) and peak flux (lower) distributions of the ejected
population at 1.0 AU with a simulated energy binning of 3 keV for the minimum
and maximum collisional region parameters (shown in each legend). The fluence
and flux distributions peak at higher energies for hotter and sparser
collisional regions similar to the region boundary results, with the largest
divergence found between minimum and maximum temperature parameter sets.

Figure 6 shows the fluence and peak flux energy spectra at 1.0 AU for
independently varied values of $L$, $T$ and $n_{e}$ (similar to Figure 3).
There are minor differences between both distributions for a small and large
collisional region, suggesting that region size is difficult to constrain
within the selected range.

Here, the most significant difference is between colder and hotter regions,
with the maximum fluence and peak flux occurring at larger energies as the
temperature increases, as expected. It also shows a significant deviation
between the low and high temperature regions in the 10-19 keV range that is
not as readily seen when other parameters are varied instead. For 20 MK, the
density comparison demonstrates a similar (if weaker) pattern, with lower
density regions allowing more of the non-thermal injected population to escape
into (and propagate through) the heliosphere.

Figure 7: Fluence (upper) and peak-electron flux (lower) spectra of the
ejected population for intermediate collisional region plasma properties and
varying analytical parameters: the distance of the virtual measurement from
the Sun $R_{i}$ (left), the total integration time of the simulation
$\tau_{i}$ (middle) and the energy bin resolution $S$ (right). Other
analytical parameters are fixed at $R_{i}=1.0$ AU, $\tau_{i}=50,000$ s and
$S=3$ keV. The fluence is effectively unchanged across the heliosphere at the
maximum integration time with greater peak flux values at all energies closer
to the Sun. Shorter integration times show a significant reduction of fluence
at 1.0 AU. Bin sizes larger than $\sim 3$ keV mask the maximum values of peak
flux and fluence and make it difficult to identify features at the low-energy
ranges ($<$ 10 keV) of both distributions.

In addition to varying the physical parameters of the collisional region, we
can also vary the parameters of the analysis itself: accounting for the
position of a virtual instrument $R_{i}$, the total simulated runtime
$\tau_{i}$ and the width of the energy bins $S$. These are displayed in Figure
7, with fixed values of $R_{i}=1.0$ AU, $\tau_{i}=50,000$ s and $S=3$ keV
unless otherwise stated.

For the maximum simulated runtime, the fluence across the heliosphere remains
relatively unchanged as the majority of electrons will cross all values of
$R_{i}$ that are being compared. The peak flux on the other hand is
significantly higher closer to the Sun, though the shape of the distribution
remains approximately the same. Scattering effects in the heliosphere delay
transport along a 1D line out to 1.0 AU so while the energy values are
unchanged and will arrive at every point along the heliosphere eventually, the
rate of arrival is spread out resulting in lower peak flux.

Reducing the simulated runtime shows a lower number of electrons arriving at
1.0 AU, reducing the overall fluence. The lowest energy electrons are expected
to arrive latest so reducing $\tau_{i}$ steadily will show a much faster
decrease in low energy fluence first. This is also reflected in the flux peak
plot though to a lesser extent.

Enlarging the bin size results in a higher fluence and peak flux per bin, as
naturally more energy is collected within a wider range. However this has the
added effect of obfuscating the location of the spectral peaks. Sufficiently
high resolution energy measurement would be required to identify and measure
this feature ($\sim<3$ keV binning at energies $<20$ keV).

Figure 8: Full collisional-region parameter analysis at 0.4 AU (upper) and 1.0
AU (lower). The energies where the maximum fluence (left) and peak flux
(right) occur are shown for 3 keV (thin line) and 1 keV (thick line)
resolutions. Ratios of the [$7-10$) to [$4-7$) keV bin heights for both
fluence (circles) and flux (crosses) are calculated and displayed in the right
column. While the size of some 1 and 3 keV bins mask the change in peaks
(fluence and peak flux) between parameter sets for regions with temperatures
above 10 MK, the bin height ratios demonstrate the same relation for a
sufficiently long integration time.

Compiled in Figure 8 are the energies where the maximum values of fluence and
peak flux occur, for electron distributions at two heliospheric positions (0.4
AU and 1.0 AU) and integration times (10,000 s and 50,000 s). We also consider
how they would manifest in 1 and 3 keV binning schemes comparable to modern
spacecraft instrumentation. In addition, we calculate the ratio of the
[$7-10$) to [$4-7$) keV bin heights for each parameter set. With the exception
of the $R_{i}=1.0$ AU, $\tau_{i}=10,000$ s case, the peaks and ratios show
relatively ‘settled’ cases where the majority of electrons have reached
$R_{i}$ and little variation can be seen between these parameter sets. At high
$R_{i}$ and low $\tau_{i}$ the maxima (fluences and peak fluxes) are biased
towards higher energies that have been able to reach 1.0 AU in the reduced
integration time. In addition, this is the only case where the fluence bin
ratios exceed those of the flux ratios, indicating that the peak flux ratios
are identifiable within a shorter simulation runtime than the fluence ratios.

With regards to the collisional region parameters, there are clear patterns
shown in both the peaks and ratios. For the high-resolution 1-keV-binned peak
cases, it can be seen that increasing the temperature of the region results in
a fluence and flux peaks occurring at a higher energy at both 0.4 and 1.0 AU,
though there is relatively little difference when increasing the density at
high temperatures.

Increasing the density at lower temperatures on the other hand shows a
significant drop in peak fluence and flux energies from $1\times 10^{9}$ cm-3
to $5\times 10^{9}$ cm-3. This indicates that a cold ($<$ 30 MK) and dense
($>$ 1x109 cm-3) collisional region is an effective decelerator of energetic
electrons and evident even out at 1.0 AU.

These patterns are somewhat more difficult to discern at 3 keV binning; with
the exception of the lowest temperature parameter sets, the high-resolution
peaks occupy the same [$4-7$) keV bin. Instead we compare the relative heights
of the [$7-10$) and [$4-7$) keV bins for fluence and peak flux, which show the
same relations as the higher-resolution maxima. As such, in the absence of
sufficiently high-resolution energy binning these ratios are our preferred
metric for constraining collisional region properties using heliospheric data.

##  4 Summary and Discussion

Locating the acceleration region is fundamental in understanding how and where
energetic particles are produced in flares, as well as the relationship
between different particle populations observed at the Sun and in
interplanetary space (e.g., plasma environment, magnetic topology). Thus,
examining whether signatures of hot, over-dense plasma exist in interplanetary
electron spectra is vital. This analysis is concerned with so-called ‘prompt’
events where energetic electrons are evidenced to be directly related to the
flare and not to any secondary acceleration mechanism such as a CME shock.
Previous studies provide conflicting evidence regarding where in-situ
electrons are energised compared to their HXR-emitting counterparts. In
certain flares, for example, emerging flux or interchange reconnection (e.g.,
Battaglia et al., 2023; Wang et al., 2016; Heyvaerts et al., 1977) and/or the
presence of a flare-jet structure (e.g., Musset et al., 2020; Krucker et al.,
2011; Bain & Fletcher, 2009) may ultimately allow loop-top electrons to
escape, while having access to hot, over-dense material usually related to
HXR-emitting electrons. Alternatively, or in conjunction, the presence of a
turbulent acceleration mechanism (e.g., Stores et al., 2021; Kontar et al.,
2017a) may both heat the surrounding plasma and energise electrons
simultaneously.

Here, we analysed how the plasma properties of a hot, over-dense coronal
region (including or close to an acceleration region) change the properties of
$\approx 1-20$ keV electrons and examine whether the signature of such plasma
properties can be extracted from the electron fluence or peak flux spectrum.
In this study, we only present very specific or ‘extreme’ plasma examples that
are easily identifiable in the electron spectrum, unlike lower temperatures
($<10$ MK) and densities ($<10^{9}$ cm-3), possibly in higher altitude
locations away from flare site producing HXR-emitting electrons.

The results suggest that if interplanetary flare-related electrons are indeed
produced in hot, over-dense regions, possibly during continued accelerated in
flares (similar to HXR-producing electrons) and within a magnetic topology
that allows the escape of such electrons, then we should be able to detect
these signatures, and even constrain the plasma properties of such a region.
We tested how different bin sizes used for plotting electron fluence or peak
flux affected the extraction of such properties, and how electron loss at
different locations in the heliosphere can also hide these signatures. Thus,
the lack of such ‘thermal’ signatures would suggest that such particles indeed
originate from ‘higher’ coronal locations, unrelated to the acceleration
region producing HXR-emitting electrons. In order to diagnose these
properties, a detailed individual flare study would have to be performed
(including a comparison of in-situ and HXR-emitting electrons), beyond the
scope of this initial modelling study.

Properties of an ‘acceleration’ region can be more narrowly constrained by
identifying signatures of the ‘thermal’ ($<20$ keV) electron datasets. Energy
of peak fluence and peak flux bands at resolutions of 1 keV both vary
significantly between different parameters sets. If these features exist, then
these properties can be readily extracted from modern in-situ instruments at
comparable resolutions, meaning that such constraining is currently possible
with existing and upcoming observational data. Such information will be
extracted by carefully examining individual flares spectra with simulation
outputs. The prospect of different plasma conditions changing the properties
of an electron population (accelerated out of the thermal plasma) at higher
energies (i.e., spectral index) will be studied in detail in Paper II.

The model we used in this preliminary analysis is rather artificial; a chosen
accelerated electron distribution (single power law) is injected and
transported through a hot over-dense region. Paper II will perform more
realistic simulation where electrons are accelerated out of different thermal
plasma using a turbulent diffusion model (similar to Stores et al., 2023),
with the resulting shape of the spectra also dependent on the plasma
properties of the acceleration region, and producing a smoother spectral
transition from thermal to non-thermal if this region exists within the $1-20$
keV data. As mentioned, we purposely used higher temperatures and higher
densities akin to hot flaring loops to show the types of spectra and
diagnostics we may expect to see if such electrons are being produced in
regions identical to HXR-emitting electrons. Moreover, we did not exhaustively
input a range of electron spectral indices or low energy cutoffs into the
simulation. We investigated the effect of independently decreasing the minimum
initial energy (down to 3 keV) and increasing the spectral index of the
injected distribution (up to $\delta=5$). The result of both changes was a
slight decrease in the energy of peak fluence at 1.0 AU, by 0.5 to 0.7 keV.
There is a similar drop in the [$7-10$):[$4-7$) keV bin ratios, by 0.1 to 0.2.
These indicate that making such changes to the injected distribution will
result in the fluence distribution at 1.0 AU shifting slightly towards lower
energies.

As discussed earlier, Kontar & Reid (2009) investigated the transport of solar
flare energetic electrons in the heliosphere taking into account the self-
consistent generation and absorption of Langmuir waves, effects of nonuniform
plasma, collisions, and Landau damping, and found that such processes lead to
a spectral break and flattening below approximately 40 keV, acting similar to
collisional effects in dense plasma, leading to an overall flatter spectra at
lower energies. For our study, modelling is required to analyse the effects
fully. However, we suggest that such processes should lead to flatter spectra
in the $1-20$ keV range, with any visible peak due to a (partially-)thermal
component possibly shifted to lower energies.

In real data the flare-accelerated population must be separated from the
different populations of the solar wind (core, halo and super-halo) (e.g.,
Feldman et al., 1975; Pierrard et al., 2001; Wang et al., 2012). Although
flares are evidenced by their increased electron flux compared to the
background, at $\approx 1$ keV the flare-spectra will start to merge back into
the solar wind halo and core (e.g., Pan et al., 1984; Lin, 1985; Wang, 2022)
and produce a spectral upturn close to merging point at $\approx 1$ keV. Thus
as above, we suggest the $1-20$ keV range is the optimal range for such
analysis. Upturns in the spectral data within the $1-20$ keV suggest the
presence of a hotter thermal component. Interestingly, in a rare published
example of electron peak intensity energy spectra i.e., (Jebaraj et al., 2023)
produced by the SolO Energetic Particle Detector (STEP, EPT and HET)
(Rodríguez-Pacheco et al., 2020) we see a flattening i.e., Kontar & Reid
(2009) but then a noticeable upturn in the energy spectrum at lower energies
(at around $10-20$ keV), which may indicate the signature of the accelerating
plasma environment. Nevertheless, this provides an excellent example of why
this spectral region deserves more attention and analysis.

## acknowledgments

RP & NLSJ gratefully acknowledge the current financial support from the
Science and Technology Facilities Council (STFC) Grant ST/V000764/1. The
authors acknowledge IDL support provided by STFC. NLSJ is supported by an
international team grant “Measuring Solar Flare HXR Directivity using
Stereoscopic Observations with SolO/STIX and X-ray Instrumentation at Earth”
from the International Space Sciences Institute (ISSI) Bern, Switzerland. The
data that support the findings of this study are available from the
corresponding author upon reasonable request. We thank Professor Eduard Kontar
for insightful comments.

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# Continuous variable quantum state tomography of photoelectrons

H. Laurell<EMAIL_ADDRESS>Department of Physics, Lund University,
Box 118, 22100 Lund, Sweden  D. Finkelstein-Shapiro  Instituto de Quimica,
Universidad Nacional Autonoma de Mexico, Circuito Exterior, Ciudad
Universitaria, Alcaldía Coyoacán C.P. 04510, Ciudad de Mexico  C. Dittel
Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-
Straße 3, 79104 Freiburg, Germany  EUCOR Centre for Quantum Science and
Quantum Computing, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße
3, 79104 Freiburg, Germany  C. Guo  Department of Physics, Lund University,
Box 118, 22100 Lund, Sweden  R. Demjaha  Department of Physics, Lund
University, Box 118, 22100 Lund, Sweden  M. Ammitzböll  Department of Physics,
Lund University, Box 118, 22100 Lund, Sweden  R. Weissenbilder  Department of
Physics, Lund University, Box 118, 22100 Lund, Sweden  L. Neoričić  Department
of Physics, Lund University, Box 118, 22100 Lund, Sweden  S. Luo  Department
of Physics, Lund University, Box 118, 22100 Lund, Sweden  M. Gisselbrecht
Department of Physics, Lund University, Box 118, 22100 Lund, Sweden  C. L.
Arnold  Department of Physics, Lund University, Box 118, 22100 Lund, Sweden
A. Buchleitner  Physikalisches Institut, Albert-Ludwigs-Universität Freiburg,
Hermann-Herder-Straße 3, 79104 Freiburg, Germany  EUCOR Centre for Quantum
Science and Quantum Computing, Albert-Ludwigs-Universität Freiburg, Hermann-
Herder-Straße 3, 79104 Freiburg, Germany  T. Pullerits  Chemical Physics and
NanoLund, Lund University, Box 124, 22100 Lund, Sweden  A. L’Huillier
Department of Physics, Lund University, Box 118, 22100 Lund, Sweden  D. Busto
<EMAIL_ADDRESS>Department of Physics, Lund University, Box 118,
22100 Lund, Sweden  Physikalisches Institut, Albert-Ludwigs-Universität
Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany

###### Abstract

We propose a continuous variable quantum state tomography protocol of
electrons which result from the ionization of atoms or molecules by the
absorption of extreme ultraviolet light pulses. Our protocol is benchmarked
against a direct calculation of the quantum state of photoelectrons ejected
from helium and argon in the vicinity of a Fano resonance. In the latter case,
we furthermore distill ion-photoelectron entanglement due to spin-orbit
splitting. This opens new routes towards the investigation of quantum
coherence and entanglement properties on the ultrafast timescale.

Thanks to the discovery of high order harmonic generation McPherson  _et al._
(1987); Ferray  _et al._ (1988), attosecond light sources were developed,
enabling the study of electron dynamics with high temporal resolution Krausz
and Ivanov (2009). By energy-time uncertainty, such attosecond light pulses,
with a central frequency in the extreme ultraviolet (XUV) range, have broad
spectral widths. Hence, their interaction with matter usually results in a
photoionization process where the ejected electron populates a broad
distribution of continuum states. The resulting electronic state may be either
pure or mixed.

The first experimental methods developed for the characterisation of
attosecond pulses relied on the coherence of the photoionization process. The
Reconstruction of Attosecond Beating By Interference of Two photon transitions
(RABBIT) Paul  _et al._ (2001), as well as attosecond streaking Hentschel  _et
al._ (2001), were initially invented to characterize the temporal properties
of attosecond light pulses. The same techniques were then applied to determine
time delays in the photoionization process Klünder  _et al._ (2011); Schultze
_et al._ (2010). More recently RABBIT was used to measure the spectral
amplitude and phase of photoelectrons in the vicinity of Fano resonances Kotur
_et al._ (2016); Gruson  _et al._ (2016); Busto  _et al._ (2018).

In general, these characterization methods are readily applicable to pure
quantum states. For mixed states of the ejected electrons, which must be
described by a density operator rather than by a state vector in Hilbert
space, they are unsuitable. Mixed states occur due to several causes, such as
decoherence processes or incomplete measurements of entangled particles or
degrees of freedom. Decoherence due to interactions with an environment can be
neglected on attosecond and few femtosecond time scales. In contrast, strong
coupling, e.g., between electronic and nuclear degrees of freedom in molecular
systems Pabst  _et al._ (2011); Nishi  _et al._ (2019); Arnold  _et al._
(2020); Vrakking (2021) has the potential to induce mixing when only the
electron’s (or only the ion’s) degrees of freedom are interrogated.

For the characterization of mixed quantum states, the gold standard is
_quantum state tomography_ (QST) Lvovsky and Raymer (2009). It aims at
reconstructing an unknown state from a series of projective measurements which
yield the state density operator – its most general quantum description – and
is widely used in quantum optics. QST has also been applied successfully in
specific instances of multidimensional spectroscopy, where pairs of coherent
pulses are used to extract the populations and coherences of the states Yuen-
Zhou  _et al._ (2011, 2014). QST has only recently been applied in attosecond
science. Trains of photoelectrons have been characterized by discrete variable
quantum state tomography, using SQUIRRELS (Spectral Quantum Interference for
the Regularized Reconstruction of free-Electron States) Priebe  _et al._
(2017) in the context of electron microscopy, and using Mixed-FROG (Frequency
Resolved Optical Gating) Bourassin-Bouchet  _et al._ (2020) for photoelectrons
created by absorption of attosecond pulse trains. Both methods rely on a
retrieval algorithm to reconstruct the photoelectron quantum state from the
measured spectrogram.

Here we propose a robust tomography protocol – Kvanttillstånds tomogRafi av
AttoseKund ElektroNvågpaket (KRAKEN, engl. “quantum state tomography of
attosecond electron wavepackets”) – that can reconstruct the photoelectron’s
quantum state without relying on a retrieval algorithm. This method, in
contrast to traditional coherent two-dimensional spectroscopy Hamm and Zanni
(2011), is based on two synchronized narrowband infrared (IR) probe fields at
different central frequencies, delayed relative to the extreme ultraviolet
(XUV) field, which creates the photoelectron. The measurements consist in
recording the photoelectron spectrum as a function of delay and frequency
difference of the probe fields. Our method allows the reconstruction of the
photoelectron’s continuous variable density matrix with, in principle,
arbitrary spectral resolution. We numerically demonstrate the KRAKEN protocol
for both pure and mixed photoelectron states, in the vicinity of autoionizing
resonances in helium and argon, and compare our results to direct calculations
of the density matrix. In the case of argon we further apply our protocol to
quantify the entanglement between ion and photoelectron.

Density matrix formulation of photoionization. Let us consider the ionization
of an atom or molecule by an XUV light pulse, creating an electron in the
continuum. We first consider a single angular momentum channel, e.g.
$1\text{s}\rightarrow\epsilon\text{p}$, so that the final state is fully
described by its energy $\epsilon$. We use the density matrix formalism, which
allows a description of mixed photoelectron states. These could arise from
experimental imperfections such as partially incoherent XUV light sources,
interactions with a fluctuating environment where different electron/ion pairs
evolve under slightly different Hamiltonians, or from an incomplete
measurement of the degrees of freedom of an entangled state, e.g. between ion
and electron.

We concentrate on the latter case, i.e. a state which exhibits entanglement
between electron and ion, while assuming a fully coherent XUV light pulse and
vanishing interactions with the environment. In this case, the quantum state
of the ion and photoelectron after the ionization is pure, and can be written
as a coherent superposition of different orthonormal ion $\ket{j}$ and
photoelectron $\ket{\epsilon}$ states,

$\displaystyle\ket{\Psi_{\mathrm{IE}}}=\sum_{j}\int\mathrm{d}\epsilon~{}c_{j}(\epsilon)\ket{j;\epsilon},$
(1)

where $\sum_{j}\int\mathrm{d}\epsilon~{}|c_{j}(\epsilon)|^{2}=1$. An
experiment that only measures the ionized electron is fully described by the
electron quantum state, that is, the reduced density matrix
${\rho}_{\mathrm{xuv}}$ obtained by tracing over the ionic degrees of freedom,

$\displaystyle{\rho}_{\mathrm{xuv}}$
$\displaystyle=\mathrm{tr}_{\mathrm{I}}\left(\ket{\Psi_{\mathrm{IE}}}\bra{\Psi_{\mathrm{IE}}}\right)$
(2) $\displaystyle=\int\
\mathrm{d}\epsilon_{1}\mathrm{d}\epsilon_{2}\sum_{j}c_{j}(\epsilon_{1})c_{j}^{*}(\epsilon_{2})\ket{\epsilon_{1}}\bra{\epsilon_{2}},$
(3)

with matrix elements
$\rho_{\mathrm{xuv}}(\epsilon_{1},\epsilon_{2})=\sum_{j}c_{j}(\epsilon_{1})c_{j}^{*}(\epsilon_{2})$.
Its purity is given by

$\displaystyle\mathrm{tr}({\rho}_{\mathrm{xuv}}^{2})$ $\displaystyle=\int\
\mathrm{d}\epsilon_{1}\mathrm{d}\epsilon_{2}\
\left|\sum_{j}c_{j}(\epsilon_{1})c^{*}_{j}(\epsilon_{2})\right|^{2}$ (4)

and is equal to one if and only if photoelectron and ion are in a separable
state, i.e., if the coefficients in Eq. (3) factorize,
$c_{j}(\epsilon)=a_{j}b(\epsilon)$, with
$\sum_{j}|a_{j}|^{2}=\int\mathrm{d}\epsilon\,|b(\epsilon)|^{2}=1$. Otherwise,
$\mathrm{tr}({\rho}_{\mathrm{xuv}}^{2})<1$. In the former case, the
photoelectron state is pure, in the latter, it is mixed. Therefore, the purity
of one party of the pure bipartite state (1) characterizes both parties’
entanglement Nielsen and Chuang (2002).

KRAKEN protocol. Without any assumptions on the quantum state of the electron
except for being a stationary state, we now show how ${\rho}_{\mathrm{xuv}}$
can be reconstructed by KRAKEN. As depicted in Fig. 1(a), we use a broadband
XUV light pulse with central frequency $\Omega$ and spectral width
$\delta\Omega$ to ionize the target atom. Two synchronised, spectrally narrow
probe pulses with central frequencies $\omega_{1}$ and $\omega_{2}$
subsequently probe the photoelectron. The absorption or emission of IR photons
leads to the formation of additional peaks in the photoelectron spectrum. The
use of a bichromatic IR pulse couples different continuum states within the
photoelectron’s bandwidth and makes them interfere in the final state.
Measurements of the resulting photoelectron spectra, as a function of the
delay between the XUV pulse and the bichromatic IR field, allow us to
characterize the coherences (i.e., off-diagonal elements) of
${\rho}_{\mathrm{xuv}}$, between states with energy difference
$\delta\omega=\omega_{1}-\omega_{2}$. A scan over $\delta\omega$, e.g. by
keeping $\omega_{1}$ constant and scanning $\omega_{2}$ from $\omega_{1}$ to
$\omega_{1}+\delta\Omega$, completes the full reconstruction of
${\rho}_{\mathrm{xuv}}$.

Figure 1: (a) Energy diagram of the KRAKEN protocol. The photoelectron is
created by absorption of a broadband harmonic with central frequency $\Omega$
and width $\delta\Omega$. The photoelectron’s coherence
$\rho_{\mathrm{xuv}}(\epsilon_{1},\epsilon_{2})$ between the continuum states
$\ket{\epsilon_{1}}$ and $\ket{\epsilon_{2}}$ is interrogated via simultaneous
coherent population transfer from $\ket{\epsilon_{1}}$ and
$\ket{\epsilon_{2}}$ into the continuum state $\ket{\epsilon_{f}}$. The
population of the latter is modulated by the phase-dependent interference term
of the two amplitudes mediated by the bichromatic IR probe’s components
$\omega_{2}$ and $\omega_{1}$, see Eq. (6).
$\rho_{\mathrm{xuv}}(\epsilon_{1},\epsilon_{2})$ is fully mapped out by
scanning $\delta\omega=\omega_{2}-\omega_{1}$ over a suitably chosen energy
range in the continuum. Panels b) and c) show the energy diagrams for the
helium 2s2p (top) and argon 3s-14p (bottom) Fano resonances, respectively.

To model the KRAKEN’s action upon ${\rho}_{\text{xuv}}$, we solve the
Liouville-von Neumann equation in first order perturbation theory (see Suppl.
Mat. (SM) for details). Assuming monochromatic probe fields, we introduce the
photoelectron’s density operator ${\rho}_{\mathrm{xuv+ir}}$ after absorbing
one photon from the bichromatic IR field (see Fig. 1(a)). For a given delay
$\tau$ between the XUV and IR fields, and frequency difference $\delta\omega$
between the spectral components of the bichromatic IR field, the probability
to find a photoelectron with energy $\epsilon_{f}$ is given by

$\begin{gathered}S(\epsilon_{f},\tau,\delta\omega)=\bra{\epsilon_{f}}{\rho}_{\text{xuv+ir}}(\tau,\delta\omega)\ket{\epsilon_{f}}.\end{gathered}$
(5)

$S(\epsilon_{f},\tau,\delta\omega)$ can be expressed in terms of the matrix
elements of ${\rho}_{\text{xuv}}$ as

$\displaystyle S(\epsilon_{f},\tau,\delta\omega)$
$\displaystyle\approx|\mu_{\epsilon_{f},\epsilon_{1}}|^{2}\rho_{\mathrm{xuv}}(\epsilon_{1},\epsilon_{1})+|\mu_{\epsilon_{f},\epsilon_{2}}|^{2}\rho_{\mathrm{xuv}}(\epsilon_{2},\epsilon_{2})$
(6)
$\displaystyle+e^{i\delta\omega\tau}\mu_{\epsilon_{f},\epsilon_{1}}\mu^{*}_{\epsilon_{f},\epsilon_{2}}\rho_{\mathrm{xuv}}(\epsilon_{1},\epsilon_{2})$
$\displaystyle+e^{-i\delta\omega\tau}\mu_{\epsilon_{f},\epsilon_{2}}\mu^{*}_{\epsilon_{f},\epsilon_{1}}\rho_{\mathrm{xuv}}(\epsilon_{2},\epsilon_{1}),$

where $\epsilon_{i}=\epsilon_{f}-\hbar\omega_{i}$ ($i={1,2}$), and
$\mu_{\epsilon_{f},\epsilon_{i}}$ are the dipole transition matrix elements
between the continuum states $\ket{\epsilon_{i}}$ and $\ket{\epsilon_{f}}$
[Fig. 1(a)]. The first two terms correspond to populations, while the last two
terms describe coherences. To extract the coherences, we Fourier transform
$S(\epsilon_{f},\tau,\delta\omega)$ with respect to $\tau$ and extract the
components oscillating at $\pm\delta\omega$, assuming that the dipole
transition matrix elements $\mu_{\epsilon_{f},\epsilon_{i}}$ between continuum
states are constant across $\delta\Omega$.

Impact of Fano resonances. As a final preparation for the analysis of our
results, we now inspect the impact of the atomic/ionic structure on the
photoelectron’s spectral density in the continuum. We examine two cases
extensively studied using RABBIT Kotur  _et al._ (2016); Gruson  _et al._
(2016); Busto  _et al._ (2018); Turconi  _et al._ (2020); Cirelli  _et al._
(2018), the 2s2p resonance in helium and the 3s-14p in argon as shown in
Figure 1(b,c), and use Fano’s description of an autoionizing resonance with
energy $\epsilon_{r}$ and width $\Gamma_{r}$. In helium, there is only one
ionic state, and Eq. (1) reads Fano (1961)

$\displaystyle\ket{\Psi_{\mathrm{IE}}}=\int\mathrm{d}\epsilon~{}E_{\text{XUV}}(\epsilon)\frac{\Delta(\epsilon)+q}{\Delta(\epsilon)+\text{i}}\ket{{}^{2}S_{1/2};\epsilon},$
(7)

where $\Delta(\epsilon)=2(\epsilon-\epsilon_{r})/\Gamma_{r}$ is the reduced
energy, $E_{\text{XUV}}(\epsilon)$ the amplitude of the XUV pulse and $q$ the
asymmetry parameter proportional to the ratio between the transition matrix
elements towards the quasi-bound state at $\epsilon_{r}$ and the continuum
state at $\epsilon$, respectively.

The case of the 3s-14p resonance in argon is more complicated for two reasons:
there are two ionic states $j\in\\{^{2}P_{1/2},^{2}P_{3/2}\\}$ separated by an
energy $\epsilon_{\text{so}}$ [see Fig. 1(c)] and the photoelectron can have
two angular momenta $\lambda\in\\{0,2\\}$. Consequently, Eq. (1) generalizes
to

$\displaystyle\ket{\Psi_{\mathrm{IE}}}=\sum_{j,\alpha}\int\mathrm{d}\epsilon~{}c_{j,\alpha}(\epsilon)\ket{j;\alpha,\epsilon},$
(8)

where $\alpha$ represents the electron quantum numbers other than its energy,
and

$\displaystyle
c_{j,\alpha}(\epsilon)=E_{\text{XUV}}(\epsilon)\frac{\Delta_{\alpha}(\epsilon)+q_{\alpha}}{\Delta_{\alpha}(\epsilon)+i}\,.$
(9)

This latter expression has to be substituted for the $c_{j}$ in (3) (with a
trivial dependence on $\alpha$ in the case of a helium target), to provide a
realistic description of ${\rho}_{\mathrm{xuv}}$.

Figs. 2(a-d) show the modulus and the phase portrait of
${\rho}_{\mathrm{xuv}}$ as given by (3) with (9), in helium (a,b) or argon
(c,d). While the helium photoelectron’s state has roughly circular support in
the energy plane, the argon state lives on a rather elliptic domain,
indicative of coherences between energetically distant states being
suppressed. Furthermore, the Fano structure induced by the autoionizing state
decaying towards one (helium – a,b) or two (argon – c,d) non degenerate ionic
states is clearly reflected by the sharp, cross-like structure(s) both in the
modulus as well as in the phase of ${\rho}_{\mathrm{xuv}}$.

Figure 2:  Intensity ($|\rho_{\mathrm{xuv}}(\epsilon_{1},\epsilon_{2})|$) and
phase profiles of the photoelectron’s (reduced) state $\rho_{\mathrm{xuv}}$ as
obtained by direct evaluation of (3) with (9) (upper panels), compared to the
output of a simulation of the KRAKEN protocol, based on the observable (6)
(bottom). The (double) cross-like structures featured by both,
$|\rho_{\mathrm{xuv}}(\epsilon_{1},\epsilon_{2})|$ and the phase, are
manifestations of the Fano profiles modulating the $c_{j}$ in (1,3,8)
according to (9), induced by the $\mathrm{2s2p}$ ($3s^{-1}4p$) resonance of
helium (argon). Black areas in (e-h) were not mapped out in our KRAKEN
simulation.

The above portraits of ${\rho}_{\mathrm{xuv}}$ are now to be compared to the
spectrograms produced by KRAKEN – which are calculated using an analytical
model for resonant two-photon transitions Jiménez-Galán  _et al._
(2016)111Note that the assumption that the continuum-continuum matrix elements
are constant across $\delta\Omega$ is still valid in the presence of the Fano
resonance in the intermediate state Jiménez-Galán  _et al._ (2016). (further
details in the SM). We choose an XUV field with a central photon energy of
60.25 eV in helium (26.70 eV in argon). In both cases, the XUV bandwidth is
0.25 eV (full width at half maximum), and the IR bandwidth is set to 1.5 nm in
helium (3 nm in argon). We vary the pulse delay between -200 fs to 200 fs in
helium (-400 fs to 400 fs in argon) with a step size of 3 fs, and the
frequency difference $\delta\omega$ is varied in steps of 0.01 eV.

We calculate spectrograms for a broad range of $\delta\omega$, to map out the
subdiagonals of the reconstructed density matrix
${\rho}_{\mathrm{xuv}}^{\text{rec}}$,and smoothen the result by a moving local
average (further details on the reconstruction procedure in the SM). The
results are shown in Figs. 2 (e,f) for helium, and (g,h) for argon,
respectively, and, by visual inspection, agree well with the direct evaluation
of (3,9) in panels (a-d). To validate this impression, we evaluate the
fidelity of ${\rho}_{\mathrm{xuv}}$ to ${\rho}_{\mathrm{xuv}}^{\mathrm{rec}}$,
according to Nielsen and Chuang (2002)

$\displaystyle
F({\rho}_{\mathrm{xuv}},{\rho}_{\mathrm{xuv}}^{\mathrm{rec}})=\mathrm{tr}\left[\left({\rho}_{\mathrm{xuv}}^{1/2}{\rho}_{\mathrm{xuv}}^{\mathrm{rec}}{\rho}_{\mathrm{xuv}}^{1/2}\right)^{1/2}\right]\,.$
(10)

We find $F=0.9997$ for helium ($0.9718$ for argon). The minor deviation from
unity is mainly caused by errors stemming from finite bandwidths of the two IR
probe pulses. The presence of two intermediate angular momentum channels
explains the slightly larger mismatch for argon.

Ion-electron entanglement. Finally, we investigate how to control and measure
the degree of electron-ion entanglement in argon, by varying the bandwidth of
the XUV radiation. To this end, recall that for pure bipartite states (1) of
electron and ion, the purity $\mathrm{tr}({\rho}_{\mathrm{xuv}}^{2})$ of the
photoelectron (reduced) state ${\rho}_{\mathrm{xuv}}$ decreases with
increasing entanglement (this simply expresses the loss of information upon
disregarding the ionic degree of freedom). The purity is related to the
concurrence, a measure of entanglement, through Mintert  _et al._ (2005)

$\displaystyle\mathcal{C}=\sqrt{2[1-\mathrm{tr}({\rho}_{\mathrm{xuv}}^{2})]}\,.$
(11)

Both quantities are plotted in Fig. 3(a) as function of the XUV bandwidth.
Solid lines result from our direct evaluation of (3), while crosses are
obtained from our simulation of the KRAKEN protocol. We see that entanglement
(purity) decreases (increases) with increasing XUV bandwidth. For XUV
bandwidths smaller than the spin-orbit splitting
$\delta\Omega<\epsilon_{\text{so}}$, the two spin-orbit components
${}^{2}P_{1/2}$ and ${}^{2}P_{3/2}$ of the ion are spectrally fully resolved,
such that there is maximal entanglement between ion and photoelectron [left of
the dashed line in Fig. 3(a)], and the detection of an electron with a given
kinetic energy allows us to determine the ionic state with certainty. On the
other hand, when the XUV bandwidth is increased, the two photoelectron spectra
associated with the spin-orbit components start to overlap. Hence, the overlap
region reveals no information about the ionic state. As a result, the degree
of entanglement between photoelectron and ion decreases. In the limit
$\delta\Omega\gg\epsilon_{\text{so}}$, the two photoelectron spectra fully
overlap, such that the energy of the electron does not reveal any information
on the ionic state and there is no entanglement.

Figure 3: Entanglement between ion and photoelectron in argon. In the top
panel (a), blue (red) solid lines show the directly calculated purity
$\mathrm{tr}({\rho}_{\mathrm{xuv}}^{2})$ (concurrence $\mathcal{C}$) as a
function of the XUV bandwidth, and blue (red) crosses correspond to the values
reconstructed via the KRAKEN protocol. The vertical black dashed line
indicates half of the spin-orbit splitting energy $\epsilon_{\mathrm{so}}$ for
the $3s^{-1}4p$ Fano resonance in argon, such that bandwidths smaller than
$\epsilon_{\mathrm{so}}/2$ resolve the spin-orbit states. On the bottom, the
modulus of the photoelectron state
$|\rho_{\mathrm{xuv}}(\epsilon_{1},\epsilon_{2})|$ as given by (3,9) is shown
for XUV bandwidths $\delta\Omega=0.14\,\mathrm{eV}$ (b),
$\delta\Omega=0.21\,\mathrm{eV}$ (c), and $\delta\Omega=0.35\,\mathrm{eV}$
(d).

To be more precise, in our model the reduced density matrix of the
photoelectron can be expressed as Turconi  _et al._ (2020)

${\rho}_{\text{xuv}}=\tfrac{1}{3}{\rho}_{1/2}+\tfrac{2}{3}{\rho}_{3/2},$ (12)

where ${\rho}_{1/2}$ and ${\rho}_{3/2}$ are identical, pure density matrices
shifted in energy by $\epsilon_{\text{so}}$, according to
$\rho_{1/2}(\epsilon_{1},\epsilon_{2})=\rho_{3/2}(\epsilon_{1}-\epsilon_{\text{so}},\epsilon_{2}-\epsilon_{\text{so}})$.
$\mathrm{tr}({\rho}_{1/2}^{2})=\mathrm{tr}({\rho}_{3/2}^{2})=1$ implies

$\displaystyle\mathrm{tr}({\rho}_{\mathrm{xuv}}^{2})=\tfrac{5}{9}+\tfrac{4}{9}\operatorname{tr}({\rho}_{1/2}{\rho}_{3/2}).$
(13)

For non-overlapping photoelectron spectra, we have
$\mathrm{tr}({\rho}_{1/2}{\rho}_{3/2})=0$, and the purity in (13) becomes
minimal, $\mathrm{tr}({\rho}_{\mathrm{xuv}})=5/9$, indicating maximal
entanglement, $\mathcal{C}=2\sqrt{2}/3\approx 0.94$. On the other hand, for
fully overlapping spectra, $\mathrm{tr}({\rho}_{1/2}{\rho}_{3/2})=1$, the
purity reaches its maximum, $\mathrm{tr}({\rho}_{\mathrm{xuv}}^{2})=1$, so
that there is no entanglement between ion and photoelectron, $\mathcal{C}=0$.

The decrease of entanglement with increasing XUV bandwidth is also visible in
the amplitude of the density matrix elements. As shown in Figs. 3(b)-(d), when
the XUV bandwidth is increased, the support of ${\rho}_{\mathrm{xuv}}$ expands
into the energy plane, since the two spin-orbit states have larger overlap. An
expanding support of ${\rho}_{\mathrm{xuv}}$ is tantamount of larger
coherences inscribed into the reduced state of the photoelectron, which
amounts to decreasing entanglement with the mother ion.

In conclusion, we presented a continuous variable quantum state tomography
protocol – KRAKEN – to characterize photoelectrons created by absorption of
XUV pulses. We tested KRAKEN in the vicinity of Fano resonances of helium and
argon. In both cases, a comparison of numerical simulations of KRAKEN output
with direct calculations of the density matrix showed excellent agreement. For
the Fano resonance in argon, we further showed how to control and measure the
degree of entanglement between the ion and the photoelectron by varying the
bandwidth of the XUV pulse. Let us stress that the proposed tomography
protocol is not restricted to Fano resonances. It is generally applicable to
photoelectrons created by absorption of high-order harmonics or free electron
laser pulses of femtosecond or attosecond duration. While we here demonstrated
that KRAKEN can be used to study entanglement properties between ion and
photoelectron due to spin-orbit interactions, it can likewise be used to
investigate other bipartite entanglement processes, e.g. regarding electron-
ion entanglement in vibrating molecules Vrakking (2021); Koll  _et al._
(2022). This opens up new routes for uncovering the role of entanglement in
atomic and molecular physics Tichy  _et al._ (2011).

This work was supported by the Swedish Research Council, the European Research
Council and the Knut and Alice Wallenberg Foundation. D.F.S acknowledges
funding from PAPIIT No. IA202821. C.D. acknowledges financial support by the
Georg H. Endress foundation. A.L. is partly supported by the Wallenberg Center
for Quantum Technologies (WACQT), funded by the Alice and Wallenberg
Foundation.

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