11institutetext: Mariano Trigo 22institutetext: Stanford PULSE Institute and
Stanford Institute for Materials and Energy Sciences (SIMES), SLAC National
Accelerator Laboratory, Menlo Park, CA 94025, USA 22email:
<EMAIL_ADDRESS>Mark P. M. Dean 44institutetext:
Condensed Matter Physics and Materials Science Department, Brookhaven National
Laboratory, Upton, New York 11973, USA 44email<EMAIL_ADDRESS>55institutetext:
David A. Reis 66institutetext: Stanford PULSE Institute and Stanford Institute
for Materials and Energy Sciences (SIMES), SLAC National Accelerator
Laboratory, Menlo Park, CA 94025, USA 66email<EMAIL_ADDRESS>
# Ultrafast x-ray probes of dynamics in solids

Mariano Trigo  Mark P. M. Dean  David A. Reis

Advances in our ability to understand and utilize the world around us have
always relied on the development of advanced tools for probing and
manipulating materials properties. X-ray matter interactions played a critical
role in the development of the modern theory of solid-state materials that has
continued over more than a century. The development of ever-brighter x-ray
sources has facilitated ever more sensitive x-ray scattering and spectroscopy
measurements that are able to probe not just the lattice structure, but also
the spectrum of elementary excitations in complex materials. The interactions
underlying the electronic, magnetic and thermal properties of solids tend to
be associated with lengthscales comparable to the atomic separation and
timescales ranging from femtosecond to picoseconds. The short wavelength,
femtosecond pulses from x-ray free-electron laser (XFEL) s therefore offer
unprecedented opportunities to probe and understand material properties on
their natural lengthscales and timescales of these processes. This chapter
reviews a number of exemplary XFEL-based experiments on lattice and electronic
dynamics, and their microscopic interactions both near and out of equilibrium.
We conclude with a brief discussion of new forms of spectroscopy enabled by
the combination of high flux and short pulse duration and give an outlook for
how the field will develop in the future.

##  1 Introduction

X-ray radiation has proven invaluable for studying the structure and dynamics
of matter on the atomic-scale. A variety of ever-more-sophisticated techniques
have been developed over the past hundred and twenty-five years concomitant
with improvements in x-ray sources, optics and detectors. These methods make
use of the short wavelength (comparable to the size of atoms), atomic and
chemical specificity, as well as the penetrating power of x rays. Each new
advance has led to important discoveries across a wide range of disciplines.
In the case of condensed matter physics, the high brightness and tunability of
storage-ring based synchrotron radiation sources enabled the development of
ultra-sensitive x-ray methods including non-resonant inelastic x-ray
scattering (IXS) krisch2007inelastic ; Baron2015  and resonant inelastic x-ray
scattering (RIXS) Ament2011resonant ; Dean2015insights . These techniques have
been transformational for the study of collective modes of quantum materials.
Here dedicated instruments have been developed that push the resolution in
energy and momentum that allow for the study of small crystals and
heterostructures, in a variety of environments. The use of such sources for
studying the nonequilibrium dynamics of quantum materials in the time-domain
is limited by the pulse duration of the sources which are typically tens to a
hundred picoseconds per burst. However, dynamics at the inter-atomic-scale,
including the highest frequency vibrations and the making and breaking of
chemical bonds occurs on a few femtosecond time-scale, while the dynamics
associated with electron correlation and exchange interaction can be even
faster.

Extremely bright femtosecond XFEL sources are transforming our ability to
follow the dynamics of solid-state materials on the relevant timescales of
their low-lying elementary excitations, as well as to study their non-
equilibrium behavior on the atomic-scalewall2016recent ; Lindenberg2017 ;
Buzzi2018probing ; Dunne2018x-ray ; Cao2019ultrafast . The first hard x-ray
XFEL, the Linac Coherent Light Source (LCLS) became operational a decade ago,
initially lasing at a wavelength of $1.5$ Å, immediately exceeding the peak
brightness of the most intense x-ray light sources in operation by 9–10 orders
of magnitude Emma2010first . This dramatic increase in peak brightness was due
to a combination of the ultrashort pulse duration of $\sim 100$ fs, large
pulse energy $\sim$ mJ and near-transform-limited transverse spatial coherence
characteristic of the high-gain self-amplified spontaneous emission (SASE)
process Kondratenko1980generating ; Bonifacio1984collective ; Bonifacio:1984aa
. Since this time a number of hard x-ray free-electron laser (FEL) facilities
have been built around the world Ishikawa2012compact ; Abeghyan2019 ;
Kang2017hard ; milne2017swissfel . The capabilities and performance of the
various XFEL s has also grown including self-seeding Amann2012demonstration ;
Inoue2019 , shorter pulses of a few tens of femtoseconds down to sub-
femtosecond (attosecond) regime Huang2017 ; Duris2020 , tunability of x-ray
energy up to $\sim$20 keV Tono_2013 , as well as two-color operation
Hara:2013aa . The next generation superconducting LINAC based XFEL s will
operate at much higher average power and repetitions rates galayda2018 ;
zhu2017sclf .

We will not discus the physics of the FEL process here, as there are a number
of excellent references for the interested reader Huang2007review ;
saldin2010statistical ; McNeil2010 ; Pellegrini2016pellegrini ; seddon2017 .
Instead we will focus on describing a sub-set of experiments that make use of
the unique properties of the XFEL for studying the ultrafast dynamics of
quantum materials. After a brief discussion of the general experimental
considerations, we focus on discussing different classes of dynamics in
optical laser-pumped materials probed by both non-resonant and resonant
scattering techniques. We begin by considering examples where non-resonant
time-resolved diffraction probes coherent excitation of lattice modes and
electron-phonon coupling in the linear-response regime, both at zone center
and in the case of diffuse x-ray scattering, for two-phonon excitations
spanning the Brillouin zone. We then go on to discuss examples where these
methods have been used to uncover the nonlinear couplings between multiple
degrees of freedom, as well as cases where photoexcitation induces excitations
well-beyond linear response using non-resonant diffuse scattering to look at
metal-insulator transitions and resonant diffraction to study changes in
charge and orbital ordering. Finally we describe experiments where time-
resolved resonant inelastic scattering provides combined momentum and energy-
resolved information on magnetic, orbital and charge excitations. We have not
attempted to be comprehensive and notably have left out a number of important
applications and techniques including lensless imaging Nugent2010coherent ,
x-ray photon correlation spectroscopy Shpyrko2014xray  and (for the most part)
x-ray nonlinear optics in solids Glover2012 ; Shwartz2014 ; Tamasaku:2014qf ;
Fuchs2015 ; Ghimire2016 ; Fuchs2018Young .

Finally we note that even before the XFEL, femtosecond duration x-ray pulses
were produced in laser-plasma interactions Murnane1991ultrafast ;
Sokolowski_Tinten_2004  laser-slicing on storage rings
Schoenlein2000generation  and via electron compression in linear-accelerator
based spontaneous synchrotron sources bentson2003 ; Lindenberg2005 ;
Cavalieri2005 . The relatively low flux and availability of these sources have
limited their utility. Nonetheless these sources were critical in developing
the nascent field of ultrafast x-ray science Rousse2001rmp ;
Cavalleri2002Review ; Reis2006ultrafast ; bargheer2006recent  including the
experiments described here.

##  2 X-ray free-electron laser-based pump-probe methods

The different experiments discussed in this chapter on solid-state dynamics
all make use of the combination of the short pulse duration and high flux
available on the XFEL. In particular, each derives its temporal-resolution
through stroboscopic, pump-probe, methods, whereby a femtosecond long-
wavelength pump (often a near-IR laser) excites the material which is
subsequently probed by the x-rays captured on a relatively slow detector. In
this manner the dynamics are built up statistically as a function of the
relative timing of the pump and probe, and the resolution is given by the
combination of the pump duration, the probe duration, and our ability to
control (or measure) the relative delay. The earlier XFEL experiments were
limited by both the timing jitter and drift between the optical and x-ray
laser, although subsequently nearly pulse-length limited resolution has been
achieved using a single-shot x-ray arrival time monitor Bionta2011spectral ;
chollet2015 . In addition, because the SASE process starts up from noise all
the various properties of the x-ray pulse fluctuate from shot-to-shot, such
that in addition to the timing, a typical experiment needs to monitor the
pulse properties and read the detectors on a shot-by-shot basis, at a few tens
to 120 Hz repetition rate. Here care needs to be taken to minimize systematic
errors over the course of an experiment, as well as to handle the processing
and analysis of the data (which can easily exceed several TB/hour).

##  3 Coherent phonon spectroscopy in linear response

We begin our discussion by considering time-domain measurements of coherently
excited collective lattice modes (phonons). Coherent phonon spectroscopy
dhar1994timeresolved ; merlin1997generating  has been an invaluable tool in
the optical domain for probing the vibrational properties of solids. An
attractive feature of time-domain, coherent phonon spectroscopy is that,
similar to nuclear magnetic resonance (NMR), the time-domain sampling of the
oscillations produced by a coherent vibration allows for frequency resolution
comparable or surpassing the natural oscillator linewidth. X-ray scattering
has several advantages over optical methods for coherent phonon spectroscopy,
including direct measurements of lattice displacements and momentum resolution
that is inaccessible with long-wavelength probing. Much of the early work on
time-resolved x-ray scattering focused on scattering from coherent acoustic
phononsRousse2001rmp ; Reis2006ultrafast ; bargheer2006recent . X-ray FELs
have the advantage of much better time resolution and flux than any previous
source, so it is no surprise that they are powerful tool for measuring
coherent phonons with much better sensitivity and from a wider range of
materials than previously possible.

[width=11.5cm]figs/Fig_SingerCDW.pdf

Figure 1.1: Example of coherent phonons probed by ultrafast x-ray diffraction.
(a) The diffraction pattern around $Q=(0\ 0\ 2q)$ of a 28 nm-thick Cr film.
The small oscillations are sidebands due to finite size. The inset shows a
detector image at the CDW wavevector. (b) Dynamics of the normalized CDW
intensity for several incident fluences. (c) Schematic of the displacive
excitation of a coherent phonon. (d) Calculation of the atomic displacement
from Eq. 1.1 using parameters from Singer2016 . The curves are displaced
vertically for clarity. (a-b) Reprinted figure elements with permission from
Ref. Singer2016  Copyright 1996 by the American Physical Society.

For example, Singer et al. used ultrafast x-ray diffraction at the LCLS to
probe the dynamics of the charge density wave (CDW) in a 28 nm-thick film of
elemental chromium (Cr) Singer2016 . This material exhibits an incommensurate
spin density wave (SDW) below $T_{N}=290$ K accompanied by a CDW at $2q$,
where $q$ is the magnitude of the SDW wavevector. In the sample studied in
Singer2016 , the CDW wavevector, $(0\ 0\ 2q)$ was perpendicular to the film.
The finite thickness of the film gives rise to finite-size side-bands of the
crystal Bragg peak as shown in Fig. 1.1(a). These finite-size satellites align
and are commensurate with the CDW wavevector due to it being pinned to the
film surface. Importantly, these finite size sidebands at $-2q$ and $+2q$
interfere constructively (destructively) with the x-ray field scattered by the
CDW, resulting in an increase (decrease) of the CDW peak intensity,
respectively. This interference, akin to a heterodyne measurement, makes the
intensity depend linearly on the atomic displacement associated with the CDW,
$\delta y$kozina2017heterodyne  and the measurement is sensitive to the sign
of the atomic motion with $\Delta I(t)=\pm 1$ corresponding to $\delta y=\pm
1$ in normalized units, respectively. The dynamics of the normalized intensity
change for the $\mathbf{Q}=(0\ 0\ k)$ peak, with $k=2\pm 2q$ are shown in Fig.
1.1(b) for several incident pump fluences. The observed oscillations, with
period $\sim 0.45$ ps (frequency $2.2$ THz) originate from the amplitude mode
of the CDW.

The CDW dynamics in Fig. 1.1(b) follow the displacive excitation of coherent
phonons (DECP) Zeiger1992theory ; merlin1997generating  common among many
charge-ordered systems under ultrafast excitation. In DECP, the coherent
motion of the mode is initiated by a sudden displacement of the parabolic
potential as shown in Fig. 1.1(c). The general solution for the atomic
displacement $\delta y(t>0)$ under a displaced harmonic potential
isZeiger1992theory

$\delta
y(t)=\frac{A}{2}\frac{\Omega^{2}}{\Omega^{2}+\beta^{2}-2\gamma\beta}\left\\{e^{-\gamma
t}\left(\cos\Omega t-\frac{\beta-\gamma}{\Omega}\sin\Omega t\right)-e^{-\beta
t}\right\\},$ (1.1)

where $\Omega$, $A$, and $\gamma$ are the frequency, amplitude and damping
constant of the oscillator, and $\beta^{-1}$ is the time scale for return to
equilibrium of the displaced harmonic potential. If $\beta-\gamma\ll\Omega$,
i.e. a harmonic system with a slow recovery of the potential compared to the
frequency, Eq. 1.1 can be simplified to

$\delta y(t)=\frac{A}{2}e^{-\gamma t}\left(\cos\Omega
t-e^{-(\beta-\gamma)t}\right)$ (1.2)

which is an oscillatory (cosine) motion with a slowly recovering equilibrium
offset. In the limit of slow relaxation of the potential $\beta\approx 0$ and
$\delta y(t)=\frac{A}{2}(e^{-\gamma t}\cos\Omega t-1)$ corresponding to a
cosine-like phase.

The opposite limit where the displaced potential recovers quickly is
$\beta\gg\gamma$ and we get

$\delta
y(t)=\frac{A}{2}\frac{\Omega\beta}{\beta^{2}+\Omega^{2}-2\gamma\beta}e^{-\gamma
t}\sin\Omega t.$ (1.3)

In this case the oscillatory motion is sine-like which behaves similarly to
that achieved by an impulsive excitation. This is the behavior of the low-
fluence trace from Singer2016  shown in Fig. 1.1. In Fig. 1.1(d) we show the
normalized dynamics (i.e. setting the amplitude $A=1$ for all traces for
clarity) of $\delta y(t)$ predicted by Eq. 1.1 with parameters corresponding
to fluences of 1, 4 and 11 mJ/cm2 (see Table S1 in the Supplementary Material
of Singer2016 ) [same color code as Fig. 1.1(b)]. The crossover between sine-
like and cosine-like motion is clear as the fluence increases and $\beta$
decreases.

t]

Figure 1.2:  Ultrafast x-ray diffraction from coherent phonons in FeSe. (a)
The top panel shows the integrated intensity of the $(0\ 0\ 4)$ Bragg peak
after ultrafast excitation with 1.55 eV photons. The lower panel shows the
oscillatory component with a slowly varying background subtracted. The inset
in the top panel shows the Fourier transform of the oscillatory component in
the lower panel. (b) The calibrated $\Delta z_{\mathrm{Se}}$ (top curve, left
y-axis); and corresponding $\Delta E_{xz/yz}$ and $\Delta E_{z^{2}}$ extracted
from time resolved ARPES (middle and lower curves, right y-axis). Also shown
is a schematic of the motion and the electronic band displacement. Adapted
from Gerber2017femtosecond . Reprinted with permission from AAAS.

Generally, under the influence of a coherent phonon the electronic bands shift
in energy due to their coupling with the lattice, quantified by the
deformation potential. Gerber et al. Gerber2017femtosecond  used time- and
angle-resolved photoemission spectroscopy (ARPES) to measure $\Delta
E_{xz/yz}$ and $\Delta E_{z^{2}}$, the shift in the $d_{xz/yz}$ and
$d_{z^{2}}$ bands of FeSe due to a coherent lattice vibration. They used
femtosecond hard x-ray pulses from the LCLS to measure precisely the
displacement of the Se atom $\Delta z_{\mathrm{Se}}$ associated with the
coherent $A_{1g}$ phonon mode induced by the pump pulse which modulates the
electronic bands. In their diffraction experiment, the phonon dynamics were
excited by an IR laser with 1.55 eV photons and 40 fs in duration and the
diffraction of the $(0\ 0\ 4)$ Bragg peak was probed by a delayed 8.7 keV
x-ray pulse of 25 fs duration from the LCLS. Figure 1.2(a) shows the dynamics
of the integrated intensity normalized to the laser-off intensity when pumped
with a fluence of 1.83 mJ/cm2. The oscillations are due to a 5.3 THz $A_{1g}$
phonon coherently modulating the $z$ position of the selenium atoms, depicted
in the inset of the top panel. The spectrum in the inset shows the Fourier
transform of the background-subtracted traces in Fig. 1.2(a). Using
information on the equilibrium atomic coordinates and the $A_{1g}$ mode
eigenvector, they relate the measured intensity to the absolute atomic motion
involving the Se atom away from the Fe, which at this fluence is $\Delta
z_{\mathrm{Se}}\sim 2$ pm near $\Delta t=0$ ps [see top curve in Fig. 1.2(b)].
In their time-resolved ARPES measurement, Gerber et al. observed that the
$d_{z^{2}}$ and $d_{xz/yz}$ bands shifted downwards in energy as the Se atom
moved away from the Fe atom. In Fig. 1.2(b) we show $\Delta z_{\mathrm{Se}}$
(top curve, left y-axis) and $\Delta E_{xz/yz}$, $\Delta E_{z^{2}}$ (middle
and lowest curve, right y-axis), as well as the schematic of the corresponding
atomic and band motion. A careful check of the linear fluence dependence
ensures that the measurements are representative of the equilibrium
properties. Thus by fitting the linear fluence dependence of both measurements
they were able to obtain the deformation potentials of the two bands $\Delta
E_{xz/yz}/\Delta z_{Se}=-13\pm 2.5$ and $\Delta E_{z^{2}}/\Delta
z_{Se}=-16.5\pm 3.2$ meV/pm. These values are significantly higher than those
predicted by DFT calculations Gerber2017femtosecond . Their interpretation is
that electronic correlations enhance the electron-phonon coupling in FeSe.
They argue that such concerted electron-electron and electron-phonon
interactions may enhance superconductivity in FeSe, which depends
exponentially in the value of the electron-phonon coupling. This work shows
that a precise measure of atomic motion in the linear response regime combined
with time-resolved ARPES enables model-free measurements of fundamental
physical quantities.

The discussion above considers coherent phonons where the macroscopic mode
amplitude $\langle u_{\mathbf{q}}\rangle(t)$, with $u_{\mathbf{q}}$ the
microscopic mode amplitude at wavevector $\mathbf{q}$, oscillates coherently
in time at the phonon frequency. Here the angle brackets correspond to
ensemble averages, appropriate for macroscopic samples measured with many
independent x-ray exposures. These coherent modes can be excited by a long
wavelength near-IR pump only at near-zero wavevector $\mathbf{q}\approx 0$
because of the small momentum of the near-IR photons. This is the case in
materials with well-defined translational symmetry (continuous or discrete).
Alternatively, in the presence of weak disorder with Fourier component at
$\mathbf{q}\neq 0,$ a coherent modes, $\langle u_{\mathbf{q}}\rangle(t)$ can
be generated as crystal momentum only needs to be conserved up to
$\hbar\mathbf{q}$. This can happen, for example when a static CDW develops,
which can provide the momentum to “fold” a large wavevector $\mathbf{q}$ back
onto the Brillouin zone center, as discussed above.

Optical excitation of a material also produces a different kind of phonon
dynamics with the average $\langle u_{\mathbf{q}}\rangle=0$ but where the
variance $\langle u_{-\mathbf{q}}u_{\mathbf{q}}\rangle(t)$ oscillates in time
at ${\emph{t}wice}$ the phonon frequency. Wavevector conservation still
applies here, but because this motion involves coherence between
$u_{\mathbf{q}}$ and $u_{-\mathbf{q}}$, the total wavevector is still
$\mathbf{q}-\mathbf{q}=0$. In this case, the coherence in the mode variance
can span the Brillouin zone. Trigo, Reis and coworkers demonstrated in 2013
that femtosecond x-ray diffuse scattering is sensitive to these dynamics in
experiments on near-infrared pumped germanium using the XPP station on
LCLSTrigo2013Fourier . They demonstrated temporal coherences due to phonons
across an extended regions of the Brillouin zone by recording the diffuse
x-ray intensity between Bragg peaks. Diffuse intensity, often associated with
an unwanted background, contains information on the non-periodicity of a
perfect crystal, as the the measured intensity is related to the projection of
the mean-square phonon displacements on the momentum transfer; in the case of
non-thermal vibrations it originates from temporal correlations in $\langle
u_{-\mathbf{q}}u_{\mathbf{q}}\rangle(t)$ induces by the pump. On the other
hand, when the correlations are thermal there is no well defined phase
relationship between the modes, and the diffuse x-ray intensity is time-
independent Warren1969Xray . Instead, in a pump-probe experiment the
correlations are time-dependent because the sudden excitation of the crystal
induces a well defined phase between $u_{\mathbf{q}}$ and $u_{-\mathbf{q}}$.

Figure 1.3:  (a) Static thermal diffuse intensity of germanium at room
temperature. The two closest reciprocal lattice points are labeled by their
Miller indices in the conventional cubic basis. (b) Oscillatory dynamics of
germanium after photoexcitation for representative pixels labeled u and v in
(a) [adapted by permission from Springer Nature Trigo2013Fourier ]. (c)
transverse acoustic phonon dispersion along the wavevectors marked on the
image on the inset. The closest high-symmetry points are labeled and the
corresponding coordinates are shown in reciprocal lattice units. The dashed
line shows a calculation of the lowest frequency TA branch in Ge. Reprinted
figure with permission from Zhu2015Phonon  Copyright 2015 by the American
Physical Society.

In Fig. 1.3(a) we show the x-ray diffuse intensity of a single crystal of
germanium with 10 keV x-ray photons at LCLS Trigo2013Fourier . The orientation
was such that multiple Brillouin zones were intercepted by the Ewald sphere
and covered by the detector. The two brightest spots correspond to regions
near the $(0\ 2\ 2)$ and $(1\ 1\ 3)$ reciprocal lattice points (conventional
unit cell), with the brightest pixels closer to the corresponding zone-center.
Importantly, the geometry was chosen such that the Bragg condition was not
satisfied, and the reciprocal lattice points are offset in the direction
perpendicular to the image Trigo2013Fourier . In Fig. 1.3(b) we show the
dynamics of the intensity at the two locations in the image labeled $u$, near,
and $v$, away, from the zone center, respectively. In Fig. 1.3(c) we show the
dispersion obtained from a Fourier transform of the time-domain dynamics along
wavevectors indicated in the inset, with labels indicating the closest high-
symmetry point in the Brillouin zone of Ge Zhu2015Phonon . The frequency in
Fig. 1.3(c) is obtained from a Fourier transform of the oscillatory signal and
the color intensity is related to the amplitude of oscillation. The number of
branches visible in this type of experiment is a function of the strength of
the coupling to the pump and the polarization sensitivity of the x-ray
scattering to the corresponding motion. The time-resolution also determines
the highest frequency that can be sampled. For the same reason, the frequency
resolution is governed by the longest delay scanned. In this way sub-meV
resolution for the excited phonons was achieved by measuring oscillations up
to 10 ps. The amplitude of the oscillations as a function of $\mathbf{q}$ is
related to how strongly the various phonon modes couple to the photoexcited
charge density, and thus provide a way to access $\mathbf{q}$-dependent
electron-phonon matrix elements.

##  4 Couplings among multiple degrees of freedom

In complex materials, the strong interaction among coupled charge, spin,
orbital and lattice degrees of freedom can lead to nearly degenerate phases
with different broken symmetries Imada1998metal . Optical excitation has been
used to alter this delicate balance and produce properties not accessible in
equilibrium. In this section we review how ultrafast x-ray scattering is used
to probe this coupling in complex materials. We focus first on the coupling
among phonon degrees of freedom, before moving onto coupled charge and spin
collective modes. While near-IR and optical excitation can transiently modify
the electronic distribution of a material, which in turn can change structural
and magnetic properties, recent development in mid-infrared lasers enable
tuning the central frequency of the incident laser pulse to be in resonance
with an optically active phonon in the THz rangeFoerst2011nonlinear ;
Mankowsky2014 ; Kozina2019 . This provides a potentially attractive avenue to
tune materials properties by interacting directly with the lattice modes.

We consider anharmonic couplings between phonon coordinates $Q_{1}$ and
$Q_{2}$. We write a generic expansion of the anharmonic potential in powers of
quasi-harmonic eigenmodes as

$V(Q_{1},Q_{2})=\sum_{n,m}g_{n,m}Q_{1}^{n},Q_{2}^{m}$ (1.4)

where some terms may be forbidden by the crystal symmetry. In the following we
discuss experimental manifestations due to anharmonic terms in this expansion
achieving mode couplings and structural control.

t]

Figure 1.4:  (a) Schematic of the single-cycle THz-pump x-ray-probe experiment
on SrTiO3 in Ref. Kozina2019 . (b) Dynamics of the intensity of the $(2\
\bar{2}\ 3)$ Bragg peak of SrTiO3 (black curve). Also shown is the electric
field of the THz pulse (red curve). (c) Fourier transform of the $(2\ \bar{2}\
3)$ Bragg peak response. The red and yellow are indicate the spectral regions
of the THz field and the soft phonon, respectively. Green and purple indicate
the peaks due to anharmonically coupled modes. (d) Illustration of the soft
mode (mode 1) and the $\sim 5$ THz mode (mode 2). Adapted by permission from
Springer Nature) Kozina2019 , Copyright (2019).

Kozina et al. used such a source of intense THz radiation to drive a
particularly anharmonic soft mode in the incipient ferroelectric SrTiO3 into a
regime of large amplitude Kozina2019 . The large motion induces upconversion
into a different phonon mode at higher frequency at the same
wavevectorJuraschek2018Sumfrequency . The displacements internal to the STO
unit cell were observed by ultrafast x-ray diffraction at the LCLSKozina2019 .
Fig. 1.4(a) shows a schematic of their pump-probe experiment. A single-cycle
THz pulse, with duration $<1$ ps and spectral content between 0.2 and 2.5 THz
is produced by optical rectification of a femtosecond Ti:sapphire pulse in a
single crystal of LiNbO3 (not shown), and focused on the sample by an off-axis
parabolic mirror. The sample, a 50 nm film of SrTiO3 (STO) on an LSAT
substrate is oriented such that the $(2\ \bar{2}\ 3)$ Bragg peak (referred to
a cubic Bravais lattice) diffracts the delayed x-ray pulse onto the detector.
Because of its strong anharmonicity the frequency of the STO soft mode is
strongly temperature dependent. Kozina et al. used this feature to tune the
frequency of the mode to the peak of their pump spectrum near $\sim 1$ THz.
Fig. 1.4(b) shows the dynamics of the integrated intensity of the $(2\
\bar{2}\ 3)$ peak (black curve) together on the measured electric field of the
THz field (red curve). The Fourier transform of these traces are shown in Fig.
1.4(c). The spectrum of both x-ray and THz field has a strong peak at low
frequencies $<1$ THz, indicating that the atomic motion follows the incident
field in phaseKozina2019 . More notably, unlike the THz field, the spectral
content of the XRD response has strong features in the range $1-2$ THz
associated with a strongly damped soft-mode and peaks at $5.15$ and $7.6$ THz
corresponding to two additional phonon modes of STO. Since the THz field has
no spectral content overlapping these modes, the excitation of these must be
due to anharmonic coupling with the soft mode at low frequency. Also, the
$7.6$ THz mode is not IR active, at least in the bulk. The mechanism was also
checked by tuning the mode frequency away from resonance with the THz field by
changing temperature. At 250 K the mode frequency is at 2.5 THz and the field
does not induce strong anharmonic motion, none of the features are present in
the Fourier transform of the x-ray diffraction at this temperature (not
shown)Kozina2019 .

While a full treatment of the potential contains multiple terms, in Ref.
Kozina2019  they also consider a simplified coupling between the low frequency
soft ferroelectric mode $Q_{1}$ and the $5.15$ THz mode, $Q_{2}$ that explains
the observed dependence on the field strength. Here, $Q_{1}$ couples directly
to the strong THz field as $\sim Z_{1}E_{\rm THz}(t)$, with $Z_{1}$ a mode
effective charge, while $Q_{2}$ couples to $Q_{1}$ through anharmonic terms in
the Hamiltonian $V(Q_{1},Q_{2})\sim
1/4g_{4,0}Q_{1}^{4}+g_{3,1}Q_{1}^{3}Q_{2}$. The equations of motion for this
model are

$\displaystyle\ddot{Q}_{1}+\gamma_{1}\dot{Q}_{1}+(\Omega^{2}_{1}+g_{4,0}Q_{1}^{2})Q_{1}$
$\displaystyle=$ $\displaystyle Z_{1}E_{\rm THz}(t)$ (1.5)
$\displaystyle\ddot{Q}_{2}+\gamma_{2}\dot{Q}_{2}+\Omega^{2}_{2}Q_{2}$
$\displaystyle=$ $\displaystyle-g_{3,1}Q_{1}^{3},$ (1.6)

where any other nonlinear terms in the equation for $Q_{2}$ are neglected.
Since for stability reasons $g_{4,0}>0$, the corresponding cubic term in the
equation for $Q_{1}$ makes the frequency higher for large amplitude motion,
making it detune from resonance as temperature increases, as observed
experimentally. The term $\sim-g_{3,1}Q_{1}^{3}$ in the equation for $Q_{2}$
produces harmonics of $\Omega_{1}$, which can overlap with $\Omega_{2}$ and
resonantly drive $Q_{2}$.

An especially intriguing example of non-linear THz excitation relates to work
on the high-temperature cuprates superconductors, whose crystal structures
feature stacks of copper-oxide square-net planes. Prior work has studied the
transient state of these systems after photo-excitation with radiation
centered at $20$ THz ($\sim 15~{}\mu$m wavelength) detecting increased
interlayer transport reminiscent of enhanced superconductivity Fausti2011light
. Mankowskii et al. have studied YBa2Cu3O6.5 after photo-excitation with
$c$-axis-polarized THz radiation Mankowsky2014 . This radiation was chosen to
maximize the interaction with a $B_{1u}$ infrared-active phonon associated
with $c$-axis atomic displacements and driven with high fluence in order to
maximize non-linear phonon interactions Foerst2011nonlinear . Figure 1.5
illustrate the concepts and data behind this work. The photo-excited $B_{1u}$
mode interacts with $A_{g}$ phonons through non-linear effects and induces
changes in the crystal structure, which were probed by ultra-fast diffraction
at LCLS through changes in Bragg intensity. By fitting different models that
consider these phonon excitations to their data, they conclude that the intra-
CuO2-plane distance is increased within the transient state at the expense of
reducing the inter-CuO2-plane distance. Such a change, would increase the
contribution of the Cu $d_{xy}$ orbital to low-energy electronic states, which
provides a possible explanation for the proposed enhancement of
superconductivity Mankowsky2014 .

Figure 1.5: Ultrafast modification of the YBa2Cu3O6.5 crystal structure after
multi-cycle terahertz excitation. (a) Illustration of coupling between
infrared and Raman phonon modes through non-linear phononics. (b) Depiction of
the intra-bilayer and inter-bilayer distortions in YBa2Cu3O6.5. (c)
Diffraction intensity of various Bragg peaks. The solid lines represent fits
to the data based on the coupling of the $B_{1u}$ phonon to $A_{1g}$ phonons.
The green line considers all the $A_{1g}$ phonons; whereas the red line only
includes the four most strongly coupled modes. Adapted by permission from
Springer Nature Mankowsky2014 , Copyright (2014).

In general the anharmonic interaction of phonons is not restricted to the
coupling between modes at the same wavevector. A classic example is the
anharmonic decay of phononsOrbach ; Klemens  which is the dominant mechanism
determining the thermal conductivity of insulators and often plays a central
role in structural phase transitions. In the context of anharmonic decay,
first principles density-functional perturbation theory (DFPT) calculations
have been used to calculate individual mode couplings for a couple of decades
Debernardi1995 ; Debernardi1998 ; Togo2015Distributions . While more
traditional inelastic neutron and x-ray scattering methods can give
information about the lifetime of modes at a given wavevector, measurements of
their underlying microscopic decay channels have not been possible before the
advent of the XFELs.

In an experiment by Teitelbaum et al., researchers directly measured the
anharmonic decay of a coherent zone center phonon in photo-excited bismuth
Teitelbaum2018 . Bismuth is the prototypical Peierls distorted material (which
can be viewed as a one-dimensional commensurate charge density wave) and one
of the first materials to show displacive-like excitation of coherent phonons
Cheng1990 ; Zeiger1992theory  the dynamics of which at zone-center has been
well studied by both ultrafast opticaldeCamp2001PRB ; hase1996 ; hase2002 ;
Murray2005 ; teitelbaum2018real  and ultrafast x-ray diffraction using plasma-
sources Sokolowski-Tinten2003 , the SPPS linac-based femtosecond spontaneous
synchrotron source Fritz2007  and laser-slicing in synchrotrons
Johnson2008nanoscale . Here the phonon-coordinate for the zone-center, fully-
symmetric mode $A_{1g}$ serves as a proxy for the Peierls distortion.
Photoexcitation tends to reduce the bandgap as it lessens the energetic
advantage for a structural distortion leading to the displacive excitation of
coherent phonons. The decay of the coherent phonon proceeds largely through
anharmonic coupling with pairs of acoustic phonons of equal and opposite
momenta through a parametric resonance process akin to parametric
downconversion Fahy2016 .

[width=7.5cm]figs/FigPhononDownconversionDiffuse.pdf

Figure 1.6: Downconversion of $A_{1g}$ mode in bismuth. (a) Phonon dispersion
relation in bismuth along the $\bm{q}=(\frac{1}{3}\xi\ \xi\ \xi)$ direction
showing one possible decay channel of the $A_{1g}$ mode into a degenerate pair
of LA phonons at $\bm{q}=\pm(0.1\ 0.3\ 0.3)$ and (b) the experimental
signature corresponding to this particular channel. The blue curve shows the
relative intensity change of the $(2\ 3\ 2)$ Bragg peak from which the
$A_{1g}$ mode amplitude is measured. The orange curve shows the relative
intensity change of the relevant region of diffuse scattering in the $(0\ 1\
1)$ zone. The black lines are simulations as described in the paper. Note the
dashed line indicates a $\pi/2$ phase shift between the $A_{1g}$ mode and the
target mode which is expected on parametric resonance. Reprinted figure with
permission from Teitelbaum2018  Copyright 2018 by the American Physical
Society.

The microscopic decay channels and their coupling constants can be measured
directly in the time domain through the build up of squeezed phonon
oscillations in femtosecond time-resolved x-ray diffuse scattering
measurements when compared to time-resolved diffraction. As described in
section 3, the diffuse x-ray intensity measures the mode variance $\langle
u_{-\mathbf{q}}u_{\mathbf{q}}\rangle(t)$, which is driven parametrically by
the large amplitude $A_{1g}$ mode. In the x-ray experiments of Teitelbaum et
al., the degenerate decay of the $A_{1g}$ oscillations are clearly resolved
across a wide-range of the Brillouin zone. One particular decay channel into
two acoustic modes at a specific $\mathbf{q}$ is shown schematically in Fig.
1.6. The coupling constant is determined from the ratio of amplitudes of the
squeezed phonon oscillations in $\langle
u_{-\mathbf{q}}u_{\mathbf{q}}\rangle(t)$ at a particular wavevector and the
amplitude of the $A_{1g}$ coherent phonon oscillation as measured
simultaneously in time-resolved diffraction at $\mathbf{q}=0$. In the linear
response regime the decay rate of a coherent phonon is expected to be the same
as the anharmonic decay rates obtained by perturbation theory, which in the
case of bismuth originates from the third-order force constants Fahy2016 . The
experimental results give a value for the coupling constant that is within an
order of magnitude of DFPT-based calculations, but systematically lower across
all the measured channels. While the reason for the discrepancy is not yet
understood the results mark the first momentum-resolved measurement of
anharmonic coupling channels and opens the door to similar studies in a broad
class of complex materials such as described above, where couplings off-zone
center can be critically important.

##  5 Beyond linear response

In this section we describe experiments where the perturbation is pushed
beyond linear response. In these experiments, the pump intensity is sufficient
to induce non-adiabatic dynamics not accessible thermally, exemplified by
lattice symmetry changes. We begin with systems that can be described by a
single or few time-dependent order parameter(s). Then we show experiments
where fluctuations from this single coordinate along multiple degrees of
freedom can be central to the physics of the material, such as the case of
VO2.

As one of the simplest examples of dynamics far from harmonic behavior we
consider the dynamics of the charge density wave order in SmTe3. In this
section we first describe the experiment in SmTe3Trigo2019Coherent  and
introduce the simplest Ginzburg-Landau formalism that results in anharmonic
dynamics. We later review experiments where this formalism is extended to
multiple degrees of freedom in Pr0.5Ca0.5MnO3, and finally discuss VO2 where
the dynamics cannot be described by a few degrees of freedom but involve a
continuum of modes.

Figure 1.7:  (a) Solid curves: dynamics of the integrated intensity of the
$(1\ 7\ q_{\mathrm{cdw}})$ of SmTe3 at room temperature for several incident
pump fluences (in mJ/cm2). Dashed lines are the dynamics computed from the
Ginzburg-Landau model described in the text for the corresponding values of
$\eta$ (see text). (b) sketch of the effective potential $V(y)$ for several
excitation levels, represented by $r$ in Eq. 1.7. Reprinted figure with
permission from Trigo2019Coherent  Copyright 2019 by the American Physical
Society.

SmTe3 exhibits an incommensurate CDW below $T=416$ K ru2008  corresponding to
a modulation of the Te-Te planes along the $c$-axis ru2008 . The dynamics of
the integrated intensity of the $(1\ 7\ q_{\mathrm{cdw}})$ CDW Bragg peak are
shown in Fig. 1.7(a) for several incident fluencesTrigo2019Coherent . Here
$q_{\mathrm{cdw}}\approx 2/7$ is the CDW wavevector. At low fluence, the
response shows oscillations at $1.6$ THz, related to the DECP of the amplitude
mode, Eq. 1.1, representing a small deviation from the equilibrium amplitude
of the CDW, $\delta y(t)$. On the other hand, for strong photoexcitation at
fluences $\geq 0.5$ mJ/cm2, the peak is strongly suppressed and the dynamics
become overdamped and the recover timescale becomes longer. This is a quite
common feature in the photoexcited dynamics of broken symmetry states.

At a phenomenological level the dynamics can be described by an extension of
the Ginzburg-Landau theory for second order phase transitions
goldenfeld1992lectures  to the time domain Trigo2019Coherent ;
Schaefer2014Collective ; Huber2014Coherent ; beaud2014time . The simplest
description considers a scalar order parameter $y$, representing both the
amplitude of the CDW lattice distortion and the charge density (or the value
of the gap), where in normalized units $|y|\neq 1$ and $y=0$ represent the
distorted low-symmetry phase or undistorted high-symmetry phase, respectively.
The potential energy is taken of the form

$V(y)=V_{0}\left(2ry^{2}+y^{4}\right).$ (1.7)

In equilibrium, the parameter $r=T/T_{c}-1$, with $T_{c}$ the transition
temperature, controls the relative stability of the two phases. The low-
symmetry phase with $y=\pm 1$ is the stable minimum of $V(y)$ for $r=-1$
$(T=0)$ (Fig. 1.7(b) (blue curve)). When $r\geq 0$ the potential has a single
minimum at $y=0$, and the symmetric phase is stable (Fig. 1.7(b). Importantly,
to lowest order the intensity of the CDW diffraction peaks is
$I_{\mathrm{cdw}}\propto y^{2}$. As expected, diffraction alone cannot
distinguish between $y=-1$ and $y=+1$, and also $I_{\mathrm{cdw}}=0$ for the
high-symmetry phase with $y=0$.

At ultrafast timescales the temperature is not a well-defined quantity,
however a reasonable assumption is that the parameter $r=r(t)$ is related to
the photoexcited charge density with a good approximation being
$r(t)=\eta\Theta(t)e^{-t/\tau}-1$ Huber2014Coherent , where $\Theta(t)$ is a
unit step function, $\tau$ is the decay constant of the force and $\eta$ is
controlled by the incident pump fluence Trigo2019Coherent . At room
temperature SmTe3 is in the low-symmetry (CDW-distorted) phase and we take
$y(t<0)=-1$ [$y(t<0)=+1$ is equivalent], representing the static distorted
lattice. After photoexcitation, $r(t>0)=\eta e^{-t/\tau}-1$, if the strength
of excitation $\eta$ is small we can consider a small perturbation of the
potential $V(y)$, which can be a DECP dynamics for the CDW
modeTrigo2019Coherent , with a DECP displacement as in Eq. 1.1. At higher
fluences (Fig. 1.7(a) green and red traces), the anharmonic behavior is most
severe when $\eta\geq 1$, ($r\geq 0$). Here the potential has a single minimum
at $y=0$ initially, and the system can reach the symmetric configuration where
the distortion associated to $y$ and the diffraction intensity are suppressed
completely. This is the essence of the high-fluence behavior in Fig. 1.7 (a)
as well as more complex materials like Pr0.5Ca0.5MnO3 to be discussed next
(shown in Figs. 1.8(a) and 1.8(b)). More sophisticated models build up from
this simple model Schaefer2014Collective ; Chuang2013RealTime , but the common
feature is some form of time dependent coefficient $r(t)$, representing a
phenomenological perturbation to the electronic properties. In all cases, the
anharmonic lattice dynamics results from nonlinear (usually quartic) terms in
the potential, Eq. 1.7.

Figure 1.8:  Dynamics of multiple orders in Pr0.5Ca0.5MnO3 at different
incident fluences. (a) Bragg peak from a crystal superstructure. (b) Orbital
order peak also associated with a Jahn-Teller distortion of the MnO6
octahedra. (c) charge order peak. Adapted by permission from Springer Nature
beaud2014time , Copyright (2014).

The Ginzburg-Landau description can be extended to systems with multiple order
parameters, for example representing coupled spin, lattice and charge degrees
of freedom Chuang2013real . Beaud and colleagues used the LCLS to probe the
dynamics of the phase transformation of Pr0.5Ca0.5MnO3 whose complex phase
diagram is linked to a coupling between lattice, spin, charge and orbital
degrees of freedom beaud2014time . Resonant diffraction, where the photon
energy is tuned near a core-hole absorption edge (of Mn in this case) can
enhance the contribution from valence electrons. This is the basis for novel
spectroscopic tools (see more in the resonant inelastic x-ray scattering
section 6) making diffraction sensitive to lattice (Fig. 1.8(a)), orbital
(Fig. 1.8(b)) and charge order (Fig. 1.8(c)). Below $5$ mJ/cm2 the charge
order is partially suppressed and the orbital and structural superlattice
peaks show strong oscillations from a 2.45 THz phonon mode oscillating
coherently. At fluence ¿ $5$ mJ/cm2 the three types of order are fully
suppressed. Notably, at this fluence the structural and orbital orders exhibit
an approximate doubling of the 2.45 THz frequency when the intensity reaches
zero, indicative of the $\sim y^{2}$ scaling of the Bragg intensity when the
relevant order parameter approaches the symmetric potential minimum near $y=0$
beaud2014time ; Huber2014Coherent . They model the structural and Jahn-Teller
dynamics using a phenomenological Ginzburg-Landau potential similar to the one
described above but with four effective lattice degrees of
freedombeaud2014time  representing combinations of phonon modes of
Pr0.5Ca0.5MnO3. Using resonance diffraction to enhance sensitivity to charge
order, this experiment showed that although the dynamics are highly out-of-
equilibrium, the system has a well defined order parameter across the symmetry
breaking phase transition beaud2014time .

In another experiment exploiting resonant scattering to enhance the
sensitivity of valence electrons, Lee et al. used resonant diffraction to
probe the dynamics of charge ordering in La1.75Sr0.25NiO4 (LSNO) WSLee2012 .
Unlike the PCMO experiment, where the coherent dynamics can be accounted for
by the amplitude of the order parameter alone, in LSNO the pump suppresses the
diffraction intensity of the charge-ordered peaks without affecting the peak
width. This contrasts with the thermodynamic transition where the peak width
(the inverse CDW correlation length) broadens with increasing temperature
through $T_{c}$. This indicates that, for the fluence range studied, the
photoexcited transient of the CDW does not contain topological defects
WSLee2012  in stark contrast to the expected thermal critical behavior. These
results in LSNO suggest that photoexcitation populates phase modes at
wavevectors other than the nominal ordering wavevector, and these suppress the
CDW diffraction peak through a Debye-Waller effect.

Figure 1.9:  (a) Structure of the $M_{1}$ and $R$ phases of VO2. The unit cell
is indicated with the grey shaded area. (b) top panel: snapshots of the pump-
induced change in diffuse intensity $\Delta I(\mathbf{Q},t)$ for
representative delays. Bragg peaks corresponding to the $M_{1}$ cell doubling
are indicated ($M_{1}$ Miller indices). (b) lower panel: dynamics of the
integrated intensity of $M_{1}$ peaks (red symbols), $R$ peaks (orange
symbols) and the integrated diffuse intensity (purple symbols). (c) the
computed V-V potential for the initial $M_{1}$ phase ($T_{\mathrm{el}}=290$ K)
and the computed potential under an electronic temperature of
$T_{\mathrm{el}}=2204$ K. (d) AIMD trajectories of the loss of V-V
dimerization. From Wall2018 . Reprinted with permission from AAAS.

Just as with any other type of disorder, phase fluctuations should manifest as
an increase in the diffuse intensity at Fourier components away from
$\mathbf{q}_{\mathrm{cdw}}$. Wall et al. used diffuse x-ray scattering at LCLS
to probe the photoinduced phase transition of VO2 Wall2018 . This materials
undergoes an insulator-metal transition accompanied by a structural change
between monoclinic (M1) and tetragonal structure (Rutile, R)Morin1959Oxides ;
Cavalleri2001Femtosecond  at $T_{c}=340$ K (Fig. 1.9(a)). The M1 has
approximately twice the volume of the Rutile unit cell, and additional
commensurate Bragg peaks indicate cell-doubling and long-range ordering of V-V
dimers Wall2018 ; Cavalleri2001Femtosecond . The V-V distances break into long
($d_{L}$) and short ($d_{S}$) bonds, which are $d_{L}=d_{S}=d_{R}$ in the R
structure, as indicated in Fig. 1.9(a). The top panel of Fig. 1.9(b) shows
representative snapshots of the differential diffuse intensity $\Delta
I(\mathbf{Q},t)$ covering multiple Brillouin zones of VO2. Also indicated are
sharp, localized features from the $(\bar{1}\ 0\ \bar{2})$, $(\bar{1}\ 2\ 2)$
and $(\bar{1}\ 2\ 0)$ peaks of the M1 phase (indices of the monoclinic
structure), which become suppressed in $\sim 140$ fs as shown in the lower
panel. The diffuse intensity integrated between the Bragg peaks,
representative of dynamic disorder of the V-V pairs associated with the M1
superstructure and peaks, increases with the same timescale (Fig. 1.9(b),
purple symbols). The distribution of diffuse intensity in $\mathbf{Q}$-space
resembles the distinct R-phase pattern Wall2018 , which originates from
transverse acoustic modes that are highly anharmonic, akin to dynamic
disorder, and account for large fraction of the lattice
entropyBudai2014metallization . The transient Rutile diffuse pattern develops
extremely fast, indicating that the lattice reaches the disorder in the
lattice expected for the Rutile phase in $140$ fs. These results were
confirmed by ab-initio molecular dynamics (AIMD), which showed that the local
potential of the V-V pairs (purple double-well trace in Fig. 1.9(c)) is
switched quickly to a nearly flat, quartic potential (pink trace in 1.9(c))
that allows the V atoms to quickly fill it by randomizing their positions. The
absorption of the near-IR pump is modeled by introducing a fictitious
electronic temperature $T_{\mathrm{el}}$. Fig. 1.9 shows individual
trajectories of this process from the AIMD simulationWall2018 . The
implications of these results extend beyond VO2: Many ultrafast experiments
focus on the behavior of Bragg peaks, while ignoring the role of disorder and
fluctuations. This work shows that disorder plays an important role in the
lattice of VO2 and ultrafast diffuse scattering not only provides a
visualization of the transformation pathway, but it can also yield an
alternative view of the thermodynamic transition.

##  6 Resonant inelastic x-ray scattering

An important subset of modern x-ray experiments take advantage of x-ray core
hole resonances in order to obtain more information than is otherwise
possible. Far from resonance x-rays interact with all the electrons in the
sample and tend to provide information about structure and structural dynamics
such as phonons. Tuning the x-ray energy to a core hole resonance opens routes
to couple to other degrees of freedom. RIXS, as illustrated in Fig. 1.10,
involves measuring the x-ray energy loss as a function of momentum in order to
characterize collective electronic excitations and gain insights into short
range correlations and interactions. This can be thought of as the inelastic
version of the resonant diffraction experiments described previously (Fig. 1.8
beaud2014time  and Ref. WSLee2012 ), where the change in x-ray energy is
directly measured in order to infer the properties of excitations beyond
simple measurements of the order parameter. Alternatively, it can be
conceptualized as an IXS experiment where an x-ray resonance is used to probe
spin, charge or orbital excitations as well as than phonons.

[width=7.5cm]figs/FigRIXS.pdf

Figure 1.10: A schematic of a direct RIXS process showing the initial state,
the intermediate state in which a core electron has been excited, and final
states in which a elementary quasiparticle has been created from left to
right. The energy and momentum change of the x-ray photon encodes the
dispersion of the quasiparticle.

The RIXS process is codified in the Kramers-Heisenberg equation, which
describes the second order perturbation theory for the interaction between the
x-ray photon and a material sakurai1967advanced . The coupling induced in the
scattering process can be expressed in terms of electric and magnetic dipole
and quadrupole operators (and in special cases to higher order still)
Joly2012resonant ; Ament2011resonant . These magnetic interactions occur due
to the fact that core-hole spin-orbit coupling can, if present, exchange the
orbital angular momentum of the photon with the spin in the valence band of
the material. For this reason, standard magnetic dipole resonant scattering
involves a rotation of the x-ray polarization – something that can be
exploited in order to identify and isolate magnetic scattering in experiments.
Other types of coupling, such as orbital scattering also occur in well-defined
channels in the x-ray polarization.

Figure 1.11: Example of ultrafast magnetism probed by RIXS. (a)&(b) show RIXS
spectra of Sr2IrO4 in equilibrium and 2 ps after photo-excitation. Magnetic
modes are seen in the 0-200 meV energy window and orbital excitations are seen
around 600 meV. The specific $Q$ points measured were (a) $(\pi,0)$ and (b)
$(\pi,\pi)$ Dean2016ultrafast . (c) The magnetic dispersion of Sr2IrO4 showing
magnons at around 200 meV at $(\pi,0)$ and almost gapless modes at
$(\pi,\pi)$. The blue arrow illustrates the decay of magnons towards the
energy-minimum at $(\pi,0)$ Mazzone2020trapped . Adapted by permission from
Springer Nature Dean2016ultrafast , Copyright (2016).

Syncrotron RIXS has grown dramatically in the past decades driven primarily by
improvements in energy resolution that have helped access the low-energy
degrees of freedom of correlated materials Ament2011resonant ;
Dean2015insights . Performing RIXS at XFELs requires the same complicated
spectrometer setup to be integrated with an XFEL often together with schemes
for photo-exciting the sample. We are, nonetheless, seeing successful examples
of time-resolved RIXS, which have targeted charge, orbital and spin degrees of
freedom Dean2016ultrafast ; Cao2019ultrafast ; dell2016extreme ;
Mitrano2019ultrafast ; Parchenko2019orbital ; Mazzone2020trapped .

Understanding transient magnetism is an especially important area for
ultrafast research Kirilyuk2010ultrafast ; Gandolfi2017emergent . Not only is
magnetism central to many phenomena including unconventional
superconductivity, charge-stripe correlations and quantum spin liquids, but it
is also one of the main means for information storage in computers. Recent
work by Dean and collaborators performed the first time-resolved RIXS
measurement of magnons Dean2016ultrafast . In this study, antiferromagnetic
Sr2IrO4 was studied after photoexcitation via a 100 fs, 2 ${\mu}$m pulse tuned
to excite carriers across the band gap. It was found that a fluence of $6$
mJ/cm2 was sufficient to almost completely suppress long-range magnetic order.
The authors proceeded to study the form of the collective magnons in this
transient state 2 ps after photoexcitation. As can be seen in Fig. 1.11, a
very similar magnon is observed at the $(\pi,0)$ reciprocal space point before
and after photoexcitation, which suggests that short-range magnetic
correlations can continue to exist even while long-range order is strongly
suppressed. Since the fluence of the initial excitation process excites a very
large fraction of the total magnetic sites, it was hypothesized that the
$(\pi,0)$ magnon might have been initiallly suppressed, but that it recovers
much faster than 2 ps. The low-energy magnon at $(\pi,\pi)$, is modified with
respect to the equilibrium configuration at 2 ps (see Fig. 1.11(b)) and
recovers of a few ps timescale. Based on these results, together with resonant
elastic scattering data, a picture of a two-stage magnetic recovery was
developed in which magnetic correlations are first recovered within the 2D
IrO2 planes of Sr2IrO4 and then the registry between planes in achieved in
order to return to the ground state. More recently, experiments have been
preformed on bilayer antiferromagnetic iridate Sr3Ir2O7. This system features
a very large spin gap and a much more gentle dispersion. In this case, magnons
at both $(\pi,0)$ and $(\pi,\pi)$ reciprocal space points were disturbed in
the transient state, which was interpreted in terms of a “spin-bottleneck”
effect in which the large spin gap trapped the transient magnons over all
reciprocal space Mazzone2020trapped .

Figure 1.12: RIXS measurements of charge density wave (CDW) dynamics in
La2-xBaxCuO4 $x=0.125$ Mitrano2019ultrafast . (a)&(b) show RIXS spectra at
$(0.23,0,1.5)$ as a function of time delay after a 0.1 mJ/cm2 optical pump
pulse represented as either (a) a colormap or (b) over-plotted spectra. The
quasi-elastic CDW scattering is seen to first be suppressed and then to
recover. (c) time traces of the recovery as a function of Q relative to the
peak in the ordering wavevector. (d) The exponential recovery parameter
$\hbar\gamma(q)$ as a function the momentum relative to to the peak in the
ordering wavevector. The gray shaded area represents the intensity of the CDW
in equilibrium. From From Mitrano2019ultrafast . Reprinted with permission
from AAAS.

RIXS is also known to be highly sensitive to diffuse charge density wave order
Ament2011resonant ; Dean2015insights ; miao2017high . Mitrano et al. exploited
this effect to analyze the CDW dynamics in cuprate superconductor La2-xBaxCuO4
$x=0.125$ as shown in Fig. 1.12. They observed that the CDW intensity returned
to equilibrium without exhibiting any oscillatory behavior. This is consistent
with over-damped, diffusive CDW dynamics. The authors outlined a scaling model
for the momentum and time dependence of the dynamics Mitrano2019ultrafast . A
small, but non-negligible change in the CDW $q$-vector was also observed,
which was interpreted in terms of a “Doppler effect”, in which the pump
imparts momentum to the CDW condensate.

To date, the majority of RIXS experiments has been analyzed assuming that the
x-rays represent a small perturbation to the material. In this approximation,
the spectrum can be interpreted in terms of a single photon interacting with
the sample, since the incident photon density is small enough that there is a
negligible probability of one photon encountering the effects of other
photons. Pioneering work has demonstrated that FELs now deliver peak photon
intensity sufficient to stimulate processes Rohringer2012atomic ;
Beye2013stimulated ; Yoneda2015atomic . Such work has progressed from
demonstrations in gases to work on solids, which are the subject of this
chapter and the energy employed in the demonstration has extended from extreme
ultraviolet energies, through soft x-ray into work with hard x-rays. Figure
1.13 reproduces work by Yoneda and collaborators, which realizes lasing with
the $K_{\alpha}$ emission line of Cu films Yoneda2015atomic . The authors used
a two-stage system to focus the beam from the SACLA XFEL to a 120 nm spot on
the sample and monitored the emitted x-rays with a flat silicon crystal
spectrometer. Amplified spontaneous emission of x-ray was shown by
demonstrating a super-linear dependence of the emitted x-ray intensity as a
function of the incident intensity, as plotted in Fig. 1.13(c). A notable
broadening of the spectrum was reported at high fluences which was attributed
to the presence of $3d$ core holes states as well as the target $1s$ core
holes excited. Further demonstration of the non-linear interactions was
provided by demonstrating seeded emission in which two different color pulses
are used, the setup for which is illustrated in Fig. 1.13(b). The results (as
shown in Ref. Yoneda2015atomic ) demonstrate that such a seeded approach can
amplify a specific energy emission line, as chosen by providing a narrow-
bandwidth seed pulse Yoneda2015atomic . As well as being of intrinsic interest
in terms of photonics, such approaches provide new possibilities to perform
more efficient measurement that mitigate beam damage or use the enhanced
efficiency to access weak multipolar order parameters Wang2017on .

Figure 1.13: Nonlinear processes in the x-ray regime. (a) The experimental
setup for demonstrating amplified spontaneous emission, by population
inversion for Cu $K_{\alpha}$ emission. X-rays are produced by an undulator
and focused to a 120 nm spot on a copper foil. (b) The setup showing amplified
emission, in which the pump pulse is accompanied by a seed pulse. (c) Measured
Cu $K_{\alpha}$ emission spectra a different pump intensities. The emitted
intensity increases exponentially with incident pump intensity. Adapted by
permission from Spinger Nature Yoneda2015atomic , Copyright (2015).

##  7 Outlook

In quantum materials, the complex microscopic interactions result in strong
competition between multiple nearly-degenerate ordered phases. Ultrafast
pulses provide a strategy for selectively perturbing microscopic interaction
parameters to transiently coerce materials into a desired
phaseBasov2017towards . XFEL sources will soon offer high-repetition rate and
high-average powergalayda2018 ; zhu2017sclf . The development of cavity-based
FELmarcus2019cavity , would allow for transform-limited pulses with narrow
bandwidth $\sim$ 1 meV in the case of an XFEL oscillator (XFELO) FELOscillator
; kim2009tunable  and higher peak power with moderate $\sim$100 meV bandwidth
huang2006fully ; marcus2019cavity in the case of an XFEL regenerative
amplifier(RAFEL). These developments would complement the broadband $\sim$10
eV pulses that allow for near transform limited attosecond pulses Huang2017 ;
Duris2020 . These developments will provide higher-stability and control over
the x-ray properties with applications that require either high spectral or
temporal resolution, as well as provide multi-color, multi-pulse capabilities
for x-ray pump and x-ray probe measurements of materials dynamics.

XFEL s will enable comprehensive microscopic characterization of
nonequilibrium dynamics in complex materials, by providing unprecedented
sensitivity to the collective modes. For example, high repetition rate sources
will facilitate visualization of nanoscale heat conduction with potential
prospects for microscopic control of energy flow. Novel approaches have
emerged for control of sound and heat exploiting (coherent) interferences of
waves within structures comparable to the wavelength maldovan . The impact of
new XFEL sources for probing nanoscale heat transport will be in probing the
dynamics of high wavevector phonons with nm wavelenths at femtosecond
timescales, yielding a microscopic view of wave propagation and heat
conduction.

In addition, the combination of high repetition-rate and tunable photon energy
is well-matched to studying novel light-driven quantum states through resonant
scattering, while multi-color, multi-pulse and attosecond operation will
enable a much deeper understanding of nonlinear light-matter interactions in
quantum materials. Time-resolved inelastic scattering measurements will be
possible in a variety of sample environments such as inside a diamond anvil
cell. This will enable detailed measurements of quantum critical fluctuations
of charge and spin degrees of freedom, their associated collective modes and
their couplings, while tuning continuously through a quantum critical point.
Such unique capabilities will improve our understanding of the connection
between superconductivity and the fluctuations in other order parameters
Joe2014emergence ; Feng2015itinerant . These are just some examples of how the
revolutionary properties of x-ray free electron lasers are likely to make an
lasting impact on important problems in materials science and condensed matter
physics.

This work was supported by the U.S. Department of Energy, Office of Science,
Office of Basic Energy Sciences through the Division of Materials Sciences and
Engineering under Contract Nos. DE-AC02-76SF00515 (MT and DAR) and DE-de-
sc0012704 (MPMD).

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