Probing light-driven quantum materials with ultrafast resonant inelastic X-ray scattering

Ultrafast optical pulses are an increasingly important tool for controlling quantum materials and triggering novel photo-induced phase transitions. Understanding these dynamic phenomena requires a probe sensitive to spin, charge, and orbital degrees of freedom. Time-resolved resonant inelastic X-ray scattering (trRIXS) is an emerging spectroscopic method, which responds to this need by providing unprecedented access to the finite-momentum fluctuation spectrum of photoexcited solids. In this Perspective, we briefly review state-of-the-art trRIXS experiments on condensed matter systems, as well as recent theoretical advances. We then describe future research opportunities in the context of light control of quantum matter.


I. INTRODUCTION
Understanding and controlling quantum materialsmaterial systems exhibiting quantum-mechanical effects over wide energy and length scales 1 -is a central challenge in modern condensed matter physics. Over the last two decades, ultrafast lasers have had a tremendous impact on quantum materials research and provided a novel route to on-demand engineering of their electronic and structural properties. They have not only allowed for tuning well-known states of matter far from equilibrium, e.g. magnetism 2-6 , charge/spin order 7-11 , and ferroelectricity 12-14 , but also led to novel dynamical phenomena, such as transient superconductivity [15][16][17][18] and Floquet topological phases [19][20][21] .
In nonequilibrium experiments, a sample is typically excited by a pump pulse and monitored by a subsequent probe. Interpreting the properties of a photoexcited material, especially when different instabilities are intertwined, requires precise knowledge of how the lattice, band structure, and collective fluctuations respond to the pump. To meet these needs, the ultrafast community developed ultrafast X-ray and electron diffraction for monitoring the crystal lattice [24][25][26][27] and time-and angleresolved photoemission (trARPES) for probing the electronic structure 7,9,28 . On the other hand, light-driven collective excitations are commonly investigated with ultrafast optical methods [29][30][31] , which however cannot probe their dispersion in reciprocal space due to the negligible momentum of optical photons. This implies that the microscopic distribution of nonequilibrium fluctuations in quantum materials is largely inaccessible to most existing experimental methods.
Time-resolved resonant inelastic X-ray scattering (tr-RIXS) is a momentum-resolved spectroscopy aimed to interrogate nonequilibrium collective modes, which has been recently enabled by the development of femtosecond X-ray free electron lasers (XFELs) 32,33 . As shown in Fig. 1a, trRIXS probes nonequilibrium dynamics by FIG. 1. Basics of a time-resolved resonant inelastic X-ray scattering (trRIXS) experiment. a A material is driven out of equilibrium by an ultrafast optical pump and then probed with short, delayed X-ray pulses with energy and momentum (k i , ωi) -resonantly tuned to a specific absorption edge -which are scattered off the sample with energy and momentum (k f , ω f ). The scattered photon energy is resolved by spatially separating different spectral components onto a charge-coupled device (CCD) detector with either a diffraction grating or a crystal analyzer. b Time-dependent sequence of the trRIXS process. Adapted with permission from Ref. 22 Copyrighted by the American Physical Society.
scattering ultrashort X-ray pulses tuned to a characteristic atomic absorption edge. Once the incoming X-ray photon is absorbed, the pump-excited material transitions to an intermediate state in which a core-level electron is transferred to (or above) the valence orbitals. Within a few fs, the highly unstable intermediate state decays and a valence electron fills the core hole by emitting a X-ray photon (see Fig. 1b and spin order (SO) peaks in La2−xBaxCuO4. The blue area represents the maximum momentum transfer achievable at the Cu L-edge (931 eV). The location of peaks in momentum space is denoted by the Miller indices (H, K, L) and expressed in reciprocal lattice units (r.l.u.) b Time-resolved resonant inelastic X-ray scattering (trRIXS) spectra of photoexcited La1.875Ba0.125CuO4 at QCO = (0.236, 0, 1.5) r.l.u. at the Cu L-edge for variable pump-probe time delay. c trRIXS spectra for a selection of time delays showing a prompt melting of the CO, while higher energy features of the inelastic spectrum, e.g. dd excitations/Cu 2+ emission and charge transfer (CT) excitations, remain unaffected. Error bars represent Poisson counting uncertainties. Adapted with permission from Ref. 23. rays are then analyzed in both momentum and energy, yielding information about the pump-induced collective dynamics. The resonance condition greatly enhances the RIXS cross section, but the many-body interactions in the intermediate state are what really makes trRIXS sensitive to a wide variety of charge, orbital, and spin excitations of the valence electrons 34,35 .
In this Perspective, we survey recent experimental and theoretical progress in trRIXS. Then, we outline future research opportunities emerging from these new spectroscopic capabilities. While trRIXS will have a tremendous cross-disciplinary impact, ranging from chemistry [36][37][38] to condensed matter physics, here we specifically focus on quantum materials' research.

II. A UNIQUE EXPERIMENTAL TOOL
The Stanford Linac Coherent Light Source (LCLS) has been at the forefront in developing trRIXS capabilities in both the soft and hard X-ray regime 32,33 . Over the last five years, these developments motivated a variety of experiments focused on probing nonequilibrium correlations in light-driven quantum materials, especially in connection to the problem of high-T c superconductivity.
Key challenges in the physics of high-T c superconductors are understanding the relationship between superconductivity and other low-temperature instabilities, as well as devising routes to further enhance T c . In the case of copper oxides, while superconductivity appears upon doping, hole-like carriers also form unidirectional charge (and sometimes spin) order (CO) modulations close to 1/8 doping and at temperatures above the superconducting T c , which result in diffraction peaks at a finite momentum Q CO 39-45 . Experimental and theoretical evidence suggest indeed that these two phases interplay and often compete 41,47,48 . Furthermore, ultrafast optical pulses have been found to enhance superconductivity while melting charge order correlations 15-17,49-51 , and vice versa 52 . Understanding how light affects the balance between these two phases and their collective dynamics with the aim of further optimizing superconductivity requires measuring the transient inelastic charge spectrum.
To this end, a recent trRIXS experiment at the Cu L-edge investigated the light-induced charge order dynamics of the prototypical stripe-ordered cuprate La 2−x Ba x CuO 4 (see Fig. 2a) 23,53 . trRIXS spectra clearly show that 1.55-eV photons, which transiently enhance interlayer superconducting tunneling 49,54 , also deplete the quasielastic charge order peak at Q CO (see Fig. 2b-c). Unlike conventional charge density waves 55 , the CO is found to undergo a sudden light-induced sliding motion 53 and exhibit an exponential recovery dominated by yet unobserved diffusive fluctuations at the sub-meV scale 23 . By applying a similar approach to other copper oxides such as YBa 2 Cu 3 O 7−δ and b-c Ir L-edge (11.215 keV) time-resolved resonant inelastic X-ray scattering (trRIXS) snapshots of the pseudospin excitations excited with a 620 meV pump pulse at negative time delay t (grey) and after 2 picoseconds (red) at momentum transfer Q1 = (0.5, 0) and Q2 = (0.5, 0.5) reciprocal lattice units (r.l.u.). The intensity is measured in counts (cts), while error bars represent Poisson counting uncertainties. The light-induced spectral weight change is highlighted by a red shading. Adapted with author's permission from Ref. 46 Nd 1+x Ba 2−x Cu 3 O 7−δ 56 , we expect trRIXS to provide new insights about the broader dynamical relationship between charge order and superconductivity.
Aside from charge dynamics, spin fluctuations (particularly near the antiferromagnetic wavevector Q AF = (0.5, 0.5) r.l.u.) are believed to participate to the superconducting pairing [57][58][59][60][61][62] . Thus, their optical excitation is a promising route to manipulate nonequilibrium superconductivity. Being sensitive to spin degrees of freedom through the intermediate state, trRIXS is the only available method for measuring the transient magnetic excitation spectrum as a function of momentum. Pioneering experiments provided a first glimpse of lightinduced spin dynamics in a Mott insulator 46,63 . Unlike Cu L-edge X-rays (see Fig.2a), photons at the L-edge of 5d transition metals carry enough momentum to fully map magnetic fluctuations throughout the Brillouin Zone (see Fig.3a). Among the 5d materials, iridates (Sr n+1 Ir n O 3n+1 ) are particularly interesting analogues of copper oxides. These compounds may give rise to unconventional superconductivity upon doping 64,65 , with pseudospin fluctuations (due to spin-orbit coupling) playing the same role as spins in cuprates. Driving onsite orbital excitations with infrared pump pulses resulted in a significant spectral weight reshaping of pseudospin excitations at the high symmetry points Q 1 = (0.5, 0) and Q 2 = (0.5, 0.5) reciprocal lattice units (r.l.u.) (see Fig.3b-c) 46,63 . These first snapshots of light-stimulated magnetic excitations at the Ir L-edge show that optical pump pulses with a near-zero momentum transfer have profound influence on the finite-momentum spin dynamics of quantum materials throughout the Brillouin zone.
These experiments, along with further studies of or-bital excitations 66,67 , demonstrate how trRIXS provides simultaneous access to charge, spin, and orbital degrees of freedom far from equilibrium. Its sensitivity to collective fluctuations at large momenta fills a long-standing gap in time-resolved experiments and makes trRIXS a unique tool to advance our microscopic understanding of light-driven phenomena in quantum materials.

III. AN EVOLVING THEORETICAL FRAMEWORK
A fundamental need for current and future trRIXS research lies in calculating the cross-section and predicting the many-body response of photoexcited quantum materials. Unlike other spectroscopies (e.g. trARPES, and non-resonant light/x-ray scattering) 68-71 , the RIXS process involves a four-time correlation function, due to the presence of the resonant intermediate state 22,72 . While this complexity entails modeling and numerical challenges, it is precisely the intermediate state dynamics which makes RIXS sensitive to a wide variety of collective excitations. Moreover, these excitations encode highorder correlations beyond the linear response, and thus play a crucial role in emergent phenomena with strong quantum fluctuations.
If the intermediate state is assumed to last for a negligibly short time [ultrashort core-hole lifetime (UCL) approximation], the dominant contribution to the equilibrium RIXS spectrum comes from the dynamical structure factors. 34,[73][74][75] . For this reason, early theories of trRIXS focused on calculating charge and spin structure factors in correlated electron models driven out of equilibrium 69,76 . For example, an early simulation of a one-dimensional Mott insulator excited by a realistic below-gap pump pulse found evidence of lightinduced Floquet replicas in both the spin and charge response. Different from an ideal Floquet picture, these light-engineered excitations persisted and evolved after the pump pulse 69 , thus indicating that light pulses are conducive to emergent Floquet dynamics at finite momentum.
While these results constitute an exciting starting point, the UCL approximation ignores physical processes faster than the timescale of the core-hole lifetime (longer than 2 fs at the Cu L edge 34,77 ) as well as highorder correlations 78 . Therefore, a simple calculation of the structure factors may not capture the full trRIXS spectrum and miss nonlinear spectral features such as bimagnons and dd excitations. In order to compute the full trRIXS cross-section, one needs to explicitly model the resonant probe process and account for the finite intermediate-state lifetime, leading to a computational complexity O(N 4 t ), where N t is the number of evolution steps 22,72 . Figure 4 shows the numerically-calculated trRIXS spectrum of a two-dimensional Hubbard model probed through an indirect scattering process (e.g. Cu K-edge) and explicitly accounting for intermediate state effects. Here, excitations are generated through interactions between the valence electrons and the core hole during the intermediate state 22 . Unlike the structure factors obtained within the UCL approximation, the trRIXS cross-section contains high-order excitations including the bimagnon at energy ∼ 3J (J = 4t 2 h /U is the spinexchange interaction, t h is the hopping amplitude, and U is the onsite Coulomb repulsion), in addition to Mottgap excitations. After being driven by a pump resonant with the Mott gap, spectral weight gets transferred from the Mott peak to in-gap excitations and bimagnons. In this case, the visibility of Floquet replicas of collective excitations is reduced when compared to Ref. 69 and 72, mainly due to the shorter Floquet state lifetime in the presence of electron-electron interactions. The predicted in-gap spectral weight transfer resembles the quasielastic scattering intensity enhancement observed in the iridate L-edge trRIXS spectra (see Fig. 3), and could be measured in future K-edge experiments on gapped correlated materials.
Intermediate state effects are not only required to capture richer physics in trRIXS simulations, but will also guide the interpretation of new types of inelastic scattering experiments. RIXS spectra collected while slightly detuning the incident X-ray pulse energy away from resonance can be used to selectively enhance the intensity of excitations involving specific intermediate states 22,79 and to extract information about momentum-resolved electron-phonon coupling of well-isolated modes 77,80,81 . Detuning experiments, in tandem with microscopic calculations, will be particularly impactful in interrogating the electron-phonon coupling in photoexcited charge density waves 11,52,55 , light-induced ferroelectricity 13 and superconductivity 82 .
In summary, advances in trRIXS experiments are accompanied by a steadily evolving theoretical framework aimed at understanding the observed light-induced dynamics. Current computational capabilites and stateof-the-art algorithms allow to perform accurate and predictive nonequilibrium simulations of the trRIXS spectrum in photoexcited correlated electron systems. In the future, we expect trRIXS theory to play an ever important role not just in interpreting time-resolved scattering experiments, but in leading the field towards new discoveries.

IV. A NEW SCIENTIFIC OPPORTUNITY
As shown in previous sections, trRIXS is rapidly growing into a major spectroscopy of light-driven quantum materials, and its future developments are critically tied to the pace of technological advances at XFEL facilities.
Owing to the small inelastic cross sections, trRIXS will benefit from a dramatic increase in the average XFEL spectral brightness from the current LCLS 10 20 to 10 24 − 10 25 photons s −1 mm −2 mrad −2 (0.1% bandwidth) −1 at 1 keV 32,83 . This brightness enhancement will be mainly achieved through increased repetition rates, e.g. at both the LCLS-II and the European XFEL, and will lead to order-of-magnitude improvements of the signal-to-noise ratio. Furthermore, higher energy resolution (especially in the soft X-ray regime) will enable observing lowenergy collective fluctuations, such as phonons and spin waves. At the time of this writing, large spectrometers are in construction at both the LCLS-II (NEH2.2/q-RIXS instrument) 83 and the European XFEL (hRIXS instrument). The target resolving power at 1 keV would be of order 3.0 · 10 4 (∼ 0.03 eV resolution), thus implying a 20x improvement with respect to the data shown in A tantalizing application of trRIXS is the study of spin excitations in light-driven quantum materials. Multiple theoretical studies have proposed that spinexchange interactions can be controlled by renormalizing the effective Hamiltonian interactions through a periodic drive (see Fig. 5a), in the spirit of the so-called "Floquetengineering" [85][86][87][88] . In this idealized scheme, photons dress the intermediate electronic states of the exchange (or superexchange) process, leading to an effective energy scale J eff (see Fig. 5b). Although the parameter renormalization is rigorous only for an infinitely-long periodic pump, it could also be achieved with an ultrashort laser pulse 71 . Alternatively, the spin excitations could be dynamically altered through other protocols, e.g. by distorting the lattice via the nonlinear phonon coupling 89,90 . Once J eff is modified by the pump, trRIXS measurements will interrogate changes in the dispersion, linewidth and spectral weight of the spin fluctuation spectrum (see Fig. 5c). A measurement of the dispersion allows us to disentangle renormalization effects of multiple coexisting exchange interactions 91 , which cannot be discerned from the bimagnon peak in the Raman spectrum. Furthermore, a lineshape analysis throughout the Brillouin zone enables exploring excitations beyond magnons, e.g. incoherent fluctuations in geometrically frustrated lattices [92][93][94][95] . Since the experimental energy resolution will be of order ∼ 0.03 eV in the near future 83,84 , this technique will be particularly effective for studying materials with relatively large exchange scales, such as cuprates, iridates and certain nickelates, where the spin excitations disperse over energies larger than 0.1 eV.
In addition to spin fluctuations, trRIXS will play as well a pivotal role as a probe of ultrafast charge and orbital dynamics. We envision here two promising research directions involving the charge sector of low-dimensional quantum materials, namely the time-resolved investigation of fractionalized excitations, and the search for new photoinduced condensation phenomena. In one dimension, electrons cannot propagate freely, but instead displace their neighbors due to electron-electron interactions. This leads to a breakdown ("fractionalization") of the electron in a variety of collective excitations propagating with different velocities 96 . trRIXS provides an opportunity to study these fundamental phenomena in real time by exciting a transient particle-hole plasma with a high photon energy pump and directly observing the time-dependent behavior of the collective modes contained in the transient RIXS spectrum. First experiments could focus on fractionalization in Sr 2 CuO 3 97 or CaCu 2 O 3 98 , and on manipulating their excitations by dynamically tuning the balance between spin-orbit coupling and crystal field 99 . Similarly, trRIXS experiments in photoexcited Mott insulators could reveal the fingerprints of new condensation phenomena, such as η-pairing 100,101 and dynamical p-wave superconductivity 102 . The ηpaired phase is a superfluid of doubly-occupied electronic states carrying finite momentum Q η and arising from a broken SU(2) symmetry of the Hubbard Hamiltonian. Since the η-pairing is an eigenstate, but not necessarily a ground state, pump light pulses open the possibility of stabilizing this yet unobserved phase [103][104][105][106][107] . A periodic pump is theoretically predicted to enhance pairing correlations and establish true off-diagonal long-range order by dynamically renormalizing the onsite Coulomb repulsion 107 . trRIXS would then search for η-pairing signatures in the finite-momentum pairing susceptibility 108 , namely a divergent quasi-elastic structure factor at Q η and a triplet of collective modes at energies ω = 0, ±(U −2µ) (µ being the chemical potential) 101 . Beyond η-pairing, resonantly driving orbital degrees of freedom in doped Mott insulators has also been proposed as a route to dynamically stabilize p-wave superconductivity 102 .
The search for new light-driven phenomena in materials dominated by local electronic correlations also calls for the development of more advanced spectroscopic methods. One such approach is trRIXS interferometry. Thanks to the local nature of the intermediate core holes and the intrinsic coherence of the scattering process, the RIXS signal can indeed exhibit interference among different intermediate states 109 . In the dimerized spinorbit coupled insulator Ba 3 CeIr 2 O 9 110 , the intermediate state involves a coherent superposition of a single core hole on either of the two atoms in the dimer. This leads to a cos 2 (Q·d/2) modulation of the RIXS intensity in momentum space (d being the intradimer distance). Importantly, the interference pattern varies in amplitude and phase depending on the symmetry of the excitedstate wavefunction 110 , thus providing an interferometric measurement of the local atomic orbitals. By sampling multiple interference fringes in the hard X-ray regime (e.g. Ir L-edge, and Cu K-edges), it would be possible to exquisitely resolve transient changes to the local electronic structure down to the picometer level with energy selectivity. Not just limited to dimerized compounds, this technique could likewise be applied to study nonequilibrium dynamics of confined electrons along one or two directions, such as in ladder compounds 111 or layered cuprates 82 and nickelates 112,113 .
In a very different context, confined electronic motion is also a defining property of edge states in light-induced topological phases, which could be revealed by trRIXS experiments 79 . Circularly polarized laser pulses have been shown to break time-reversal symmetry and induce transient states with non-trivial Chern numbers 21,[114][115][116] . An intriguing application of this experimental approach is the creation of tunable Floquet topological insulators (FTIs), in which topology can be manipulated by varying pump amplitude, energy, and polarization [117][118][119][120][121] . As shown in Fig. 6a, a possible route for creating a FTI in two dimensions starts from a material with bulk massless Dirac fermions. The circularly-polarized pump induces a gap opening at the Dirac point and chiral edge modes at the sample boundary, which disperse across the lightinduced bandgap (see Fig. 6b). Detecting these edge states entails probing either the transient band structure with trARPES, or their collective modes in the dynamic structure factor through trRIXS, depending on the experimental constraints. However, trRIXS offers a crucial advantage. By resonantly tuning the incident photon energy to transitions from core states into the light-induced bandgap, this technique can boost the visibility of the topological edge states over the bulk signal, and hence distinguish their dispersion from other bulk collective modes 79 (see Fig. 6c). Future developments in nano-trRIXS may enable the direct spatial imaging of edge state dynamics and, thus, further increase their visibility over bulk excitations (although these experiments would require a special handling of X-ray irradiation effects). Finally, an additional area of interest (particularly for hard X-ray trRIXS) is the light-control of candidate topological superconductors under high pressures [122][123][124][125] , which are inaccessible to photoemission experiments. This short, and by no means complete, array of examples underscores how trRIXS measurements, alongside new theoretical methods, will play an essential role in detecting and understanding new dynamic phenomena in light-controlled quantum materials. Increased XFEL FIG. 6. Generation and detection of topological collective modes. a Circularly-polarized pump pulses can drive bulk Dirac carriers (blue) and b open a gap at the Dirac points by breaking time-reversal symmetry. E and k denote energy and momentum of the band structure, respectively, while ωin stands for the incoming X-ray photon energy. In addition to the light-induced Floquet bandgap, the system exhibits chiral edge states (red) which give rise to dispersing collective modes in the time-resolved resonant inelastic X-ray scattering (trRIXS) spectrum c. These collective modes are linearly dispersing and only present at overlap with the pump excitation, unlike the particle-hole continuum due to the bulk states. ω and q indicate energy loss and momentum transfer, while kF represents the Fermi wavevector.
beamtime availability, X-ray brightness and spectrometer performance will enable more sophisticated trRIXS experiments with higher energy resolution and polarization control. The possibilities opened by these advances are difficult to grasp, but are certainly positioning trRIXS on the leading edge of a decade of discovery.