# On Ultrafast X-ray Methods for Magnetism

R. Plumley  Stanford Institute for Materials and Energy Science, Stanford
University and SLAC National Accelerator Laboratory, Menlo Park, California
94025  Linac Coherent Light Source, SLAC National Accelerator Laboratory,
Menlo Park, CA 94720  Physics Dept., Carnegie Mellon University, Pittsburgh,
PA 15213  S. R. Chitturi  Linac Coherent Light Source, SLAC National
Accelerator Laboratory, Menlo Park, CA 94720  Department of Materials Science
and Engineering, Stanford University, Stanford, CA 94305  C. Peng  Stanford
Institute for Materials and Energy Science, Stanford University and SLAC
National Accelerator Laboratory, Menlo Park, California 94025  T. A. Assefa
Stanford Institute for Materials and Energy Science, Stanford University and
SLAC National Accelerator Laboratory, Menlo Park, California 94025  Division
of Synchrotron Radiation Research, Department of Physics, Lund University,
Solvegatan 14, 22100 Lund, Sweden  N. Burdet  Stanford Institute for Materials
and Energy Science, Stanford University and SLAC National Accelerator
Laboratory, Menlo Park, California 94025  Linac Coherent Light Source, SLAC
National Accelerator Laboratory, Menlo Park, CA 94720  L. Shen  Stanford
Institute for Materials and Energy Science, Stanford University and SLAC
National Accelerator Laboratory, Menlo Park, California 94025  Division of
Synchrotron Radiation Research, Department of Physics, Lund University,
Solvegatan 14, 22100 Lund, Sweden  A. H. Reid  Linac Coherent Light Source,
SLAC National Accelerator Laboratory, Menlo Park, CA 94720  G. L. Dakovski
Linac Coherent Light Source, SLAC National Accelerator Laboratory, Menlo Park,
CA 94720  M. H. Seaberg  Linac Coherent Light Source, SLAC National
Accelerator Laboratory, Menlo Park, CA 94720  F. O’Dowd  Linac Coherent Light
Source, SLAC National Accelerator Laboratory, Menlo Park, CA 94720  S. A.
Montoya  Center for Memory and Recording Research, University of
California–-San Diego, La Jolla, California 92093  H. Chen  Physics
Department, Northeastern University, Boston, MA 02115  Stanford Institute for
Materials and Energy Science, Stanford University and SLAC National
Accelerator Laboratory, Menlo Park, California 94025  Linac Coherent Light
Source, SLAC National Accelerator Laboratory, Menlo Park, CA 94720  A. Okullo
Department of Physics and Astronomy, Howard University, Washington DC, 20059
S. Mardanya  Department of Physics and Astronomy, Howard University,
Washington DC, 20059  S. D. Kevan  Department of Physics, University of
Oregon, Eugene, OR 97401  Advanced Light Source, Lawrence Berkeley National
Laboratory, Berkeley, CA 94720  P. Fischer  Materials Sciences Division,
Lawrence Berkeley National Laboratory, Berkeley, CA 94720  E. E. Fullerton
Center for Memory and Recording Research, University of California–-San Diego,
La Jolla, California 92093  Department of Electrical and Computer Engineering,
University of California–-San Diego, La Jolla, California 92093  S. K. Sinha
Department of Physics, University of California–-San Diego, La Jolla,
California 92093  W. Colocho  Linac Coherent Light Source, SLAC National
Accelerator Laboratory, Menlo Park, CA 94720  A. Lutman  Linac Coherent Light
Source, SLAC National Accelerator Laboratory, Menlo Park, CA 94720  F.-J.
Decker  Linac Coherent Light Source, SLAC National Accelerator Laboratory,
Menlo Park, CA 94720  S. Roy  Advanced Light Source, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720  J. Fujioka  Department of Applied
Physics and Quantum-Phase Electronics Center (QPEC), University of Tokyo,
Hongo, Tokyo 113-8656, Japan  Graduate School of Pure and Applied Science,
University of Tsukuba, Tsukuba, Ibaraki, 305-8573, Japan  Y. Tokura
Department of Applied Physics and Quantum-Phase Electronics Center (QPEC),
University of Tokyo, Hongo, Tokyo 113-8656, Japan  RIKEN Center for Emergent
Matter Science (CEMS), Wako 351-0198, Japan  M. P. Minitti  Linac Coherent
Light Source, SLAC National Accelerator Laboratory, Menlo Park, CA 94720  J.
A. Johnson  Swiss Light Source, Paul Scherrer Institut (PSI), 5232 Villigen
PSI, Switzerland  M. Hoffmann  Linac Coherent Light Source, SLAC National
Accelerator Laboratory, Menlo Park, CA 94720  M. E. Amoo  Department of
Mechanical Engineering, Howard University, Washington DC, 20059  A. Feiguin
Physics Department, Northeastern University, Boston, MA 02115  C. Yoon  Linac
Coherent Light Source, SLAC National Accelerator Laboratory, Menlo Park, CA
94720  J. Thayer  Linac Coherent Light Source, SLAC National Accelerator
Laboratory, Menlo Park, CA 94720  Y. Nashed  SLAC National Accelerator
Laboratory, Menlo Park, California 94025  C. Jia  Stanford Institute for
Materials and Energy Science, Stanford University and SLAC National
Accelerator Laboratory, Menlo Park, California 94025  A. Bansil  Physics
Department, Northeastern University, Boston, MA 02115  S. Chowdhury
<EMAIL_ADDRESS>Department of Physics and Astronomy, Howard
University, Washington DC, 20059  A. M. Lindenberg<EMAIL_ADDRESS>Stanford Institute for Materials and Energy Science, Stanford University and
SLAC National Accelerator Laboratory, Menlo Park, California 94025  Department
of Materials Science and Engineering, Stanford University, Stanford, CA 94305
M. Dunne  Linac Coherent Light Source, SLAC National Accelerator Laboratory,
Menlo Park, CA 94720  E. Blackburn  Division of Synchrotron Radiation
Research, Department of Physics, Lund University, Solvegatan 14, 22100 Lund,
Sweden  J. J. Turner<EMAIL_ADDRESS>Stanford Institute for Materials
and Energy Science, Stanford University and SLAC National Accelerator
Laboratory, Menlo Park, California 94025  Linac Coherent Light Source, SLAC
National Accelerator Laboratory, Menlo Park, CA 94720

###### Abstract

With the introduction of x-ray free electron laser sources around the world,
new scientific approaches for visualizing matter at fundamental length and
time-scales have become possible. As it relates to magnetism and ‘magnetic-
type’ systems, advanced methods are being developed for studying ultrafast
magnetic responses on the time-scales at which they occur. We describe three
capabilities which have the potential to seed new directions in this area and
present original results from each: pump-probe x-ray scattering with low
energy excitation, x-ray photon fluctuation spectroscopy, and ultrafast
diffuse x-ray scattering. By combining these experimental techniques with
advanced modeling together with machine learning, we describe how the
combination of these domains allows for a new understanding in the field of
magnetism. Finally, we give an outlook for future areas of investigation and
the newly developed instruments which will take us there.

###### Contents

  1. I Introduction
  2. II Ultrafast experimental methods
    1. II.1 THz / X-ray pump-probe scattering
    2. II.2 X-ray Photon Fluctuation Spectroscopy
    3. II.3 Ultrafast Diffuse X-ray Scattering
  3. III Theory
    1. III.1 Magnetic cross-section and resonant XPFS
    2. III.2 Density Functional Theory
    3. III.3 Numerical methods
  4. IV Machine Learning
    1. IV.1 Training Methodology
    2. IV.2 Metrics
    3. IV.3 Single photon detection
  5. V Outlook
    1. V.1 Scientific Prospects
    2. V.2 Special Accelerator Modes
      1. V.2.1 Fresh-slice x-ray pairs
      2. V.2.2 Two-bunch x-ray pairs
    3. V.3 State-of-the-art Instrumentation
      1. V.3.1 The chemRIXS instrument for materials
      2. V.3.2 Future capabilities
  6. VI Conclusion
  7. VII Acknowledgements

##  I Introduction

In recent years, remarkable new phases of matter have been both predicted and
measured, such as quantum spin liquids, skyrmions, strongly spin-orbit coupled
materials, quantum spin Hall insulators, and helical topological
superconductors. These phases often arise due to delicate combination of
multiple interactions such as quantum confinement or magnetic frustration, and
can display magnetic features with both short-range and long-range order
ranging from sub-nanometer to micron length scales Keimer and Moore (2017).
Quantum materials exhibiting this behavior hold promise for highly efficient
electronic- and spin-transport as well as tunability for technological
applications Giustino  _et al._ (2021); Cao  _et al._ (2018). One such field
is that of spintronics, where the magnetic degree of freedom can be used as a
knob for new functionalities and enable new abilities to control materials
Žutić, Fabian, and Sarma (2004); Hirohata  _et al._ (2020).

A common theme in “quantum engineering” of materials and devices is the
ability to temporarily drive one phase into another, usually by the
introduction of a symmetry-breaking mechanism or radiative excitation Du  _et
al._ (2021); Gong and Zhang (2019); Bogdanov and Rößler (2001); Sie  _et al._
(2019); Qian  _et al._ (2014); Liu  _et al._ (2014); Oka and Aoki (2009), or
to create new transient states of matter Wandel  _et al._ (2022); Tengdin  _et
al._ (2018); Shan  _et al._ (2021). This necessitates a robust theoretical
understanding of how systems respond to stimulation at ultrafast time-scales.
One such example in the field of magnetism is the 2D Van der Waals materials
Joy and Vasudevan (1992); Wildes  _et al._ (2015). Here, theoretical models
can be tested directly, such as the Berezinskii-Kosterlitz-Thouless transition
which predicts topological order. In the 2D limit, magnetic fluctuations can
become dominant, and can also act to mediate the formation of other types of
hidden quantum phases Burch, Mandrus, and Park (2018). Probing such
fluctuations requires the development of new experimental tools with
sensitivity on the requisite time- and length- scales.

Another rapidly growing field of study is the understanding and control of
discrete topological objects, such as magnetic skyrmions. These have been
shown to respond to small magnetic fields with incredible implications for
technological applications such as computer memory Finocchio  _et al._ (2016);
Giustino  _et al._ (2021). While the motion of individual spin-moments can be
described on the nanosecond time-scale by the Landau-Lifshitz-Gilbert
equation, descriptions of spin-dynamics alone are insufficient for getting the
full picture, since the emergence of skyrmion spin-textures arise from
complicated competitions between a host of magnetic interactions, and can
involve quasi-particles which are many lattice units in size. These structures
exhibit dynamics that range across many time-scales from microseconds to the
ultrafast, and can span length-scales much larger than the unit cells of their
parent lattice Mühlbauer  _et al._ (2009); Bogdanov and Hubert (1994);
Bogdanov and Rößler (2001); Li  _et al._ (2021).

Many powerful methods exist and have continued to be developed for
understanding dynamics related to novel magnetic phenomena. Nanoscale textures
such as those seen in skyrmion crystals for instance are often studied using
Lorentz Transmission Electron Microscopy (LTEM), a powerful tool for analyzing
magnetic structure with down to 2 nm resolution Tang  _et al._ (2019).
However, LTEM is not sensitive to the rapid changes in the magnetic structure
discussed above due to long integration times lasting seconds to minutes.
Despite challenges associated with the low readout-rates of LTEM detectors,
efforts to push LTEM to the ultra-fast regime using stroboscopic methods are
currently being pursued Möller  _et al._ (2020); da Silva  _et al._ (2018).
Small-Angle Neutron Scattering (SANS) and Neutron Spin-Echo spectroscopy (NSE)
are additional important methods for accessing the dynamic structure factor
down to nanosecond time-scales with atomic precision, but can sometimes be
challenging compared to x-ray sources due to the low absorption of neutrons in
some materials, as compared to electromagnetic radiation Nakajima  _et al._
(2020); Kawecki  _et al._ (2019). This aspect can hinder the capability of
neutron methods for studying thin samples and systems undergoing rapid time-
development, where neutron signal flux has to be averaged over long times. In
order to make continued progress, novel methods are needed to access magnetic
structure and excitations at relevant length-scales as above, but that
additionally can access dynamic phenomena at ultrafast timescales.

Complementary tools now are available which can target some of the modern
topics in magnetism for the study of dynamics at the atomic scale using X-ray
Free-Electron Lasers (X-FEL). This effort has a long history Kirilyuk, Kimel,
and Rasing (2010), from the first demonstrated rapid quenching of magnetic
order Beaurepaire  _et al._ (1996) to the early use case of using the Stanford
Linear Accelerator to create short magnetic field pulses to read and write in
magnetic recording media Siegmann  _et al._ (1995), and now these new machines
are creating enhanced capabilities in this area (For some good reviews, see
Malvestuto  _et al._ (2018); Jeppson and Kukreja (2021); Dürr (2020)). These
type of sources have had impact in many scientific research areas Bostedt  _et
al._ (2016), notably in wide-ranging areas such as atomic and molecular
science March  _et al._ (2010); Gomez  _et al._ (2014); Glownia  _et al._
(2010), astrophysics Bernitt  _et al._ (2012); Vinko  _et al._ (2012),
condensed matter and materials physicsBeaud  _et al._ (2014); Kubacka  _et
al._ (2014); Wandel  _et al._ (2022); Mankowsky  _et al._ (2014), and
structural biology Boutet  _et al._ (2012); Chapman  _et al._ (2011). This
versatility is mainly owing to the ability to provide femtosecond x-ray pulse
durations, large pulse energies, and repetition rates on the order of MHz
Bostedt  _et al._ (2016); White, Robert, and Dunne (2015); Altarelli (2011);
Schoenlein  _et al._ (2017). The photon energy is also typically tunable,
allowing for element-specific resonant x-ray scattering for instance, which
can isolate and enhance the magnetic structure signal from the sample Decker
_et al._ (2022a); Caviglia  _et al._ (2013); Lee  _et al._ (2012); Chuang  _et
al._ (2013); Först  _et al._ (2014, 2011); Forst  _et al._ (2015); Shen  _et
al._ (2020). These next generation light sources hold vast potential for
driving the field of magnetism to new heights.

It is in this context that we provide a forward-looking perspective in the
field of magnetism focused on methods based on X-FEL radiation, especially as
it relates to the ultrafast regime. We specifically focus on three methods
which could be transformative in this area. With a close coupling between
theory, combined with recent machine-assisted analysis techniques, we outline
recent progress which has occurred by optimizing this overlap. After a
description of the experimental methods, we report results on a cross-section
of magnetic systems. We support this with theoretical tools which have
potential for advanced time domain studies, as well as machine-learning
developments focused on the single photon measurement. Finally, we provide an
outlook for spin-sensitive studies in the ultrafast regime. We conclude with a
discussion on the scientific prospects which will be available with the latest
accelerator modes and new instruments being constructed at the LCLS-II.

Figure 1:  Amplitude spectrum of the THz pump pulse obtained from the Fourier
transform of its time trace. (b) Temporal evolution of the 800 nm optical
reflectivity (left axis) at 50 K (blue) and 300 K (black) after p-polarized
THz photoexcitation of the manganite single crystal, Nd1-xSr1+xMnO4 ($\it{x}$
= 2/3). The lower temperature is below the Neel temperature in this crystal
and is likely demonstrating enhanced sensitivity to the magnetic order. The
time trace of a single THz pulse is shown in red (right axis).

##  II Ultrafast experimental methods

###  II.1 THz / X-ray pump-probe scattering

One area of focus combines femtosecond x-ray probes with THz excitation of
materials, especially in the area of magnetism Hoffmann and Turner (2012).
Using resonant x-ray scattering, the magnetic structure can be directly
probed, such as the $L-$edge, responsible for $p$-to-$d$ transitions of the
magnetic ion. A powerful use of X-FELs is to combine this sensitivity of the
magnetic structure with THz excitation. With short-pulsed THz radiation, low
energy modes can be directly excited for spin relaxation, spin enhancement, or
coherent spin control Kubacka  _et al._ (2014). For instance, ultrashort THz
pump pulses can directly couple to electromagnons Kubacka  _et al._ (2014). By
using soft x-ray scattering at the Mn $L-$edge to probe the spin state in
TbMnO3, strong field THz was used to both excite and study the response of an
electromagnon excitation Dakovski  _et al._ (2015a); Turner  _et al._ (2015).
With current developments underway at the LCLS-II (see Sec.V.3.1 and
Sec.V.3.2), THz pumping will soon be possible while probing with high-
repetition rate soft x-rays for direct sensitivity to different types of
electronic ordering.

Another example is in non-resonant THz pumping, where THz excitation has been
shown to have spin sensitivity. Preliminary measurements were carried out
Schlotter  _et al._ (2012) on a manganite single crystal of Nd1-xSr1+xMnO4
($\it{x}$ = 2/3) consisting of non-resonant THz pumping the system, and x-ray
or optical probe. This sample was grown by the floating-zone method Kimura
_et al._ (2001) and polished along the (110) direction. Complementary
measurements of optical conductivity Fujioka  _et al._ (2010), 800 nm pump-
probe measurements with soft x-ray resonant probe Shen  _et al._ (2020), and
time-resolved optical reflectivity measurements Shen  _et al._ (2020) have all
been carried out. A high-power Ti:sapphire-based laser (1.55 eV) with a 50-fs
pulse duration and 120-Hz repetition rate which was split and cross-polarized
for the pump pulses, which gives a temporal resolution of about 75 fs. THz
generation was produced through non-linear rectification using a organic salt
crystal (DAST) to generate field strengths in excess of 300 kV/cm (See
Fig.1a). By cooling the sample with a liquid He cryostat in UHV Doering  _et
al._ (2011); Turner  _et al._ (2015), the crystal could be studied below the
Neel temperature of 90 K Shen  _et al._ (2020).

The main THz results demonstrate how sensitive the magnetic ordering is to THz
excitation, and are shown in Fig.1b. This illustrates the reflectivity
response to high-field, short-pulse THz radiation centered at around 1.5 THz,
the frequency spectrum of which is shown in Fig.1b. The curves show the THz
pulse trace (red), the negligible response to IR light above the ordering
temperature at 300K (black), and the response of the spin state which occurs
below the Neel temperature (blue). When the crystal is magnetically ordered,
the THz response shows a much more dramatic response compared to above room
temperature. This is reminiscent of other effects that seem to be enhanced
with THz radiation, such as the surprising sensitivity to the superconducting
condensate in the presence of charge rather than magnetic order, in the high-
temperature superconductor YBCO Dakovski  _et al._ (2015b). These results
point to the value of exploiting the use of the strong THz response to
magnetic systems with different types of ordering while using the high
repetition rate capability at X-FELs to map out the ultrafast magnetic
response in fresh detail.

###  II.2 X-ray Photon Fluctuation Spectroscopy

Another area of anticipation is in using x-ray pulses with different
separation times between pulses, to perform ‘probe-probe’ measurements to
study magnetic fluctuations, sometimes referred to as X-ray Photon Fluctuation
Spectroscopy’ (XPFS) Shen  _et al._ (2021a). This is similar to X-ray
intensity fluctuation spectroscopy (XIFS) Sutton (2008) or X-ray photon
correlation spectroscopy (XPCS) Sinha, Jiang, and Lurio (2014), but instead of
correlating scattered x-ray speckle patterns between shots, the shots are
added together and the contrast is extracted from the pulse pair, when the
pulses within each pair are finely spaced. These methods take advantage of the
high degree of coherence of the x-rays and advanced light sources to produce
‘speckle’ patterns, where scattered photons create a complex pattern based on
the exact configuration of the system, the configuration which is typically
not detected but averaged over when the degree of coherence is not high. By
monitoring this speckle pattern, fluctuations of the structure can be measured
and related back to theory to directly access the interactions between
constituents within the system.

XPFS is an ultrafast version of XIFS or XPCS and is in the spirit of speckle
visibility spectroscopy, where the contrast is analyzed rather than the
intensity-intensity autocorrelation function, but the contrast is obtained
from a pair of summed pulses Gutt  _et al._ (2009a). This benefit provides
technical motivation because in this case, the area detector collecting images
does not have to be read out at the rate of the pulse separation, but rather
the repetition rate of the pulse pairs. This capability provides an incredible
potential for access to much shorter times. Importantly, the information can
still be captured with the summation of the individual pulses, when keeping
track of the number of pulses added and with the caveat that the signal-to-
noise is lower for multi-shot images.

Figure 2:  The time dependent structure factor $S(\mathbf{q},t)$ of the
skyrmion lattice of FeGd recorded at the skyrmion lattice scattering peak.
This is a multi-layered system which forms a lattice and spontaneous dynamics
can be measured with XPFS on nanosecond times. The correlation function as a
function of amplitude ratio $\Lambda=20\%$ (pink), $12.5\%$ (green), and
$10\%$ (blue). The curves are plotted along with an pure exponential decay of
the same time constant (dotted lines) are offset by 0.05 for clarity.

By using XPFS, ground state fluctuations can be studied of a magnetic texture
in its natural state, without inducing the non-equilibrium nature which is
typically associated with the ultrafast. This also requires ultra-short pulses
to take snap shots of the magnetic state for different times, but the focus is
to make a comparison of the system for different times. For instance, this has
been carried out in the multilayered system FeGd Seaberg  _et al._ (2017);
Esposito  _et al._ (2020); Seaberg  _et al._ (2021), which has been shown to
form a skyrmion lattice Lee  _et al._ (2016); Montoya  _et al._ (2017a, b).
Here by varying the pulse separation, a damped ‘phonon-like’ mode was measured
of the skyrmion lattice. This is reminiscent of the Goldstone mode, and was
observed by directly probing the resonant scattering from the magnetic quasi-
particles.

One point to note is that, in using this method, equal pulses are needed to
extract the fluctuations in the system being studied. We demonstrate this by
showing new data from the damped oscillatory fluctuations from the skyrmion
lattice mentioned above, in the FeGd system in Fig. 2. In a Self-Amplified
Spontaneous Emission (SASE) process such as that produced at the LCLS, each
pulse amplitude generated in the accelerator can vary dramatically. This can
be captured in the nanosecond regime by using a fast digitizer as a diagnostic
Seaberg  _et al._ (2017); Sun  _et al._ (2018); Esposito  _et al._ (2020);
Assefa  _et al._ (2022). In Fig. 2, we analyze pulses that have the pulse
ratio between the first and second pulses $\Lambda_{12}=a_{1}/a_{2}-1\leq
10\%$, or vice versa $\Lambda_{21}=a_{2}/a_{1}-1\leq 10\%$, shown in the blue
curve, as reported by Seaberg et al. Seaberg  _et al._ (2021). This was
determined to be the closest in amplitude one could analyze, while leaving
reasonable statistics for this data set. In addition, we show plots with
$\Lambda\leq 12.5\%$ (green), and $\Lambda\leq 20\%$ (pink). These plots
demonstrate both the washing out of the oscillation amplitude in the
correlation function as well as the larger background at long times. Both of
these effects are expected as a result of adding pulse pairs with more
‘single-pulse’ characteristics. For instance, in the extreme case, where one
pulse is large while the other is negligible, the contrast will approach
$C(\mathbf{q},\tau)\sim 1$, the value of a single shot – regardless of the
state of the system. This behavior underpins the importance of this type of
diagnostic in XPFS and the need to produce equal pulses in this mode. Future
work could correct for this amplitude ratio and retain the largely asymmetric
data to improve statistics further.

Moreover, recent work has focused on changing the ratio of the two probes to
increase the first well beyond the second pulse by adjusting the pulse
energies generated from the machine (Sec. V.2.1). The idea of this option is
to perform x-ray pump / x-ray probe by selectively x-ray pumping the system to
study the response with soft x-rays. This is quite different in nature to XPFS
and instead has the goal of exciting the system at very high energies.

Finally, we point out the natural extension of this, where one combines XPFS
with an optical/THz pump pulse to probe non-equilibrium fluctuations in the
excited state. Here the idea would instead be to not keep the system in
equilibrium, but understand how the dynamical heterogeneity takes place during
the excitation process. This is more sophisticated then a typical pump-probe
study because much more information about the length-scale is available than
in the scattered intensity. In other words, the monitoring of the contrast can
be used for going beyond the coherent response to probe disorder and
heterogeneity in non-equilibrium systems as well.

###  II.3 Ultrafast Diffuse X-ray Scattering

A third method that will be critical to exploit in the field of ultrafast
magnetism is in the study of diffuse magnetic scattering, where weakly or less
well defined long-range scattering can be measured. Features that often define
the most interesting properties are in regions where materials exhibit well-
defined structure and excitations, and can be resolved in reciprocal space and
can even be controlled externally (see Sec. II.1). However, sometimes
considerable information is contained away from the well-defined structural
responses of the system, in the shorter range interactions, which disturb the
density-density correlations and lead to diffuse scattering near the peaks
corresponding to long-range order. For example, as the thermal motion of atoms
about their sites generates thermal diffuse scattering that can be used to
obtain information about phonons Trigo  _et al._ (2013), there is analogous
information which will shed light on the magnetic structure as well. In this
case, scattering from different components can overlap in reciprocal space,
but can be unraveled by observing different signatures in the time domain.

For example, in the spin-glass systems, structural or charge scattering can
obscure magnetic scattering directly related to the spin structure. In thin
films of CuMn, this has been observed by comparing non-resonant to resonant
scattering at the Mn-edge as a function of momentum transfer. This was shown
to be successful in the measurement of the 4-spin correlation function which
displays dynamics on very slow timescales, at the level of hundreds to
thousands of seconds, to determine the Edwards-Anderson order parameter Song
_et al._ (2020). Diffuse scattering was furthermore measured in a forward
scattering geometry at the LCLS, shown in Fig. 3. This shows a resonant
coherent speckle pattern of the diffuse scattering around $q=0$ using the XPFS
prototype instrument at the LCLS Assefa  _et al._ (2022). It was shown that
the large amount of diffuse scattering in the spin glass state could be
measured out to large $q$, up to almost 100 mÅ-1, and could furthermore be
captured on the order of one pulse width, about 100fs. Here, instead of using
resonant enhancement to separate the charge and magnetic scattering for
extraction of pure spin dynamics, the dynamical signatures of the scattering
could be used to measure the dynamic spin component, on top of the static
charge.

Figure 3: Resonant speckle pattern collected at an X-FEL which can be used to
study nanosecond scale spontaneous fluctuations of the prototypical spin-glass
system CuMn. The image shows the spin-glass speckle over a 500$\times$500
pixel image using the XPFS prototype instrument Assefa  _et al._ (2022). The
image was collected at the $L-$edge resonance for Mn in a forward scattering
geometry and demonstrates the degree to which magnetism can be extracted with
a large amount of diffuse scattering and on short timescales, of order one
pulse length of about 100fs. The center of the peak located $q=0$ is
positioned to the right of the sensor region.

Another path for magnetic diffuse x-ray scattering is working at a suitable
resonant edge, which under the right conditions is advisable to boost the
observable magnetic signal. To push to the true ultrafast regime requires some
of the techniques discussed earlier in this section (see Sec. II.2 for
instance), especially with new electron accelerator technology Decker  _et
al._ (2022b) (see Sec. V.2.1). A celebrated example of this in magnetism is
the observation of the so-called “pinch points” that are characteristic of the
spin ice state Fennell  _et al._ (2007). Diffuse scattering studies using
neutrons for instance have been carried out, where the spin-$\frac{1}{2}$
neutrons interact directly with the magnetic fields generated by magnetic
ions. Such measurements typically take a long time due to the limited neutron
flux available, and can usually be considered as long-time averages of the
short-range order. As we might expect from the fluctuation-dissipation
theorem, this short-range scattering may display dynamics on a range of time
scales, depending on the specific origin of the diffuse scattering. This is
well studied in soft condensed matter, for example regarding the glass
transition in polymers Hoshino  _et al._ (2020), but less so in crystalline
materials.

On longer time scales, diffuse scattering from neutron scattering on
frustrated magnets has been observed to vary on time scales of 10s of minutes,
with changes failing to stabilize over several hours in the Ising spin chain
Ca3Co2O6 Agrestini  _et al._ (2011). In a similar type of system,
$\gamma$-CoV2O6, similar changes occur but on a much shorter time scale, with
relaxations observed by muon spin rotation decaying over several
$\mathrm{\mu}$s Shen  _et al._ (2021b). Using neutrons, the timescale can be
pushed shorter by using the neutron spin-echo technique Gardner  _et al._
(2020), where timescales ps to 100s of $\mu$s can be probed. This has also
been used extensively in the study of polymers, but again is very flux hungry,
with a limited number of instruments around the world.

Just as with polymer glasses, magnetic glasses, quantum spin glasses, and
frustrated magnets are classes of materials where diffuse scattering is vital
in being able to unravel the nature of the interactions, which are frustrated
either geometrically Gardner, Gingras, and Greedan (2010) or by introducing
site or bond disorder to randomize the exchange interactions Binder and Young
(1986). Neutron spin echo has been used to study a variety of spin glasses,
often showing that significant dynamical processes exist on sub-picosecond
timescales Pappas, Ehlers, and Mezei (2006), but no ultrafast version of any
of these neutron scattering methods exists. By targeting the many areas which
have been addressed above in magnetic neutron diffuse scattering with
ultrafast x-rays, a wide ranging and fertile ground is available for
exploration.

##  III Theory

To take full advantage of the research available with the expanding X-FEL
capabilities, certain theoretical tools are vital to tie the methods as
outlined above together. Especially due to the notorious scarcity of X-FEL
beamtime and experimental complexity of the methods involved. Strong
predictive models are needed before and during the experiment to ensure the
allocated time is used optimally and efficiently. Furthermore, because of the
rich variety of interactions between x-rays and materials, a theoretical
perspective is crucial when analyzing data in order to distinguish signal from
background. Since the focal point here is on resonant x-ray scattering, we
start with an overview of the theoretical background for the magnetic cross-
section, with an added discussion of how this relates directly to XPFS. This
is followed by a discussion of Density Functional Theory (DFT) and the
numerical methods we focus on for magnetism, Exact Diagnolization (ED) and the
Density Matrix Renormalization Group (DMRG).

###  III.1 Magnetic cross-section and resonant XPFS

The main mechanism we will focus on in this article is resonant magnetic
scattering. In an experiment this is achieved by tuning the incoming x-ray
photon energy to an absorption edge of the metallic ion carrying the magnetic
spin-moments originating from unpaired valence electrons. To model this, we
typically focus on dipole transitions, though the quadropolar channel can also
be studied. The full cross-section for resonant scattering for the electric
dipole transitions was first worked out by M. Blume Blume (1985) and is given
by:

$f=f_{c}-if_{m1}({\epsilon_{f}^{*}}\times\epsilon_{i})\cdot\textbf{s}+{f_{m2}(\epsilon_{f}^{*}\cdot\textbf{s})(\epsilon_{i}\cdot\textbf{s})}$
(1)

where $\epsilon$ represents the incoming and final polarization states, $s$ is
the spin of the atom, and the $f_{i}$’s are charge, first, and second order
magnetic, frequency-dependent scattering amplitudes.

Typically, in a scattering experiment, we project the cross-section into
components that are either in the scattering plane, or orthogonal to the
scattering plane. For our purposes discussed in this paper, we tend to focus
on the second term, which for small angle scattering gives only off-diagonal
matrix elements for the scattering process Hill and McMorrow (1996):

${\epsilon_{f}^{*}}\times\epsilon_{i}=\left(\begin{array}[]{cc}{\epsilon_{\sigma}^{*}}\times\epsilon_{\sigma}&{\epsilon_{\sigma}^{*}}\times\epsilon_{\pi}\\\
{\epsilon_{\pi}^{*}}\times\epsilon_{\sigma}&{\epsilon_{\pi}^{*}}\times\epsilon_{\pi}\end{array}\right)=\left(\begin{array}[]{cc}0&\bf{k_{i}}\\\
\bf{-k_{f}}&0\end{array}\right)$

In this case, the cross-section goes as $\sim k\cdot s$ and is optimized for
spins pointing out of the sample plane, or parallel to the incoming beam. The
resonant intensity enhancement can be several hundredfold that of the non-
resonant magnetic contribution, but is still often small compared to the
intensity originating from charge-scattering. In practice, it is best to
choose Bragg reflections which are forbidden by the space-group of the
chemical lattice but allowed by the magnetic sublattice, so that the first
term of 1 can be ignored and only the magnetic terms remain.

In order to observe the dynamics associated with the magnetic scatterers as
described in II.2 the resultant scattering intensity from 1 is measured as a
time-series at a region of interest in reciprocal space $q$. For a typical
XPCS experiment, the intensity-intensity autocorrelation function can be
calculated:

$g_{2}(q,\tau)=\frac{\left<I(q,t)I(q,{t+\tau})\right>}{\left<I(q,t)\right>^{2}}$
(2)

where $\tau$ is the time difference between intensities at different times,
and the brackets designate an average over $t$. Importantly, this can be cast
in terms of the intermediate scattering function by the Siegert relation
Siegert (1943) as:

$g_{2}(q,\tau)=1+A{[S(q,\tau)/S(q)]}^{2}$ (3)

where the intermediate scattering function is equal to the field-field
correlation function, $g_{1}(q,\tau)$. This holds as long as the scattering
being observed is a Gaussian process, with the phase having an equal
probability on the range of $\phi\in[0,2\pi]$ Goodman (2007). One note here is
that when representing magnetic x-ray scattering, this is not the spin-spin
correlation function, which is typically calculated from theory, but the
squared amplitude of the spin-spin correlation function, or a 4-spin
correlation function. However, one is also able to carry out so-called
‘heterodyne’ measurements, and so provide access directly to the spin-spin
correlation function.

For the XPFS measurements mentioned in Sec. II.2, one additional element is
the relationship of the contrast, which can be directly calculated from the
summed pulses, and the correlation function above. An important development in
this area was in the demonstration of the equivalence of these two quantities
to within a multiplicative factor Gutt  _et al._ (2009b). This indicates that
a contrast measurement is able to retrieve the equivalent information about
the intermediate scattering function, as in XPCS.

Lastly, typical experiments rely on both the large pulse energies and the
pulse structure of the beam in time, but the scattered photons are usually
collected in the sparse limit. In this case, photon counting is necessary to
evaluate the contrast. Because the beam is fully coherent, the contrast can be
determined by fitting the distribution of photon counts per speckle to the
negative binomial distribution, which relates the contrast $C=C(q,\tau)$ to
the probability of $k-$events for a given average intensity, $\overline{k}$:

$P(k)=A_{0}(k,M)\left(\frac{{\overline{k}}}{\overline{k}+M}\right)^{k}\left(\frac{M}{\overline{k}+M}\right)^{M},$
(4)

where $A_{0}(k,M)$ is a normalization constant which depends on the contrast
and the speckle photon density, given by:

$A_{0}(k,M)=\frac{\Gamma(k+M)}{\Gamma(M)\Gamma(k+1)}$ (5)

In Eq. 4 and Eq. 5, the dependence is expressed as the number of degrees of
freedom of the speckle pattern $M$, or $M=1/\sqrt{C(q,\tau)}$. Since this
equation can not be solved analytically, the parameters of the distribution
are usually estimated by invoking estimators that are valid in the low
$\overline{k}$ limit Sun  _et al._ (2020) or by using maximum likelihood
estimation Chitturi  _et al._ (2022). Under certain conditions, an analytical
solution has been shown to exist when multiple $k-$events are observed Shen
_et al._ (2021a).

###  III.2 Density Functional Theory

Next, we outline our theoretical approach, which starts with first-principles
density functional theory (DFT) based modeling. The recipe proceeds along the
following steps: (1) Advanced density functionals Hohenberg and Kohn (1964);
Kohn and Sham (1965), such as the recently constructed SCAN functional Sun,
Ruzsinszky, and Perdew (2015), are used to gain a handle on the ground-state
electronic, magnetic, and topological structure. Spin-orbit coupling (SOC)
effects can be accounted for in these computations Zhang  _et al._ (2020). (2)
First-principles spin-resolved band structures and wavefunctions are then used
to evaluate magnetic anisotropy energies, magnetic moments, exchange
parameters, anisotropic exchange coefficients, and the Dzyaloshinskii-Moriya
interaction (DMI) Moriya (1960); Dzyaloshinsky (1958); Skyrme and Brown
(1994). (3) Informed by first-principles results, material-specific, effective
tight-binding model Hamiltonians can be constructed for incorporating effects
of electron-electron interactions in our modeling. (4) Using atomistic spin
dynamics (ASD) simulations with our model Hamiltonians, we can investigate the
evolution of skyrmion states in various materials Müller  _et al._ (2019);
Mentink  _et al._ (2010). Finally, (5) the preceding steps can be repeated
after revealing effects of strains and magnetic fields on effects such as
skyrmion formation.

The modeling of skyrmion structures within the DFT framework is quite
challenging due to the large size of the magnetic unit cells involved.
Therefore a combination of DFT and ASD simulations are needed to study DMI-
induced skyrmions in the presence of an external magnetic field. This has
previously been used in CrI3 Behera, Chowdhury, and Das (2019). It involves
finding the solution to the Landau-Lifshitz-Gilbert (LLG) equation Mentink
_et al._ (2010):

$\frac{dm}{dt}=-|\gamma|m\times\mathcal{H}_{eff}+\alpha(m\times{\frac{dm}{dt}})$
(6)

where $\gamma$ is the gyromagnetic ratio, $m$ is the magnetic moment vector
for the magnetic atom, $\alpha$ is the Gilbert damping coefficient and
$\mathcal{H}_{eff}$ is the effective magnetic field,

$\mathcal{\mathcal{H}}_{eff}(x)=-\nabla\mathcal{H}(x),$ (7)

where $\mathcal{H}$ is the Hamiltonian of the system, which can be written as:

$\mathcal{H}=\mathcal{H}_{ex}+\mathcal{H}_{ani}+\mathcal{H}_{z}+\mathcal{H}_{DMI}$
(8)

Here $\mathcal{H}_{ex}$, $\mathcal{H}_{ani}$, $\mathcal{H}_{z}$, and
$\mathcal{H}_{DMI}$ are the exchange, anisotropy, Zeeman, and DMI terms,
respectively. This can be expressed more specifically as:

$\mathcal{H}=-\sum_{<ij>}J_{ij}n_{i}\cdot
n_{j}-\sum_{ij}K_{j}(\hat{K}_{j}\cdot n_{i})^{2}-\sum_{i}\mu_{i}B\cdot
n_{i}-\sum_{<ij>}D_{ij}\cdot(n_{i}\times n_{j})$ (9)

where $J_{ij}$ is the Heisenberg symmetric exchange, $K_{j}$ is the single-ion
magnetic anisotropy, $B$ is the external magnetic field, and $D_{ij}$ is the
DMI vector. Note that $\hat{K}_{j}$ denotes the direction of the anisotropy
and $m_{i}=\mu_{i}\cdot n_{i}$ is the magnetic moment.

The first step of this approach is to obtain the electronic structure of a
given magnetic material within the DFT framework. We will then use this
electronic structure to build a highly accurate real-space low-energy model
using Wannier functions as the basis implemented in the Wannier90 code Marzari
_et al._ (2012); Mostofi  _et al._ (2008). Once we achieve the low-energy
model, we can employ Green’s functions and magnetic force theorem to
systematically calculate the exchange parameters of the Heisenberg Hamiltonian
following the Korotin approach Korotin  _et al._ (2015). Following this
approach, we can get the isotropic exchange parameters, DMI, and the
anisotropic exchange between two sites i,and j from the following equations.

$J_{ij}=Im(A_{ij}^{00}-A_{ij}^{xx}-A_{ij}^{yy}-A_{ij}^{zz}),$ (10)
$J^{ani}=Im(A^{uv}+A^{vu})$ (11)
$\vec{D}_{ij}^{u}=Re(A_{ij}^{0u}-A_{ij}^{u0}),$ (12)

where
$A_{ij}^{uv}=-\frac{1}{\pi}\int_{-\infty}^{E_{F}}Tr\\{\textbf{p}_{i}^{z}\textbf{G}_{ij}^{u}\textbf{p}_{j}^{z}\textbf{G}_{ji}^{v}\\}d\epsilon$,
${u,v}=\\{0,x,y,z\\}$, $\textbf{G}_{ij}$ is the Greens function, and
$\textbf{p}_{i}=\textbf{H}_{ii}(R=0)$ is the intra-atomic part of the
Hamiltonian. These magnetic exchange parameters are then used in the ASD
simulation to predict the presence of skyrmions. To our satisfaction, this
method has been extensively tested for various magnetic materials and
predicted their magnetic properties. For example, we have tested this method
to obtain exchange parameters for NiPS3 with a zig-zag AFM structure. The
first, second, and third nearest neighbor exchange parameters are obtained as
1.2917 meV, 0.1044 meV, and -4.089 meV, respectively, which are similar to the
previous data-, 1.093 meV, 0.613 meV, and -3.882 meV, respectively.

###  III.3 Numerical methods

The heart of this process after using first-principles DFT computations as the
starting point input, is in the strongly correlated numerical methods.
Studying the ground state of an effective model Hamiltonian derived from DFT
computations can uncover the low-energy physics in quantum magnetism. The spin
interactions in the Hamiltonian are typically strongly correlated, which
requires associated methods, such as exact diagonalization (ED)Weiße and
Fehske (2008), variational Monte CarloHu  _et al._ (2013); Choo, Neupert, and
Carleo (2019); Chen  _et al._ (2022), and density matrix renormalization group
(DMRG)White (1992), to simulate an accurate ground state of super-exchange
electrons.

Since quantum Monte Carlo suffers from the severe “sign problem”Loh Jr  _et
al._ (1990); Troyer and Wiese (2005) at low temperatures and the possible
system size can be limited in EDRichter and Schulenburg (2010), we point out
that DMRG can have an impact on the ultrafast study of quantum magnets. DMRG
is an unbiased numerical method that is widely used for calculating quantum
systems. The second generation DMRG McCulloch (2007); Pirvu  _et al._ (2010);
Schollwöck (2011); White (2005); Dolfi  _et al._ (2012); Stoudenmire and White
(2013) is based on the matrix product state (MPS) and matrix product operator
(MPO), which for a given Hamiltonian, such as that defined in Eq. (9), can
precisely encode a MPO. This allows the algorithm to target the true ground
state by minimizing the variational ground state energy.

Searching for this ground state of a given Hamiltonian can be costly due to
the exponentially increasing Hilbert space with system size. DMRG is a one-
dimensional (1D) algorithm, as initially proposed, but has been used to
simulate two-dimensional (2D) systems. This is possible by numbering each site
of an interacting spin in real space into a sequence so that some of the
nearest neighbor spins in real 2D become long-range pairs in the sequence. The
approximation is inevitable in DMRG because it prioritizes short-range
entanglement. However, the long-range entanglement, if not dominant, can also
be truncated to minimize the computational cost. Therefore, DMRG always
utilizes a cylinder geometry (as shown in Fig.4) with periodic boundary
conditions in one direction to approximate a 2D system so that the nearest
neighbor sites will not be separated by a large distance when transforming to
the numbered sequence. The realisitc 2D system is approached by expanding the
cylinder width.

Figure 4:  The quasi-one-dimensional system is used for the DMRG calculations.
(a) An example of a 4-leg ladder, with extension in the vertical direction.
(b) For the computation to mimic the 2D condition, a cylinder geometry allows
for a periodic boundary condition in one dimension.

Considering conserved quantum numbers can further reduce the computational
cost when approaching larger cylinders with higher accuracy. The constraints
rule out unwanted states, such as violation of particle number conservation,
so that one can search the variational ground state in a subspace faster. The
ground state properties of quantum materials can suffer from finite-size
effects, low computational accuracy, or both, so it is highly nontrivial to
push the limit of large-scale DMRG calculations to achieve higher accuracy
with large system sizes. Further speedup is possible through parallel
computing.

Moreover, the dynamical density-matrix renormalization group (DDMRG)Kühner and
White (1999) is a variational method for calculating the zero-temperature
dynamical properties in 1D and quasi 1D quantum many-body systems. Typically,
it is given by a dynamical correlation function, defined as

$G_{X}(\omega+i\eta)=-\frac{1}{\pi}\langle\psi_{0}|\hat{X}^{{\dagger}}\frac{1}{E_{0}+\omega+i\eta-H}\hat{X}|\psi_{0}\rangle$
(13)

Where, $|\psi_{0}\rangle$ is the ground state and $\hat{X}$ is a quantum
operator. However, consider $W_{X,\eta}(\omega,\psi)$ substantively with the
formula

$W_{X,\eta}(\omega,\psi)=\langle\psi|(E_{0}+\omega-H)^{2}+\eta^{2}|\psi\rangle+\eta\langle
X|\psi\rangle+\eta\langle\psi|X\rangle$ (14)

Where $|X\rangle=\hat{X}|\psi_{0}\rangle$. Then minimizing
$W_{X,\eta}(\omega,\psi)$ yields the imaginary part of the dynamical
correlation function for $\eta\rightarrow 0$, i.e.,
$I_{X}(\omega+i\eta)=\text{Im
}G_{X}(\omega+i\eta)=-W_{X,\eta}(\omega,\psi_{min})/\pi\eta$. The variational
problem for solving $|\psi_{min}\rangle$ is achievable with the standard DMRG.

The zero-temperature time-dependent correlation function, defined as
$G_{X}(t\geq 0)=\langle\psi_{0}|\hat{X}(t)\hat{X}(0)|\psi_{0}\rangle$, is
solvable through the Laplace transform of a spectral function
$G_{X}(\omega+i\eta)$, or using ED together with time-dependent DMRG Cazalilla
and Marston (2002). Comparing the calculated results through the numerical
techniques discussed here with real experimental data, can help confirm or
further modify the microscopic model provided by DFT. Significantly, this type
of modeling of magnetic systems has fresh potential for original contributions
in further understanding the results using the experimental X-FEL methods in
the ultrafast regime.

##  IV Machine Learning

In this section, we present a machine learning (ML) model that uses predictive
classification to effectively replace the more traditional droplet algorithms
that are widely used in XPFS-type measurements. Based on fully convolutional
neural networks (CNNs), we show the algorithm is able to photonize raw input
XPFS patterns and extract important information under significant levels of
electron cloud smearing. We demonstrate that the CNN is able to obtain
discrete photon maps from our procedure, which is used to extract the contrast
in this method. We previously reported the success of this pipeline using a
regression based ML-approach in which the discrete number of photon counts was
estimated as a continuous value Chitturi  _et al._ (2022). Here, we present an
alternative approach in which photon assignments are classified into different
discrete bins via optimization with the categorical cross-entropy loss
function.

###  IV.1 Training Methodology

To train the model, we simulated a training dataset of 100,000 XPFS patterns
with contrasts between 0.7 and 0.9 as well as their corresponding photonized
labels. This data was obtained from an accurate simulator developed for the
Linac Coherent Light Source (LCLS) on previous XPFS data, reported in previous
work.Burdet  _et al._ (2021) Each input pattern has dimensionality 90x90x1 and
corresponds to the raw ADU intensity profile of the x-ray measurment. The
relevant simulation parameters are the photon energy (340 ADU), the baseline
gaussian noise ($N(0,15)$ ADU) and the probabilistic charge cloud generation
(cloud radii $\sigma_{g}$ values: [0.1,0.25,0.35,0.45,0.55,0.6]). The labels
for the machine learning task correspond to reduced images with dimensionality
30x30x1 which contain the true photonized maps. Here, each pixel in the
reduced image is matched to the speckle size of the raw data, and stores the
number of photons detected per speckle.

The photonizing task can be formulated as a semantic segmentation computer
vision problem Long, Shelhamer, and Darrell (2015). Specifically, for each
pixel in a given XPFS frame we aim to predict a corresponding integer count
for the number of photons contained in that pixel. This can be conceptualized
as a classification problem where the classes are the photon counts: (0, 1, 2,
3, …, 8, 8+). To represent each class, we use a one-hot encoding of the
class.Goodfellow, Bengio, and Courville (2016) For example, class 3
(representing a 2 photon event) is mapped to [0 0 1 0 0 0 0 0 0]. Therefore,
the dimensionality of the output labels changes from 30x30x1 to 30x30x9 in the
one-hot encoding representation. We use a U-net neural network model
Ronneberger, Fischer, and Brox (2015); Chitturi  _et al._ (2022) with the last
layer predicting softmax probabilities corresponding to each photon count
level and for each pixel $\hat{p}_{i}$ in the resultant photon map, i.e.
$\hat{p}_{i,j}$ represents the probability that pixel $i$ is from class $j$.
To train the model, we use a cross-entropy loss function
${L}(p_{i},\hat{p}_{i})$ to measure the difference between the predicted
probabilities and true one-hot encoded label ($p_{i}$), averaged over all
pixels in all frames (N).

$L(p_{i},\hat{p}_{i})=-\sum_{i=1}^{\text{N}}p_{i}\cdot\log\hat{p}_{i}$ (15)

We obtain an estimate of the contrast by using a per-image estimate for
$\bar{k}$ and using the maximum likelihood procedure Roseker  _et al._ (2018).
Error bars for the contrast follow the asymptotic 95$\%$ CI interval:
$\hat{\beta}\pm\frac{1.96}{\sqrt{nI(\hat{\beta})}}$. The trained neural
network model is freely available at https://github.com/src47/CNN_XPFS.

The ML model discussed here was shown to perform comparably with even the most
advanced droplet algorithms, such as the GGG Burdet  _et al._ (2021), on
typical simulated XPFS data with a number of evaluation metrics. In addition,
since we train our models on simulated data with a known ground truth, we
furthermore are able to evaluate the accuracy of the trained CNN, relative to
the known absolute contrast value.

###  IV.2 Metrics

The metrics used to evaluate the photonizing task are an important part of our
evaluation and are described here. For example, the overall accuracy is not a
good metric because of data sparsity. In other words, a model which predicts 0
for each pixel will show a uninformatively high accuracy, but this is clearly
undesirable. Therefore, we report a set of metric precision scores
$\mathcal{P}$, recall $\mathcal{R}$, and the per-class $\mathcal{F}1$ which
can be obtained from the number of true positives ($T_{P}$), false positives
($F_{P}$) and false negatives ($F_{N}$), defined as:

$\begin{split}&\mathcal{P}=\frac{\textit{\text{$T_{P}$}}}{\textit{\text{$T_{P}$}}+\textit{\text{$F_{P}$}}}\\\
&\textit{{$\mathcal{R}$}}=\frac{\textit{{\text{$T_{P}$}}}}{\textit{\text{$T_{P}$}}+\textit{\text{$F_{N}$}}}\\\
&\textit{\text{$\mathcal{F}1$
}}=\frac{2*\textit{\text{$\mathcal{P}$}}*\textit{\text{$\mathcal{R}$}}}{\textit{\text{$\mathcal{P}$}}+\textit{\text{$\mathcal{R}$}}}\end{split}$
(16)

As an example, the per-class precision for the 2-photon class is the number of
true 2-photon events divided by the sum of the true number of 2-photon events
and the number of 2-photon events that are incorrectly predicted. In Table 2,
we show the results for three different metrics to evaluate the contrast based
on an average contrast of 0.7. The droplet algorithm is compared to the CNN
using all three of these metrics, and the results are shown for different
photon number events. For this work, we used an average simulated contrast
value of between 0.7 and 0.9, as this is what was measured in a previous
experiment Seaberg  _et al._ (2017) and was used as a basis for the modeling
Burdet  _et al._ (2021). In Table 2, the same data are displayed for an
average simulated contrast value of 0.9.

Figure 5: Example of a representative Ground Truth photon map (a), CNN photon
map (b) and Droplet photon map (c). On close inspection, it is clear the CNN
gets more of the minor features of the photon map predicted correctly. Table
1:  Per-class Precision, Recall and F1-score for the CNN and Droplet methods
(contrast = 0.7).

Photon Count111Performance on 5, 6, 7, 8, 8+ photon events is not shown as there is insufficient testing data for these classes. | Precision (Droplet, CNN) | Recall (Droplet, CNN) | F1-score (Droplet, CNN) | Data Volume
---|---|---|---|---
0 | 1.00, 1.00 | 1.00, 1.00 | 1.00, 1.00 | 3950406
1 | 0.97, 0.97 | 0.96, 0.97 | 0.96 , 0.97 | 488364
2 | 0.90, 0.91 | 0.90, 0.93 | 0.90, 0.92 | 54530
3 | 0.81, 0.83 | 0.83, 0.87 | 0.82, 0.85 | 5992
4 | 0.73, 0.74 | 0.74, 0.77 | 0.74, 0.74 | 637

Table 2:  Per-class Precision, Recall and F1-score for the CNN and Droplet
methods (contrast = 0.9).

Photon Count222Performance on 5, 6, 7, 8, 8+ photon events is not shown as there is insufficient testing data for these classes. | Precision (Droplet, CNN) | Recall (Droplet, CNN) | F1-score (Droplet, CNN) | Data Volume
---|---|---|---|---
0 | 1.00, 1.00 | 1.00, 1.00 | 1.00, 1.00 | 3957196
1 | 0.96, 0.97 | 0.96, 0.97 | 0.96 , 0.97 | 472173
2 | 0.91, 0.92 | 0.90, 0.93 | 0.90, 0.92 | 60729
3 | 0.84, 0.86 | 0.83, 0.87 | 0.84, 0.87 | 8536
4 | 0.75, 0.73 | 0.75, 0.77 | 0.75, 0.75 | 1140

In all test cases here, the results of the ML algorithm based on
classification due to the discrete number of photon events was shown to work
just as well as the droplet algorithm. In addition to being much faster, a
critical benefit is that there is no user input needed for this algorithm, as
is the case for the droplet algorithm, as the model has fully learned how to
interpret sparse speckle patterns.

###  IV.3 Single photon detection

A representative example of a true label photon map, the corresponding CNN,
and the GGG droplet algorithm predictions are shown in Figure 5. Here, it is
clear that the CNN and the droplet algorithm agree for most pixels and give
visually similar profiles. Although these two methods give similar profiles,
small differences in photon count predictions can lead to very different
contrast estimates. This is especially true in cases where the counts of high
photon events are incorrectly estimated. In Figure 6, we compare contrast
estimates derived from both the optimized GGG droplet algorithmBurdet  _et
al._ (2021) and our CNN model (Figure 6). For the ranges which match the
experimental data (contrast values from 0.7 - 0.9), we find that the CNN and
the droplet models agree well (Figure 6).

Figure 6:  Estimated contrast (the GGG droplet algorithm and CNN) versus the
true contrast for both types of photon maps. The value of contrast was chosen
to mimic previous experimental contrast value from previous XPFS experiments.

In order to determine which algorithm gives superior performance in the single
photon detection regime, we evaluate the full performance using the metrics as
outlined in Equation 16 for photon maps obtained from both the droplet
algorithm and the CNN for the same contrast levels of 0.7 and 0.9. The results
are summarized in Table 2-2. In conclusion, the CNN performs comparably, or
outperforms, the droplet algorithm in 97% of the cases across the various
metrics.

Figure 7:  Two bucket correlation. The pulse energy of each pulse in a two-
bucket approach. The red is plotted negative for convenience. The two peaks
are aligned at the same energy, showing they have the same color and can
arbitrarily be delayed by integer bucket values of 350ps.

As briefly mentioned in Section IV.2, it is likely that the CNN model
outperforms the GGG algorithm because it it able to learn a variety of filters
from labeled datasets constructed using physical knowledge of the scattered
photon radiation at the detector. Furthermore, the increased representational
power of neural networks allows for the possibility of learning different
filters which can handle experimental subtleties, such as variable charge
cloud sizes or a wide range of average photon energies Chitturi  _et al._
(2022). In contrast, the droplet algorithm is unsupervised and requires
substantial tuning ‘by-hand’ to produce optimal results. The results outlined
here based on a new ML algorithm to handle XPFS data will be able to
accelerate progress in this field and also points to the potential for
bridging other efforts in this area, such as the continuous photon modeling
approach utilized previously Chitturi  _et al._ (2022). Moreover, this also
serves to bring more advanced X-ray methods such as this to a wider range of
scientific areas, easing the obstacles for general practitioners to execute
complex experiments with user-friendly computational tools.

Figure 8:  The chemRIXS Instrument for Solids. An image showing the chemRIXS
endstation with the enhancements designed for XPFS measurements of solids. The
main sample chamber (left) houses the liquid jet system or solid material
sample. The electromagnet is inserted perpendicular to the beam horizontally,
while the detector and mask assembly (right) are placed in the forward
scattering direction for transmission geometry measurements. All dark blue
color-coding designate components that were designed for the thin film,
ultrafast magnetism experiments.  Figure 9:  ChemRIXS Instrument Close-up A
close-up image inside the sample chamber. This chamber was modified to house
an electromagnet and cryostat sample holder to perform XPFS studies on solids.

##  V Outlook

###  V.1 Scientific Prospects

With the ultrafast experimental methods outlined here, the prospects for new
routes towards understanding magnetism are numerous. THz excitation enables a
variety of states to be excited and probed directly, especially with resonant
x-ray scattering. By new developments with the pulse structure, x-ray
scattering pulses can be compared to study spontaneous fluctuations in the
natural ground state of magnetic systems. Finally, diffuse scattering on
ultra-fast timescales can also offer new features for the field of magnetism
by extending the studies to short range ordered structures, especially in
systems that host a large degree of magnetic frustration, or those with short
correlation lengths, such as in the spin-glass field. To maximize the
potential of these techniques, we discuss in the following special modes of
ultra-fast x-ray experiments. We conclude the future outlook with a
description about novel instrumentation, and current instruments presently
being constructed.

###  V.2 Special Accelerator Modes

Generating different x-ray pulse separation times for performing probe-probe
such as XPFS, or x-ray pump / x-ray probe studies as mentioned earlier, is
important for executing the ultrafast methods outlined in this paper. This can
be accomplished by either optics or special pulse modes developed at the
accelerator. Though a large effort has been put into the use of x-ray optics
for this task and research progress is tremendous Sun  _et al._ (2019), here
we outline the two most valuable modes in the soft x-ray regime which will be
important for the scientific outlook in ultrafast magnetism.

####  V.2.1 Fresh-slice x-ray pairs

The Fresh-slice scheme Lutman  _et al._ (2016) can produce pairs of high-power
femtosecond x-ray pulses, with simple control of their wavelength and delay.
The scheme is based on controlling which temporal slice of an electron bunch
lases in a section of the undulator line. This is typically achieved by
tilting the electron bunch and subsequently controlling the bunch trajectory.
Lasing slices can be a couple of femtosecond short, and wavelengths are
controlled by the undulator strength in each section, enabling color
separation ranging from being at the same wavelength to larger than a factor
2. The maximum delay depends on the strength of an intra-undulator line
magnetic chicane. Practically, the scheme has been demonstrated for delays up
to 1 picosecond. Temporal coincidence, or exchanging the arrival time of the
two pulses, is possible if the slice on the tail is set to lase in the
upstream undulator section.

####  V.2.2 Two-bunch x-ray pairs

For longer delays, ranging from hundreds of picoseconds to hundreds of
nanoseconds, two separate electron bunches are extracted at the cathode,
accelerated and compressed in the linear accelerator and lase in the undulator
line Decker  _et al._ (2022b). The unitary time separation depends on the
accelerator RF frequency; for the LCLS S-Band linac, it is close to 350
picoseconds. Performance for each bunch can be similar to that of a regular,
single bunch SASE pulse. Typically, the two-bucket scheme is set up to have
both pulses lasing at the same wavelength, but a small color separation is
achievable by having the two electron bunches at slightly different energies.
For instance, Fig. 7 shows the pulse energy of each pulse in a dispersive
location of the accelerator, with a positive correlation. This can be used to
adjust the color of each pulse to have the same wavelength as well as equal
pulse energies.

Figure 10:  XPFS Instrument. The image shows a rendition of the XPFS chamber
housing the fast, pixelated detector, on its own movable stand. The scattering
path is evacuated at UHV conditions after precision alignment. The whole
apparatus mounts to the qRIXS sample chamber, and is part of the NEH 2.2
instrument at the LCLS-II, shown here installed on the beamline. The
instrument is still currently under design.

###  V.3 State-of-the-art Instrumentation

Currently, instruments are being developed to take full advantage of the
capabilities at these types of sources in the area of magnetism. For instance,
at the SLAC National Accelerator Laboratory, the new accelerator for the LCLS-
II will soon provide up to nearly $\sim$1 MHz repetition rate with an array of
possible pumping schemes, including THz, with new instruments currently being
designed, constructed, and commissioned. Some efforts have been made to
retrofit current x-ray instruments, as well as the development of full-scale
instruments designed for the purpose of carrying out XPFS measurements in the
soft x-ray regime for a variety of different geometries. We outline two such
cases here. The first is the modification of the so-called chemRIXS station
recently constructed to carry out XPFS studies on materials, while the second
is a future instrument currently under construction.

####  V.3.1 The chemRIXS instrument for materials

As a first test case, we have developed a capability to take advantage of the
recent beamline developed as part of the LCLS-II suite of instruments focused
on ultrafast chemical science, specifically in x-ray absorption spectroscopy
and resonant inelastic x-ray scattering, the so-called “chemRIXS” beamline.
This is specifically focused on using liquid jet or liquid sheet jet systems
for the study of ultrafast physics and chemistry experiments Kunnus  _et al._
(2012). The spectrum of the molecules illuminated in the solvent can be
optically excited and the chemical structure evolution can be followed on the
time-scale of these chemical changes Wernet  _et al._ (2015).

While not designed for solid material samples, some instrumentation
enhancements were made such that first XPFS experiments could be performed on
the newly designed beamline. This was carried out on thin film magnets by
designing and fabricating a solid sample holder to replace the liquid jet, a
mounted CCD detector, together with a detector mask, and a manipulator to
control the placement of an electromagnet. The mask was used to limit the area
of illumination on the detector chip to enable higher speed readout Seaberg
_et al._ (2017). This feature can be adjusted, i.e. to collect a larger
fraction of the speckle pattern and a slower rate, depending on the
experimental system. The detector was placed in a forward scattering geometry,
compatible with the chemRIXS set-up. This detector scheme implemented for this
set of experiments has been described in detail elsewhere as a prototype XPFS
instrument Assefa  _et al._ (2022), but was here used in conjunction with the
liquid jet endstation. An overview of this set-up in this configuration is
shown in Fig. 8. For a close-up view inside the system, a rendition of the
chemRIXS setup from inside the chamber, emphasizing the cryostat and the
electromagnetic for magnetic field dependent studies, is shown in Fig. 9.

####  V.3.2 Future capabilities

Looking forward to future capabilities, LCLS-II beamlines are already in use
with new instruments on the horizon. For THz pump/ magnetic scattering probe
studies or ultrafast magnetic diffuse x-ray scattering for instance, the
latest capability will occur with the installation of the qRIXS instrument.

The qRIXS instrument will host a sample chamber and a rotatable spectrometer
consisting of grating and high-speed 2D detector covering the range of
scattering angles from 40-150 degrees in the horizontal scattering geometry.
The spectrometer is designed to achieve an energy resolution of $\sim$ 20 meV
at 1 keV. The sample chamber is designed to support elastic soft x-ray
diffraction. The chamber is equipped with an in-vacuum diffractometer with a
6-axis degree of freedom of motion. Sample cooling down to about 25 K will be
possible with the cryogenic sample installation. The ability to introduce
over-sized optics for long-wavelength THz beams, will be accommodated in the
near future Kubacka  _et al._ (2014); Turner  _et al._ (2015); Hoffmann and
Turner (2012).

In addition the current plan, we are developing another capability to be able
to carry out XPFS measurements in the soft x-ray regime as well (see Fig.10).
The focal point of this instrument will be the in-vacuum, long distance, and
movable detector motion. This will incorporate the ‘ePixM’ series designed at
Stanford University and SLAC National Laboratory. This will be a large,
pixelated detector that can count single photons down to the carbon edge
($\sim$ 285 eV) and will operate at the full repetition rate of the new,
superconducting accelerator, at 929 kHz. The instrument, which is still under
design currently, will be capable of moving the detector about a limited
scattering angle range while in UHV conditions, with the use of precision
laser trackers. It will be housed in a small vacuum chamber and stand with air
bearings for chamber motion. An advanced laser tracker system will place the
detector coordinates in the correct geometry at a sample-to-detector distance
of about 3 m. Once aligned, a scattering path length drift tube will be
attached from the sample chamber, using a rotary seal, to the detector
chamber. It will be evacuated to UHV conditions to accommodate soft x-ray
scattering in the area of quantum materials. Once complete, this will offer
the potential for first XPFS measurements with soft x-rays at a arbitrary
scattering angle, important for a myriad of quantum materials and other types
of studies on solids.

##  VI Conclusion

In conclusion, with the latest developments at XFEL sources, studies in
ultrafast magnetism are ripe for a renewed focus across many areas of
research. We have targeted three new areas that are continued to be developed
and will help to capitalize on fresh capabilities to spur magnetic studies in
both equilibrium and non-equilibrium investigations. Combining these advanced
methods with the latest progress in condensed matter theory and machine
learning, the synergy and progress capable in the future for ultrafast
magnetism is bright.

##  VII Acknowledgements

This work was supported by the U.S. Department of Energy, Office of Science,
Basic Energy Sciences under Award No. DE-SC0022216. Portions of this work were
also supported by the U. S. Department of Energy, Office of Science, Basic
Energy Sciences, Materials Sciences and Engineering Division, under Contract
DE-AC02-76SF00515. The use of the Linac Coherent Light Source (LCLS), SLAC
National Accelerator Laboratory, is also supported by the DOE, Office of
Science under the same contract. S. A. M. acknowledges support by the U. S.
Office of Naval Research, In-House Lab Independent Research program. S.K.,
P.F., and S.R. acknowledge support by the U.S. Department of Energy, Office of
Science, Office of Basic Energy Sciences, Materials Sciences and Engineering
Division under Contract No. DE-AC02-05-CH11231 (NEMM program MSMAG). The
research at UCSD was supported by the research programs of the U.S. Department
of Energy (DOE), Office of Basic Energy Sciences (Award No. DE-SC0003678). J.
J. Turner acknowledges support from the U.S. DOE, Office of Science, Basic
Energy Sciences through the Early Career Research Program.

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