††thanks: J.H. and U.B. contributed equally to this work.††thanks: J.H. and
U.B. contributed equally to this work.

# Witnessing Light-Driven Entanglement using Time-Resolved
Resonant Inelastic X-Ray Scattering

Jordyn Hales  Department of Physics and Astronomy, Clemson University,
Clemson, SC 29634, USA  Utkarsh Bajpai  Department of Physics and Astronomy,
Clemson University, Clemson, SC 29634, USA  Tongtong Liu  Department of
Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA  Denitsa R. Baykusheva  Department of Physics, Harvard University,
Cambridge, Massachusetts 02138, USA  Mingda Li  Department of Nuclear Science
and Engineering, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA  Matteo Mitrano  Department of Physics, Harvard
University, Cambridge, Massachusetts 02138, USA  Yao Wang  Department of
Physics and Astronomy, Clemson University, Clemson, SC 29634, USA

###### Abstract

Abstract: Characterizing and controlling entanglement in quantum materials is
crucial for the development of next-generation quantum technologies. However,
defining a quantifiable figure of merit for entanglement in macroscopic solids
is theoretically and experimentally challenging. At equilibrium the presence
of entanglement can be diagnosed by extracting entanglement witnesses from
spectroscopic observables and a nonequilibrium extension of this method could
lead to the discovery of novel dynamical phenomena. Here, we propose a
systematic approach to quantify the time-dependent quantum Fisher information
and entanglement depth of transient states of quantum materials with time-
resolved resonant inelastic x-ray scattering. Using a quarter-filled extended
Hubbard model as an example, we benchmark the efficiency of this approach and
predict a light-enhanced many-body entanglement due to the proximity to a
phase boundary. Our work sets the stage for experimentally witnessing and
controlling entanglement in light-driven quantum materials via ultrafast
spectroscopic measurements.

##  I Introduction

Quantum materials are systems featuring collective electronic behavior [1]
with broad technological applications, such as superconductivity [2],
topological order [3], and quantum spin liquidity [4, 5]. Underlying these
emergent phenomena is the presence of entanglement among subparts of the
electronic wavefunctions [6, 7], which has important fundamental and applied
consequences. On one hand, quantum fluctuations caused by entanglement play an
important role in the appearance of quantum phase transitions characterized by
unconventional behavior [8, 9, 10, 11, 12, 13]. On the other hand,
entanglement constitutes a precious resource for material-based quantum
computing, where information is encoded and manipulated via arbitrary
entangled or even multipartite entangled states [14, 15]. Therefore,
accurately characterizing and controlling entanglement in the solid state is a
key step towards the realization of future quantum technologies [16].

The many-body wavefunction of a highly entangled quantum system cannot be
expressed as the direct product of multiple single-particle states in any
basis. Entangled wavefunctions in synthetic few-body quantum simulators can be
experimentally characterized [17, 18, 15, 19, 20] through the Rényi entropy
[21, 22, 23, 24, 25] and multi-point correlations [26, 27, 28, 29, 30, 31, 32,
33]. However, the measurement complexity increases with the Hilbert-space
dimension and scales exponentially with the system size. As solid-state
measurements are restricted to a limited number of macroscopic observables, a
tomography of electronic wavefunctions in quantum materials becomes
impractical [34, 35, 36, 37].

A more efficient approach for investigating entanglement in real materials
relies on determining the ``entanglement depth'' [38, 39, 40, 41], defined as
the minimum number of entangled modes required to construct a specific many-
body state. The bounds on entanglement depth of a quantum system can be
quantified through expectation values of specific operators called
entanglement witnesses [42, 43, 44, 45, 46]. Recent developments along this
direction have been enabled by use of the quantum Fisher information (QFI)
[47, 48, 49], which can be quantified from equilibrium spectroscopy [50]. By
extracting the QFI from the imaginary part of the dynamical susceptibility, it
is possible to witness a lower bound for the entanglement depth without
relying on a full tomography of the wavefunction [51, 52, 53]. This approach
has been successfully applied to the study of magnetic materials [54, 55, 56],
demonstrating its feasibility in solid-state experiments at equilibrium.

Diagnosing entanglement in quantum materials out of equilibrium would be
particularly impactful, as ultrafast lasers have led to the synthesis of
nontrivial states of matter without equilibrium analogues [57, 58, 59].
Dynamical entanglement has been theoretically demonstrated in the
wavefunctions of solvable toy models [44, 60, 61], but so far it has been
unclear how to experimentally characterize this phenomenon in nonequilibrium
spectroscopic experiments. Inspired by quantum metrology, entanglement can be
quantified by the fluctuations of local probes at a given distance or a finite
momentum, and the recently developed time-resolved resonant inelastic x-ray
scattering (trRIXS) technique [62, 63, 64] — sensitive to collective charge,
orbital, spin, and lattice excitations [62, 65, 66, 67, 68] — opens new
opportunities for diagnosing entanglement in light-driven systems.

Witnessing entanglement with trRIXS requires measuring the scattering cross-
section of a light-driven material, evaluating the transient dynamical
structure factors, calculating the QFI, and determining relevant quantum
bounds signaling the presence of entanglement for a certain observable (see
Fig. 1). Each step encapsulates a distinct challenge, and we illustrate them
by focusing on the spin degrees of freedom. First, estimating accurate spin
fluctuations using trRIXS requires separating the contributions of high-order
excitations mediated by the intermediate state of the scattering process.
Second, competing time and energy resolution effects imply that the QFI for
the instantaneous wavefunctions [69] cannot be directly extracted from the
spectrum at a single time delay and must be deconvolved on the time axis.
Third, assessing the entanglement depth requires calculating suitable
theoretical bounds for the QFI operator. While the connection between QFI and
multiparticle entanglement is well established for spin operators in magnetic
materials and gapped fermionic lattice modes [70], this connection is unclear
for interacting fermions with dopant carriers, which is particularly relevant
to the study of light-driven materials.

Figure 1: Probing light-driven entanglement in quantum materials. An intense
pump laser drives a material out of equilibrium and its time-dependent
collective excitations are probed by time-resolved Resonant Inelastic X-ray
Scattering (trRIXS). One witnesses many-body entanglement by first extracting
the nonequilibrium dynamical structure factor from the trRIXS response
function, then calculating the quantum Fisher information associated with a
specific operator using the transient dynamical structure factor, and,
finally, comparing the transient quantum Fisher information with operator-
specific quantum bounds.

Here, we address these challenges and develop a procedure to witness
entanglement using trRIXS. As a brief summary of the main results, we quantify
the impact of the finite core-hole lifetime on the evaluation of dynamical
structure factor, develop a self-consistent protocol to calculate the
instantaneous QFI, and assess the multipartite entanglement dynamics using the
QFI and appropriate theoretical bounds. Through this procedure, we focus on
the study of light-induced entanglement when a system is close to a phase
boundary and choose a one-dimensional (1D) extended Hubbard model (EHM). The
EHM under consideration, with repulsive on-site interaction and attractive
nearest-neighbor interaction, has been recently identified as the underlying
microscopic description of 1D cuprate chains (e.g., Ba2-xSrxCuO3+δ) [71]. By
comparing its dynamics with that of a pure Hubbard model, we find that the
nonlocal interaction plays a crucial role on the nonequilibrium enhancement of
many-body entanglement.

##  II Results

###  II.1 Quantifying the Time-dependent Quantum Fisher Information from
trRIXS

For a time-dependent wavefunction $|\psi(t)\rangle$ and in the Schrödinger
picture, the instantaneous QFI density associated with a local spin operator
is defined as [47, 48, 49]

$\displaystyle f_{\mathrm{Q}}(q,t)$ $\displaystyle=$
$\displaystyle\frac{4}{N}\sum_{ij}e^{iq(r_{i}-r_{j})}\left[\left\langle\psi(t)\left|{S}_{i}^{z}{S}_{j}^{z}\right|\psi(t)\right\rangle\right.$
(1)
$\displaystyle\left.-\left\langle\psi(t)\left|{S}_{i}^{z}\right|\psi(t)\right\rangle\left\langle\psi(t)\left|{S}_{j}^{z}\right|\psi(t)\right\rangle\,\right]\,.$

where $r_{i}$ is the real-space position vector of site $i$, $S_{j}^{z}$ is
the localized spin operator ($z$ component), and $N$ is the total number of
sites. The transient QFI $f_{\mathrm{Q}}(q,t)$ is completely determined by the
instantaneous wavefunction $|\psi(t)\rangle$, or an ensemble represented by
the density matrix $\rho(t)$. However, different from the integral relation
for equilibrium states [50], one cannot obtain the $f_{\mathrm{Q}}(q,t)$ by
directly integrating an instantaneous pump-probe spectrum at a certain time
delay $t$, due to the convolution with finite-width probe pulses. Such a
finite width contributes to the energy resolution of time-resolved
spectroscopy experiments and is usually comparable to the natural timescales
of the investigated systems. We describe here a strategy to extract the time-
dependent QFI from a sequence of nonequilibrium spectra.

Let us first assume that the dynamical structure factor $S({q},\omega,t)$ is
experimentally accessible, leaving the discussion of its extraction from x-ray
measurements to Sec. II.2. The $S({q},\omega,t)$ is defined as [72]

$\displaystyle S(q,\omega,t)$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
N}\sum_{ij}e^{-iq(r_{i}-r_{j})}\iint\displaylimits_{-\infty}^{+\infty}dt_{1}dt_{2}\,g\left(t_{1};t\right)g\left(t_{2};t\right)$
(2)
$\displaystyle\left\langle\hat{S}_{i}^{z}\left(t_{1}\right)\hat{S}_{j}^{z}\left(t_{2}\right)\right\rangle
e^{i\omega(t_{1}-t_{2})},$

where $g(\tau;t)$ denotes the temporal probe envelope, usually approximated
with a Gaussian profile [73]

$\displaystyle
g(\tau;t)=\frac{1}{\sigma_{\mathrm{pr}}\sqrt{2\pi}}e^{-(\tau-t)^{2}/2\sigma_{\mathrm{pr}}^{2}}\,,$
(3)

with pulse duration $\sigma_{\mathrm{pr}}$. This duration is physical,
primarily set by the spectral content of the laser pulses but also
renormalized by the instruments and the material's self-energy. We express the
pump-probe spectra in the interaction picture (same below), where the operator
$\hat{\mathcal{O}}(t)=\mathcal{U}(-\infty,t)\mathcal{O}\mathcal{U}(t,-\infty)$
evolves via the unitary operator
$\mathcal{U}(t,t_{0})=\mathcal{\hat{T}}_{t}\left[e^{-i\int_{t_{0}}^{t}\mathcal{H}(t^{\prime})dt^{\prime}}\right]$.
Here, the time-dependent Hamiltonian $\mathcal{H}(t)$ only includes the pump
(and not the probe) field.

Due to the effects of the probe pulse profile on time and energy resolution,
the spectrum $S({q},\omega,t)$ at time $t$ is determined not only by the
instantaneous wavefunction $|\psi(t)\rangle$, but also by earlier or later
wavefunctions in a finite time window. Therefore, the QFI density
$f_{\mathrm{Q}}(q,t)$, which diagnoses the entanglement of the instantaneous
wavefunction $|\psi(t)\rangle$, cannot be evaluated by simply integrating the
$S({q},\omega,t)$ along the energy axis. As we show in Supplementary Note 1,
the relationship between the time-dependent QFI $f_{\mathrm{Q}}(q,t)$ and the
transient structure factor $S({q},\omega,t)$ becomes an implicit integral
equation

$\displaystyle\int\displaylimits_{-\infty}^{+\infty}d\tau
g(\tau;t)^{2}f_{\mathrm{Q}}(q,\tau)=4\int\displaylimits_{-\infty}^{+\infty}d\omega\,S(q,\omega,t)\,.$
(4)

Note that we have assumed the absence of long-range magnetic order at the
specific momentum $q$, which is the case for the simulations in this paper. If
a long-range order is present, one should further subtract the elastic peak
from the structure factor, whose intensity corresponds to the disconnected
part (second term) of Eq. (1). In the limit of ultrashort probe pulses, i.e.,
$\sigma_{\rm pr}$ smaller than any nonequilibrium physical timescale of the
system, the envelope $g(\tau;t)^{2}$ can be approximated by a delta function
$\delta(\tau-t)$, leading to an explicit solution consistent with the
equilibrium sum-rule integral [50]. However, when the probe pulse has a finite
time duration (as in spectroscopy experiments with high energy resolution),
this approximation breaks down. In order to extract the instantaneous QFI from
this implicit Eq. (4), we expand its left-hand side and convert the equation
into a self-consistent integro-differential problem. As detailed in
Supplementary Note 1, this leads to

$f_{\mathrm{Q}}(q,t)=8\sigma_{\mathrm{pr}}\sqrt{\pi}\int\displaylimits_{-\infty}^{+\infty}d\omega\,S(q,\omega,t)+\sum_{m=1}^{\infty}\frac{\mathcal{C}_{m}}{(2m)!}\frac{\partial^{2m}f_{\mathrm{Q}}}{\partial
t^{2m}},$ (5)

where
${\mathcal{C}_{m}=-(1/\sigma_{\mathrm{pr}}\sqrt{\pi})\int_{-\infty}^{\infty}e^{-x^{2}/\sigma_{\mathrm{pr}}^{2}}}x^{2m}dx={-(\sigma_{\mathrm{pr}}^{2m}/\sqrt{\pi})\Gamma(m+1/2)}$.
In the presence of time-translational invariance, the infinite series on the
right-hand side of Eq. (5) vanishes and we reproduce the equilibrium relation
$f_{\mathrm{Q}}^{\rm
eq}(q,\tau)=8\sigma_{\mathrm{pr}}\sqrt{\pi}\int\displaylimits_{-\infty}^{+\infty}d\omega\,S^{\rm
eq}(q,\omega,t)$ [50]. However, high-order derivative terms on the right-hand
side can play a significant role far from equilibrium, as discussed in Sec.
II.3.

The self-consistently calculated $f_{\mathrm{Q}}(q,t)$ serves the purpose of
witnessing entanglement in a transient $k$-partite quantum state when
exceeding its operator-specific boundary [50]. While we use the pure-state
notation in the derivation of the QFI sum rule and choose a pure initial state
in our simulations, this approach applies to both pure and mixed initial
states (see Supplementary Note 1 for further details). This generalization
relies on considering $\langle\cdots\rangle$ as an ensemble average and on the
linearity of Eqs. (4) and (5). We note that, as specified by Hauke et al. in
Ref. [50], this protocol requires a careful normalization of the scattering
cross-section into absolute values by properly accounting for the trRIXS
matrix elements.

Figure 2: Overview of time-resolved Resonant Inelastic X-ray Scattering
(trRIXS) spectra. a-c Snapshots of the trRIXS spectra with $q=\pi/6$ and for
selected pump-probe time delays. The red arrow indicates the resonance
incident energy ${\omega_{\rm i}}$ used to analyze the structure factors. The
upper insets show the time relative to the pump pulse, whose amplitude is
$A_{0}=1$ and frequency is $\Omega=10t_{\mathrm{h}}$. The core-hole lifetime
is set as $\tau_{\mathrm{core}}=1/3\,t_{\mathrm{h}}$.

###  II.2 Time-Resolved Resonant Inelastic X-Ray Scattering

Recent experimental work on diagnosing entanglement in the solid state focused
on inelastic neutron scattering of low-dimensional spin systems [55, 56, 54],
as the neutron scattering cross section is directly proportional to the
dynamical spin structure factor [74]. While there is not yet an ultrafast
incarnation of inelastic neutron scattering, the recent development of trRIXS
provides an alternative pathway to access nonequilibrium dynamical structure
factors of spin and charge degrees of freedom [62, 63, 65, 64]. As detailed in
Methods, trRIXS is a photon-in-photon-out x-ray scattering process involving
an intermediate state with a finite lifetime. Due to the spin-orbit coupling
at the core level (e.g., the $2p$ orbitals for the transition-metal $L$-edge
RIXS), this intermediate state can involve spin flip events and couple to
magnetic excitations of the valence band [75]. Therefore, trRIXS is sensitive
to spin excitations and can be used to probe the nonequilibrium spin dynamics
of light-driven materials [76].

The trRIXS cross-section, denoted as $\mathcal{I}(q,{\omega_{\rm
i}}-\omega,{\omega_{\rm i}},t)$ in Methods, depends on energy, momentum, and
polarization of both the incident and scattered photons. In our simulation, we
select a scattering geometry with $\pi$-polarized (parallel to the scattering
plane) incident photons and $\sigma$-polarized (perpendicular to the
scattering plane) scattered photons, which maximizes the spin-flip
contribution to the trRIXS cross-section [75, 76]. Due to our focus on spin
entanglement, we keep this polarization configuration fixed throughout the
paper and omit the polarization subscripts in $\mathcal{I}$. The trRIXS
spectrum comes with two energy axes for the incident photon energy
${\omega_{\rm i}}$ and the energy loss $\omega$ (difference between incident
and scattering photon energies). Figure 2 shows sample trRIXS spectra for a
driven extended Hubbard model (see Sec. II.3) and for a range of incident
energies ${\omega_{\rm i}}$. We select the resonance ${\omega_{\rm i}}$ by
maximizing the trRIXS intensity. At fixed ${\omega_{\rm i}}$, the
nonequilibrium dynamical structure factor $S(q,\omega,t)$ can be estimated by

$S(q,\omega,t)\approx\frac{\mathcal{I}(q,{\omega_{\rm i}}-\omega,{\omega_{\rm
i}},t)}{\tau_{\rm core}^{2}|{M}^{\rm(in)}_{{q_{\rm i}}{{\bm{\varepsilon}}_{\rm
i}}}{M}^{\rm(out)}_{{q_{\rm s}}{{\bm{\varepsilon}}_{\rm s}}}|^{2}}\,,$ (6)

which implies replacing the excitation operator
$\hat{\mathcal{D}}^{\dagger}_{{q}_{s}{\epsilon}_{s}}(t_{1}^{\prime})\hat{\mathcal{D}}_{{q}_{i}{\epsilon}_{i}}(t_{1})$
with a spin-flip operator $\hat{S}^{+}(t_{1})$. Here,
$\hat{\mathcal{D}}_{{q}_{i}{\epsilon}_{i}}$ and
$\hat{\mathcal{D}}^{\dagger}_{{q}_{s}{\epsilon}_{s}}$ denote the dipole-
transition operators for the incident and scattering processes, respectively;
${M}^{\rm(in)}_{{q_{\rm i}}{{\bm{\varepsilon}}_{\rm i}}}$ and
${M}^{\rm(out)}_{{q_{\rm s}}{{\bm{\varepsilon}}_{\rm s}}}$ indicate the
corresponding matrix elements and $\tau_{\rm core}$ is the core-hole lifetime.
The definition of these factors and the full expression of the trRIXS cross-
section $\mathcal{I}(q,{\omega_{\rm i}}-\omega,{\omega_{\rm i}},t)$ are
explained in the Methods section. The estimation of Eq. (6) is analogous to
the ultrashort core-hole lifetime (UCL) approximation in equilibrium [75, 77],
and it can be proven that Eq. (6) becomes exact in the $\tau_{\rm
core}\rightarrow 0$ limit.

###  II.3 TrRIXS and QFI in a Driven Extended Hubbard Model

In this work, we aim to witness entanglement with trRIXS in a prototypical
correlated electron system. Given prior equilibrium RIXS experiments [78], we
consider 1D cuprate chains as an ideal platform for the experimental
verification of our results. Recent experiments in Ba2-xSrxCuO3+δ have
identified the EHM with mixed-sign interactions [71] as their underlying model
Hamiltonian (see Methods) and here we investigate its light-driven dynamics.
Throughout this paper, we set the on-site ($U$) and the nearest-neighbor ($V$)
interactions to $U=8t_{\rm h}$ and $V=-t_{\rm h}$, respectively, corresponding
to the characteristic values for cuprate chains [71, 79]. The existence of
this nearest-neighbor term $V$ is crucial for the presence of many-body
entanglement, as we discuss in Sec. II.4.

We introduce the pump excitation in our second-quantized electrons through the
standard Peierls substitution. The pump laser pulses are described by a vector
potential in the form of an oscillatory Gaussian
${A}(t)=A_{0}\,e^{-t^{2}/2\sigma_{\mathrm{pump}}^{2}}\cos(\Omega t)$ with
fixed width $\sigma_{\mathrm{pump}}=3t_{\mathrm{h}}^{-1}$, variable amplitude
$A_{0}$, and frequency $\Omega$. The ground state of the EHM is calculated by
the parallel Arnoldi method with Paradeisos acceleration [80, 81] and the time
evolution is evaluated by the Krylov subspace technique [82, 83]. We adopt a
12-site chain with periodic boundary conditions and quarter filling throughout
this paper, due to its proximity to the triplet-pairing phase [84, 85]. We
employ the ground state at zero temperature as the initial state, due to the
computational complexity of simulating the trRIXS cross-section of an
ensemble. The generality of this method is further discussed in Sec. II.1 and
the SI.

Figure 3: Momentum distribution for time-resolved Resonant Inelastic X-ray
Scattering (trRIXS) spectra. From left to right: trRIXS spectra for the 1D
momentum $q$ ranging from 0 to $\pi$, at $t=-5t_{\mathrm{h}}^{-1}$, $t=0$, and
$t=5t_{\mathrm{h}}^{-1}$, respectively. The incident energy ${\omega_{\rm i}}$
is set as 1.8$t_{h}$ and the upper insets show the time relative to the pump
pulse, whose condition is the same as Fig. 2.

Figure 2a shows selected theoretical $\pi$-$\sigma$-polarized trRIXS spectra.
The incident energy ${\omega_{\rm i}}=1.8t_{\mathrm{h}}$ (defined relative to
the absorption edge) is determined according to the equilibrium resonance
profile for the singly-occupied initial state. By varying the momentum
transfer $q$, the equilibrium RIXS spectrum of a quarter-filled EHM displays a
two-spinon continuum ($t\\!=\\!-5t_{h}^{-1}\\!<\\!-\sigma_{\rm pump}$), as
shown in Fig. 3. Different from undoped antiferromagnets [56], low-energy spin
excitations of a doped extended Hubbard model mainly lie near the nesting
vector $q=2k_{F}$ of spinon Fermi surface [86]. At large momenta
($q\geq\pi/2$), the trRIXS spectral weights are gradually suppressed by the
x-ray scattering matrix elements [87, 77], which we divide out as shown in Eq.
(6) when evaluating the spin QFIs. When calculating the time-dependent QFI, we
focus on a specific momentum, $q=\pi/6$. Distinct from undoped
antiferromagnets previously employed to witness equilibrium entanglement [56],
this small-momentum wavevector exhibits the most evident spectral changes and
captures the longest-range correlations supported by our system size. The 1D
system simulated in this paper does not have a spontaneous symmetry breaking,
which simplifies the entanglement analysis of the spin fluctuations. At the
selected resonance and momentum transfer, we calculate the time-dependent
$S(q,\omega,t)$ from the trRIXS intensity following Eq. (6), as shown in Fig.
4a. At the center of pump pulse, i.e., $t=0$, the excitation spectrum
experiences an overall softening [also see Fig. 4b] due to a Floquet
renormalization of the spin-exchange energy, and a slight broadening of the
spectral peak [88, 76]. These spectral changes persist long after the pump
pulse as a result of strong correlation effects [72]. However, different from
the case of light-driven spin spectra at the top of a magnon band [76], the
softening here is accompanied by an increase of the spectral intensity. To
analyze the nonequilibrium entanglement, the evolution of the intensity is
more relevant than that of the peak position.

Figure 4: Nonequilibrium dynamical structure factor and transient quantum
Fisher information (QFI). a Evolution of the dynamical spin structure factor
$S(q,\omega,t)$ estimated from $\mathcal{I}(q,{\omega_{\rm
i}}-\omega,{\omega_{\rm i}},t)$ by fixing the incident energy at ${\omega_{\rm
i}}=1.8t_{h}$, as in Fig. 2. b Spectral distribution for $t=-5t_{h}^{-1}$, 0,
and $5t_{h}^{-1}$, respectively. c Time-dependence of the snapshot quantum
Fisher information (QFI) density $\bar{f}_{\mathrm{Q}}(q,t)$ (blue curve)
evaluated by integrating the instantaneous spectral cuts [e.g., each curve in
panel (b)] and of the exact QFI (black dashed line). d Time dependence of the
QFI density ${f}_{\mathrm{Q}}(q,t)$ (red curve) evaluated by the self-
consistent iteration in Eq. (5) and comparison with the exact QFI (black
dashed line). The core-hole lifetime for these data is
$\tau_{\mathrm{core}}=1/3\,t_{\mathrm{h}}$, while the momentum transfer is
fixed to $q=\pi/6$.

To test our framework for evaluating the transient QFI of light-driven spin
degrees of freedom, we first calculate the exact values of
${f}_{\mathrm{Q}}(q,t)$ [dashed curves in Figs. 4c and d] by plugging the
simulated time-dependent wavefunctions $|\psi(t)\rangle$ into Eq. (1). Note
that this direct evaluation is theoretically possible due to the access to the
instantaneous wavefunctions, which cannot be measured in experiments. We then
proceed to evaluate the QFI as it would be done in real trRIXS experiments,
i.e., by only assessing the sequence of trRIXS snapshots without additional
theoretical knowledge about the state of the driven system. A direct extension
of the equilibrium formula in Ref. 50 entails treating each single time delay
as an equilibrium spectrum. To distinguish it from the nonequilibrium
definition in Eq. (5), we rename it as ``snapshot QFI''

$\displaystyle\bar{f}_{\mathrm{Q}}(q,t)=8\sigma_{\mathrm{pr}}\sqrt{\pi}\int
d\omega\,S(q,\omega,t)\,.$ (7)

Compared with the exact QFI densities, the $\bar{f}_{\mathrm{Q}}(q,t)$ (blue
curve) overestimates the transient increase near $t\sim 0$ and does not
capture the oscillations after the pump ends [see Fig.4c]. Such a deviation at
ultrafast timescales reflects the presence of convolution effects caused by
the finite probe width.

Figure 5: Core-hole lifetime contribution to the quantum Fisher information
(QFI) evaluation. a Evolution of $S(q,\omega,t)$ extracted from the trRIXS
intensity for core-hole lifetime set as $\tau=1/2t_{h}$. b Comparison of the
exact results of instantaneous QFI (black dashed line), the snapshot QFI
density $\bar{f}_{\mathrm{Q}}(q,t)$ (blue curve) evaluated the direct
integration, and the QFI density ${f}_{\mathrm{Q}}(q,t)$ (red curve) evaluated
by the self-consistent iteration. c,d Same as a,b but for the zero core-hole
lifetime situation, where trRIXS is identical to $S(q,\omega,t)$. The upper
insets show the evolution of the pump-field vector potential.

The failure of the snapshot QFI in capturing the exact QFI evolution requires
the introduction of the full self-consistent iteration Eq. (5). Compared to
the snapshot QFI, the latter contains high-order time-derivatives, which are
non-negligible when the spectrum varies rapidly in time. We evaluate these
high-order derivatives using finite difference methods, starting from the time
sequence of trRIXS spectra, and then solving the self-consistent equations.
This procedure leads us to the ${f}_{\mathrm{Q}}(q,t)$ indicated by the red
curve in Fig. 4d. The full self-consistent QFI is closely aligned with the
exact QFI behavior and captures the time-dependent oscillations induced by the
pump for $t>5t_{h}^{-1}$ [see Fig. 4d]. This implies that the self-consistent
calculation of the QFI is essential for capturing fast coherent dynamics and
it is not sufficient to approximate each time delay as a quasi-equilibrium
spectrum. The remaining deviation with respect to the exact evolution of QFI,
which occurs both at equilibrium and at the center of the pump, can be
attributed to the fact that the trRIXS spectrum is not identical to the spin
structure factor $S(q,\omega,t)$. Such a discrepancy is known in equilibrium
RIXS, which captures the poles of $S(q,\omega,t)$ but is less accurate in
yielding its spectral weight [77]. Physically, the intermediate state in
trRIXS has a finite lifetime and contains additional dynamics, such as multi-
magnon or spin-charge excitations, besides instantaneous spin-flip events in
the UCL limit. Due to the admixture with these excitations, Eq. 6 is only a
good approximation but not an identity. The trRIXS spectrum underestimates the
QFI at equilibrium, where the local moment is maximal and other processes are
irrelevant (see Supplementary Figure 2), while overestimating the QFI at the
center of the pump pulse where there are more charge carriers induced by the
laser.

To quantify the effect of a finite core-hole lifetime, we compare simulations
with $\tau_{\rm core}=1/2t_{h}$ and $\tau_{\rm core}=0$. As shown in Fig. 5,
the QFIs extracted from the trRIXS spectra converge towards the exact
calculations with decreasing core-hole lifetime. In the limit of an
infinitesimal $\tau_{\rm core}$ [see Figs. 5d], the self-consistently
calculated $f_{\mathrm{Q}}(q,t)$ precisely matches the exact QFI density
obtained from the instantaneous wavefunctions. In contrast, the ``snapshot
QFI'' $\bar{f}_{\mathrm{Q}}(q,t)$ (blue curve) still deviates from them,
reflecting the intrinsic error caused by the finite time resolution due to the
probe pulse. When the lifetime is not negligible, the conversion of the
equilibrium RIXS intensity into the $S(q,\omega)$ requires a more systematic
approach. One method consists of calculating the four-particle response
function as the lowest-order perturbative expansion of the lifetime $\tau_{\rm
core}$ [77]. For trRIXS, one can also correct for the finite lifetime effects
by using an overall scaling factor determined by the equilibrium RIXS
intensity and $S(q,\omega)$, both of which can be independently measured. As
discussed in Sec. II of the SI, this correction provides a good approximation
of the long-term dynamics even for a large core-hole lifetime.

Apart from accurately describing the wavefunction entanglement encoded in the
pump-probe spectrum, the self-consistent Eq. (5) also provides an alternative
way to interpret the light-induced spin fluctuation dynamics. We note that the
leading order $m=1$ term in the series of Eq. (5) is the second-order time
derivative of $f_{\mathrm{Q}}$, which reflects an inertia of the underlying
wavefunction toward the change. This inertia enables self-driving of the spin
fluctuations and explains why the QFI in Fig. 5 continues to oscillate for
$t\gtrsim 5t_{\mathrm{h}}^{-1}$ even when the pump field vanishes. In
contrast, trRIXS spectra for non-interacting fermions (and accordingly any
quantities extracted from them) completely recover to the initial equilibrium
spectra after the pump is gone [89, 90]. Therefore, the self-driving
wavefunctions are a unique feature of correlated systems with interactions and
including these high-order time-derivatives in Eq. (5) is crucial to correctly
capture the nonequilibrium QFI dynamics. The ``snapshot QFI''
$\bar{f}_{\mathrm{Q}}(q,t)$, on the other hand, serves as a good approximation
for the $f_{\mathrm{Q}}$ at the pump arrival, but starts to deviate at later
time delays where the pump tails off.

Following the pump pulse, the trRIXS spectral weight at large momenta and high
energies is transferred into small momenta [see Fig. 3]. This transfer
suggests the onset of long-wavelength spin fluctuations, although the overall
magnetic moment decreases with the generation of doublon-hole fluctuations.
While we focus on the smallest momentum $q=\pi/6$ of the cluster in this work,
the efficiency of the self-consistent approach is not restricted to any
specific momentum. As shown in Supplemental Note 3, the evaluated QFIs
accurately match those calculated through the wavefunction evolution.
Therefore, the time-dependent QFIs for different momenta witness the transfer
of entanglement at different length scales.

###  II.4 Light-Enhanced Entanglement

A reliable extraction of time- and momentum-resolved QFI $f_{\mathrm{Q}}(q,t)$
allows us to witness the entanglement depth of the driven EHM. As originally
investigated in pure spin systems [48, 49, 91], the QFI informs us about the
presence of an entangled many-body state if its value exceeds a minimum value
derived from the quantum Cramér-Rao bound. To witness entanglement dynamics in
the spin sector, we determine a quantum bound suited to diagnose multipartite
entanglement after obtaining the time-dependent QFI. Since the doped fermionic
model has a local magnetic moment $\langle m_{z}^{2}\rangle<1$, we normalize
the QFI bound by reducing the total spin $S$ by the doping concentration. This
implies that, for a $k$-producible state, the QFI obtained by a RIXS spectrum
is bounded by

${f_{\mathrm{Q}}(q,t)}\leq 4kn^{2}S^{2}\,$ (8)

where $n=\langle\sum_{i\sigma}n_{i\sigma}\rangle/N$ is the average electron
density per site and $S=1/2$ in the single-band system. Here, we neglected the
reduction of the normalization factor resulting from double occupation (which
was discussed in Refs. 92, 93) since the latter is minimal in our quarter-
filled system [see Supplementary Note 3]. It follows that the upper bound of
$f_{Q}$ for a 1-producible state ($k=1$) is 0.25, as the average density is
$n=0.5$ at quarter filling, and any value of $f_{Q}>0.25$ signals the presence
of at least bipartite entanglement (i.e., $k\geq 2$).

Figure 6: Light-manipulation of the quantum Fisher information (QFI) for
various pump conditions and models. a,b The transient QFI density $f_{Q}$ in a
light-driven EHM [i.e., Eq. (11) with $V=-t_{\mathrm{h}}$] for selected a pump
amplitudes at fixed energy $\Omega=10t_{h}$ and b pump energies at fixed
amplitude $A_{0}=0.6$. c,d Same as a,b but for a Hubbard model without
nonlocal interactions [i.e., $V=-t_{\mathrm{h}}$]. The bar to the right
denotes the range and boundary for multiparticle entanglement with a
$k$-producible state. The upper insets are cartoons of the Hamiltonian terms
contained in each model, while the middle insets show the time evolution of
the pump vector potential.

As shown in Fig. 6a, the QFI $f_{Q}$ increases following the excitation with a
$\Omega=10t_{h}$ laser pulse. As the pump amplitude increases, $f_{Q}$ is
monotonically pushed towards the boundary and exceeds 0.25 for $A>0.6$ (i.e.,
a 50% enhancement), signaling that the pump-induced state is at least
bipartite entangled. Note that time evolution is nonmonotonic, indicating the
presence of oscillations of the many-body states which are not governed by
thermalization. (This has been demonstrated by comparing nonequilibrium and
finite-temperature spectra in Ref. 72.) We now investigate the pump frequency
dependence of the light-enhanced entanglement. As shown in Fig. 6b, the light-
induced changes are weak for $\Omega<10t_{h}$. Since the particle-hole
excitations in our EHM cost $U-2V=10t_{h}$, low-frequency pump pulses do not
have enough energy to create doublon-hole pairs and scramble the spin
configurations. Thus, the transient increase of $f_{Q}$, which disappears when
the pump is over, can be attributed to Floquet engineering of the spin
exchange interaction via virtual processes [88]. The maximal enhancement is
achieved for $\Omega\sim 10-11t_{h}$, resonant with the doublon-holon
excitation energy. In this case, long-wavelength spin fluctuations are
generated through the creation and annihilation of doublon-hole pairs, which
contribute to a more entangled many-body state. This process is slightly
reduced when the pump photon energy is higher than the doublon-holon
excitation energy. We argue that the light-enhanced entanglement likely
reflects the proximity to a quantum phase transition. Although our trRIXS
simulation is restricted to a small cluster and cannot rigorously determine
phase boundaries, we compare the QFI dynamics under the same excitation
conditions in the EHM and in a doped Hubbard model (without the attractive
$V$). While the two Hamiltonians only differ by a $V\sim-t_{h}$ term, much
smaller than the dominant on-site interaction $U=8t_{h}$, previous studies
have shown the presence of a triplet superconducting phase for moderate
attractive near-neighbor interactions [94, 84, 95, 96, 97]. This phase
transition boundary has been explored through exact numerical methods in the
thermodynamic limit, suggesting that, for $U=8t_{h}$, triplet
superconductivity develops for $-1.7t_{h}\lesssim V\lesssim-1.1t_{h}$ [85].
Therefore, the EHM with $V=-t_{h}$ is expected to display a stronger light-
induced entanglement, compared with a pure Hubbard model (without $V$), due to
the proximity to this phase boundary. As shown in Figs. 6c and d, the QFI of
the driven Hubbard model is much smaller than that of the EHM at same pump
conditions, consistent with the absence of such quantum phase transition. In
other words, the nearest-neighbor interaction, whose feature was recently
identified in 1D cuprate chains but the phonon-mediated mechanism widely
exists in transition-metal oxides, is crucial towards achieving light control
of entanglement in correlated materials.

While here we mainly aim to witness spin-mode multipartite entanglement, one
might also explore entanglement depth and quantum bounds associated with
fermions. In Supplementary Note 5, we report an extension of our calculations
to the single-particle fermionic modes of the driven EHM following the
approach in Ref. 70, with the $U(1)\times U(1)$ symmetry. The obtained bounds
are weaker than Eq. (8) and they could be constrained by additional fermionic
symmetries and more complex fermionic operators [see discussion in
Supplementary Note 5]. A comprehensive study of basis-independent entanglement
witnesses for indistinguishable fermions is beyond the scope of this work.

##  III Discussion

In this work, we have connected the trRIXS spectrum of a light-driven material
to the entanglement depth of its time-dependent wavefunction. Our calculations
explicitly account for the experimental time resolution and finite core-hole
lifetime. We have developed a self-consistent procedure to extract the time-
dependent QFI from a realistic pump-probe spectrum. With the full information
about the x-ray probe pulse, one can reproduce the ultrafast dynamics of
equal-time observables beyond the limitations set by the time resolution. The
core-hole lifetime is an intrinsic property of materials and introduces a
quantitative but not qualitative deviation between the extracted QFI and the
exact values, which can be corrected by using equilibrium spectroscopy data.

Through the time-dependent QFI, we determine a small-wavevector enhancement of
multipartite entanglement in the driven state of the EHM, which becomes at
least bipartite entangled after the pump arrival. This finding is in contrast
with the time evolution of a simple Hubbard model with the same pump
parameters, in which the QFI never exceeds the minimum bound for a separable
state. We interpret this difference in terms of light-enhanced quantum
fluctuations due to the EHM proximity with a phase transition boundary between
Luttinger liquid and triplet superconductivity. The predicted enhanced
entanglement depth could be measured via future trRIXS experiments on doped
quasi-1D cuprates (e.g. Ba2-xSrxCuO3) and other low-dimensional correlated
oxides. Due to the crucial role of the attractive interactions on the light-
enhanced entanglement, these experiments will in turn help detecting nonlocal
interactions, usually challenging to characterize in quantum materials.
Additionally, since the nonequilibrium control of entanglement was recently
discussed also in the context of spintronic devices [98], our self-consistent
trRIXS approach could eventually be applied to probe time-dependent
entanglement in non-optically-driven systems.

Beyond experimentally diagnosing transient entanglement dynamics in driven
quantum materials, we anticipate the need for further theoretical
developments. Here, we mainly discuss multipartite entanglement and quantum
bounds in the spin basis. However, it is also important to characterize the
entanglement depth of fermionic degrees of freedom via observables specific to
indistinguishable particles. The intrinsic entanglement of a many-body state
is a basis-independent property [99], i.e., it is invariant under a unitary
transformation over all modes on the particle creation/annihilation operators.
In a few-body system, the Slater rank number or fermionic concurrence [100,
101, 99] can serve as basis-independent quantification of entanglement. In a
many-body material system, it is impractical to measure these quantities, but
one can construct different witnesses to set tight bounds among various
entangled states and to fit observables accessible by solid-state
measurements. A promising witness for indistinguishable fermions is operators
sensitive to paired states [102] and constructed by at most two creation and
two annihilation operators.

##  IV Methods

###  IV.1 Time-Resolved Resonant Inelastic X-Ray Scattering

The trRIXS is a photon-in-photon-out scattering process with a resonant
intermediate state, whose cross-section reads as [89, 76]

$\displaystyle\mathcal{I}({q},{\omega_{\rm s}},{\omega_{\rm i}},t)=$
$\displaystyle\frac{1}{2\pi
N}\\!\iiiint\\!dt_{1}dt_{2}dt_{1}^{\prime}dt_{2}^{\prime}e^{i{\omega_{\rm
i}}(t_{2}-t_{1})-i{\omega_{\rm s}}(t_{2}^{\prime}-t_{1}^{\prime})}$
$\displaystyle\times\,g(t_{1};t)g(t_{2};t)l(t_{1}^{\prime}-t_{1})l(t_{2}^{\prime}-t_{2})$
$\displaystyle\times\langle\hat{\mathcal{D}}^{\dagger}_{{q}_{i}{\epsilon}_{i}}(t_{2})\hat{\mathcal{D}}_{{q}_{s}{\epsilon}_{s}}(t_{2}^{\prime})\hat{\mathcal{D}}^{\dagger}_{{q}_{s}{\epsilon}_{s}}(t_{1}^{\prime})\hat{\mathcal{D}}_{{q}_{i}{\epsilon}_{i}}(t_{1})\rangle$

where $q={q_{\rm i}}-{q_{\rm s}}$ ($\omega={\omega_{\rm i}}-{\omega_{\rm s}}$)
is the momentum (energy) transfer between incident and scattered photons, and
$l(t)\\!=\\!e^{-t/\tau_{\rm core}}\theta(\tau)$ the core-hole decay lifetime.
For a direct transition, the dipole operator reads as

$\displaystyle\mathcal{D}_{q{\bm{\varepsilon}}}=\sum_{{i}\alpha\sigma}e^{-iq\cdot{r}_{i}}(M_{\alpha{\bm{\varepsilon}}}c_{{i}\sigma}^{\dagger}p_{{i}\alpha\sigma}+h.c.)\,,$
(10)

where $c_{{i}\sigma}^{\dagger}$ ($c_{{i}\sigma}$) and
$p_{{i}\alpha\sigma}^{\dagger}$ ($p_{{i}\alpha\sigma}$) denote the creation
(annihilation) operators for valence and core-level electrons at site $i$ with
spin $\sigma=\uparrow,\downarrow$. Since the transition-metal $L$-edge usually
involves multiple core-level $p$ orbitals, we label them by $\alpha$, for
which $M_{\alpha{\bm{\varepsilon}}}$ is the matrix element of the dipole
transition between each core level and the valence band via an
${\bm{\varepsilon}}$-polarized photon. For a transition-metal $L$-edge trRIXS,
the full derivation of the matrix elements is reported in Ref. 76. While these
dipole transitions preserve the total spin, the pair of photon absorption and
emission events, described by
$\hat{\mathcal{D}}^{\dagger}_{{q}_{s}{\epsilon}_{s}}(t_{1}^{\prime})\hat{\mathcal{D}}_{{q}_{i}{\epsilon}_{i}}(t_{1})$
in Eq. (IV.1), may flip a spin due to the spin-orbit coupling of the core
levels. This spin-flip process is maximized for the $\pi-\sigma$ polarization
and, therefore, provides a good estimate of the dynamical structure factor
[75, 103, 104, 77, 105]. Throughout this paper we exclusively employ such a
polarization configuration.

###  IV.2 Extended Hubbard Model

The extended Hubbard model Hamiltonian reads as

$\displaystyle{\mathcal{H}}$ $\displaystyle=$ $\displaystyle-
t_{\mathrm{h}}\sum_{i\sigma}\left[c_{i\sigma}^{\dagger}c_{i+1,\sigma}+\mathrm{H.c}\right]+\,U\sum_{i}n_{i\uparrow}n_{i\downarrow}$
(11)
$\displaystyle+V\sum_{i,\sigma,\sigma^{\prime}}n_{i\sigma}n_{i+1,\sigma^{\prime}}+\mathcal{H}_{\rm
core},$

where $c_{i\sigma}$ ($c_{i\sigma}^{\dagger}$) annihilates (creates) a valence
electron and $n_{i\sigma}=c_{i\sigma}^{\dagger}c_{i\sigma}$ is the number
operator. The valence electrons form a single band with nearest-neighbor
hopping amplitude $t_{\mathrm{h}}$, on-site Coulomb repulsion $U$, and
nearest-neighbor interaction $V$. Although the formalism is general, we choose
here model parameters capturing the physics of cuprate chain compounds such as
Ba2-xSrxCuO3+δ, namely $U=8t_{\rm h}$ and $V=-t_{\rm h}$ [89, 79]. The ground
state of EHM with these parameters at quarter filling was suggested to reside
in proximity to a spin-triplet superconducting state [94, 84, 95, 96, 97, 85].

To account for the x-ray absorption and emission processes, the full
Hamiltonian in Eq. (11) also contains terms involving the core holes

$\displaystyle\mathcal{H}_{\rm core}$ $\displaystyle=$
$\displaystyle\sum_{i\alpha\sigma}E_{\rm
edge}(1-n^{(p)}_{i\alpha\sigma})-U_{c}\sum_{i\alpha\sigma\sigma^{\prime}}n_{i\sigma}(1-n^{(p)}_{i\alpha\sigma^{\prime}})$
(12)
$\displaystyle+\lambda\sum_{i\alpha\alpha^{\prime}\atop\sigma\sigma^{\prime}}p_{i\alpha\sigma}^{\dagger}\chi_{\alpha\alpha^{\prime}}^{\sigma\sigma^{\prime}}p_{i\alpha^{\prime}\sigma^{\prime}}.$

Here, $p_{i\alpha\sigma}$ ($p_{i\alpha\sigma}^{\dagger}$) annihilates
(creates) a core-level electron with multiple degenerate orbitals labeled by
$\alpha$, corresponding to the $2p_{x,y,z}$ orbitals in transition-metal
$L$-edge RIXS, and
$n^{(p)}_{i\alpha\sigma}=p_{i\alpha\sigma}^{\dagger}p_{i\alpha\sigma}$ is the
core-level electronic number operator. The potential $U_{c}$ describes the
attractive interaction between the core-hole holes and valence-level electrons
and is fixed at 4$t_{h}$ for all the 2$p$ orbitals [106, 87, 77]. The edge
energy $E_{\rm edge}$ is selected as 938 eV to represent the Cu $L$-edge x-ray
absorption and the spin-orbit coupling $\lambda$ of the core states is set to
13 eV [106, 107].

## Data Availability

The numerical data that support the findings of this study are available from
the corresponding authors upon reasonable request. The data generated in this
study are provided in the figshare repository

## Code Availability

The relevant scripts of this study to reporoduce all figures are available at
the figshare repository. Other codes are available from the corresponding
authors upon reasonable request.

## Acknowledgements

We acknowledge insightful discussions with R.C. de Almeida, S. Ding, P. Hauke,
and P. Laurell. J.H. and Y.W. acknowledge support from U.S. Department of
Energy, Office of Science, Basic Energy Sciences, under Early Career Award No.
DE-SC0022874. D.R.B. and M.M. are primarily supported by from U.S. Department
of Energy, Office of Science, Basic Energy Sciences, under Early Career Award
No. DE-SC0022883. D.R.B. also acknowledges funding by the Swiss National
Science Foundation through Project No. P400P2$\\_$194343\. T.L. and M.L.
acknowledge the support from U.S. Department of Energy (DOE), Office of
Science, Basic Energy Sciences (BES), award No. DE-SC0021940. This research
used resources of the National Energy Research Scientific Computing Center
(NERSC), a U.S. Department of Energy Office of Science User Facility located
at Lawrence Berkeley National Laboratory, operated under Contract No. DE-
AC02-05CH11231 using NERSC award BES-ERCAP0020159.

## Author contributions

Y.W. and M.M. conceived the project. J.H., U.B, D.R.B., and Y.W. performed the
calculations and data analysis. All the authors contributed to the
interpretation of the results and the manuscript writing.

## Competing interests

The authors declare no competing interests.

##  Appendix A Derivation of the Self-Consistent Equation

The nonequilibrium dynamic spin structure factor $S({q},\omega,t)$ defined in
Eq. (2) of the main text reads as

$\displaystyle S({q},\omega,t)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}{\sigma_{\mathrm{pr}}^{2}}N}\iint\displaylimits_{-\infty}^{+\infty}d\tau
d\bar{\tau}\,e^{-(\bar{\tau}-t)^{2}/\sigma_{\rm
pr}^{2}}e^{-{\tau}^{2}/4\sigma_{\rm pr}^{2}}e^{i\omega\tau}$ (13)
$\displaystyle\left\langle\hat{\rho}_{-q}^{\mathrm{s}}\left(\bar{\tau}+\frac{\tau}{2}\right)\hat{\rho}_{q}^{\mathrm{s}}\left(\bar{\tau}-\frac{\tau}{2}\right)\right\rangle,$

where we use the Wigner transformation of the time variables
$\bar{\tau}=(t_{1}+t_{2})/2$ and $\tau=t_{1}-t_{2}$. Here, the average
notation is not restricted to a pure state, but can also be generalized to a
thermal ensemble

$\displaystyle\left\langle\hat{O}_{1}\left(t_{1}\right)\hat{O}_{2}\left(t_{2}\right)\right\rangle$
(14) $\displaystyle=$
$\displaystyle\textrm{Tr}\left[\frac{e^{-\beta\mathcal{H}}}{\mathcal{Z}}\hat{U}(-\infty,t_{1})\hat{O}_{1}\hat{U}(t_{1},t_{2})\hat{O}_{2}\hat{U}(t_{2},-\infty)\right]\,.$

The $\hat{U}(t_{1},t_{2})$ is the unitary time evolution operator,
$\mathcal{Z}$ is the partition function of the equilibrium state (at
$t=-\infty$), and $\beta$ is the inverse temperature. To simplify the
derivation, we define the momentum-space spin excitation operator

$\displaystyle\hat{\rho}^{\mathrm{s}}_{{q}}=\sum_{i}\hat{S}^{z}_{i}e^{iq\cdot
r_{i}}\,.$ (15)

Note that the $\hat{\rho}_{q}$ in Eq. (13) is written in the interaction
picture as defined in the main text Sec. II and evolves according to the
driven EHM Hamiltonian after the Peierls substitution. Therefore, Eq. (13)
does not obey time-translational invariance and the $\bar{\tau}$ cannot be
separated. Integrating in $\omega$ the left- and right-hand sides of Eq. (13)
and Taylor expanding $\hat{\rho}^{s}_{q}$ in $\tau$ lead to the identity

$\int\displaylimits_{-\infty}^{+\infty}d\omega\,S({q},\omega,t)=\frac{1}{2\pi
N\sigma_{\mathrm{pr}}^{2}}\int\displaylimits_{-\infty}^{+\infty}e^{-\frac{(t-\tau)^{2}}{\sigma_{\mathrm{pr}}^{2}}}\langle\hat{\rho}^{\mathrm{s}}_{-{q}}({\tau})\hat{\rho}^{\mathrm{s}}_{{q}}({\tau})\rangle\,d{\tau}.$
(16)

This equality states that the energy integral of the nonequilibrium dynamic
spin structure factor at a given momentum $q$ is a convolution of the two-time
spin correlation functions with the time-dependent profile of a probe with
finite pulse duration. Note that the QFI $f_{\mathrm{Q}}({q},t)$ is defined as
$4\langle\hat{\rho}^{\mathrm{s}}_{-{q}}(t)\hat{\rho}^{\mathrm{s}}_{{q}}(t)\rangle/N$,
which is an equal-time measurement for an instantaneous wavefunction at time
$t$ [47, 48, 49]. Therefore, without time-translational invariance in
nonequilibrium systems, one cannot obtain the correct time evolution of the
QFI by just integrating the dynamic spin structure factor.

However, the information about the time sequence of the QFI
$\\{f_{Q}(q,t)|-\infty<t<\infty\\}$ is encoded in the entire time evolution of
$S(q,\omega,t)$, although without snapshot-to-snapshot correspondence.
Therefore, one can apply self-consistent iterations to deconvolve $f_{Q}$ from
the integral equation. By introducing a change of variable $x=\tau-t$ we can
Taylor expand $f_{Q}(\tau)=f_{Q}(t+x)$ around $t$ in Eq. (16) to get

$\int\displaylimits_{-\infty}^{+\infty}d\omega
S({q},\omega,t)=\frac{1}{8\pi\sigma_{\mathrm{pr}}^{2}}\sum_{m=0}^{\infty}\frac{1}{m!}\frac{\partial^{m}f_{\mathrm{Q}}}{\partial
t^{m}}\int\displaylimits_{-\infty}^{+\infty}e^{-x^{2}/\sigma_{\mathrm{pr}}^{2}}x^{m}dx.$
(17)

Noting that the integral on the right-hand side is nonvanishing only for even
values of $m$, we can rewrite this equation in the form of Eq. 5 of the main
text

$f_{\mathrm{Q}}({q},t)=8\sigma_{\mathrm{pr}}\sqrt{\pi}\int\displaylimits_{-\infty}^{+\infty}d\omega\,S({q},\omega,t)+\sum_{m=1}^{\infty}\frac{\mathcal{C}_{m}}{2m!}\frac{\partial^{2m}f_{\mathrm{Q}}}{\partial
t^{2m}},$ (18)

where
$\mathcal{C}_{m}=-(1/\sigma_{\mathrm{pr}}\sqrt{\pi})\int_{-\infty}^{\infty}e^{-x^{2}/\sigma_{\mathrm{pr}}^{2}}x^{2m}dx=-(\sigma_{\mathrm{pr}}^{m-1/2}/\sqrt{\pi})\Gamma(m+1/2)$.
Since simulating correlated electrons using the Krylov-subspace requires very
small time steps, $S(q,\omega,t)$ is evaluated over a fine time grid and
enables the reliable calculation of high-order derivatives.

Eq. (18) can be numerically solved by a self-consistent iteration scheme where
we truncate the derivative series to some finite $m=M$ and start with the
snapshot QFI as

$f_{\mathrm{Q}}^{(0)}({q},t)=\bar{f}_{\mathrm{Q}}({q},t)=8\sigma_{\mathrm{pr}}\sqrt{\pi}\int
d\omega\,S({q},\omega,t)\,.$ (19)

Then the $k$-step iteration depends on the lower orders as

$f_{\mathrm{Q}}^{(k)}({q},t)=8\sigma_{\mathrm{pr}}\sqrt{\pi}\int\displaylimits_{-\infty}^{+\infty}d\omega\,S({q},\omega,t)+\sum_{m=1}^{M}\frac{\mathcal{C}_{m}}{2m!}\frac{\partial^{2m}f_{\mathrm{Q}}^{(k-1)}}{\partial
t^{2m}},$ (20)

The convergence is reached by satisfying the criterion
$|f^{(k)}_{\mathrm{Q}}-f^{(k-1)}_{\mathrm{Q}}|<\delta$, where $\delta$ is a
small number. Once convergence is reached, time-dependent QFI is reasonably
well approximated by $f_{\mathrm{Q}}({q},t)\approx
f^{(k)}_{\mathrm{Q}}({q},t)$. In practice, we employ the Fourier method to
solve this iterative equation, which does not require an artificial
termination.

##  Appendix B Correction of trRIXS with Long Core-Hole Lifetime

Figure 7: QFI dynamics evaluated using the simulated trRIXS with large core-
hole lifetime ($\tau_{\rm core}=1/2t_{h}$, red curve) and the corrected trRIXS
spectra scaled by an overall factor of 1.163 (purple curve) determined by the
ratio of equilibrium $S(q,\omega)$ and RIXS, compared with the exact QFI
evolution obtained by instantaneous wavefunctions (black dashed curve) .

Due to the finite core-hole lifetime, RIXS does not reflect the exact
$S(q,\omega)$. As shown in the Fig. 5 of the main text, trRIXS captures the
oscillation of QFI for the wavefunction dynamics, but there is a finite offset
between the values extracted from trRIXS and those evaluated exactly using the
wavefunctions. It is important to note that this offset does not increase in
time compared with that at equilibrium. This phenomenon indicates that finite
lifetime effect, which causes the deviation between RIXS and $S(q,\omega)$, is
insensitive to whether the initial state is equilibrium.

Therefore, we introduce a correction of the trRIXS intensity by a constant
factor determined by comparing the equilibrium RIXS spectrum and the
$S(q,\omega)$ (which can be measured with inelastic neutron scattering). As
shown in Fig. 7 for $\tau_{\rm core}=1/2t_{h}$, the QFI extracted from
equilibrium RIXS spectrum aligns with that from the $S(q,\omega)$ after
scaling by an overall factor 1.163. After this rescaling, the QFI extracted
from trRIXS is sufficiently accurate throughout the entire dynamics. The
efficiency of this correction in turn demonstrates that trRIXS probes the spin
excitations out of equilibrium with limited errors. This statement holds not
only for the excitation energy, but also for the spectral intensity after
accounting for this constant correction factor.

##  Appendix C Momentum Dependence and the Evolution of the Local Moment

To further test the self-consistent approach, we apply Eq. (5) of the main
text to all momenta. As reflected by the comparison between dashed and solid
lines in Fig. 8, the QFI evaluated from the trRIXS spectra and self-consistent
iteration agrees well with that calculated using instantaneous wavefunctions,
throughout the entire time evolution. This comparison verifies the general
applicability of our approach discussed in Sec. II.

Figure 8: QFI density for various momenta. The colored curves represent
$f_{q}$ evaluated from the trRIXS spectra using the self-consistent approach
[i.e., main text Eq. (5)]. The corresponding dashed black lines represent the
exact QFI density calculated using the instantaneous wavefunction in the main
text Eq. (1).

The light-induced doublon-hole fluctuations are reflected by the time-
dependent evolution of $\langle\psi(t)|m_{z}^{2}|\psi(t)\rangle$, defined as

$m_{z}^{2}=\sum_{i}(n_{i\uparrow}-n_{i\downarrow})^{2}\,.$ (21)

As shown in Fig. 9, the equilibrium local moment at quarter filling is
$\langle n\rangle\sim 0.5$. The computed value at negative time delays
slightly differs from $0.5$ due to the presence of fluctuating double
occupancies (0.0076 per site) in the quarter-filled system. At the pump
arrival, the light-driven motion of charge carriers generates additional
doublons and holes and reduces the magnetic moments. Different from the
Hubbard model ($V=0$), the magnetic moments are largely preserved in the EHM
due to the presence of a nonlocal attractive interaction $V$, as shown in Fig.
9. Since this interaction favors adjacent singly-occupied states over doublons
and holes, the local moment $\langle m_{z}^{2}\rangle$ quickly recovers after
the pump for the EHM, in contrast to the persistent decrease observed in the
Hubbard model.

This post-pump recovery of magnetic moments causes the transfer of spin
fluctuations among different wavevectors. As shown in the black dashed curves
of Fig. 8, the compression of spin fluctuations at large momenta, inherent
from the equilibrium Luttinger instability, is partially compensated by the
enhancement at small ones ($q=\pi/6$ in this system) in the EHM. Such an
enhancement is minor for the Hubbard model, as shown in the main text Fig. 6c.
Therefore, the presence of the nonlocal interactions and the proximity to a
phase boundary are crucial for the light-induced entanglement.

Figure 9: Time dependence of the local magnetic moment. Evolution of $\langle
m_{z}^{2}\rangle$ in a light-driven EHM (left) and Hubbard model (right) for
various pump conditions, as in main text Figs. 6a,b.

##  Appendix D Nonequilibrium Entanglement Entropy

In this section, we compare the nonequilibrium QFI dynamics with the
entanglement entropy, which has been widely employed as an entanglement
measure in quantum information. We consider the 1D extended-Hubbard model with
the same parameters as the main text. The Rényi entropy for an instantaneous
nonequilibrium state is defined as

$\mathcal{S}_{\mathrm{ent}}(t;\alpha)=\frac{1}{1-\alpha}\ln\mathrm{Tr}_{A}\big{[}\mathrm{Tr}_{A^{c}}\ket{\psi(t)}\bra{\psi(t)}\big{]}^{\alpha}\,.$
(22)

Here we consider the case $\alpha=1$. The label $A$ refers to the label of the
subsystem $A$, defined as half of the chain, and $A^{c}$ refers to the
remaining subsystem.

In order to compare the time-dependent entanglement entropy with the QFI
dynamics, we adopt the same pump condition as Fig. 4 of the main text, i.e.
$A_{0}=1$ and $\Omega=10t_{h}$. As shown in Fig. 10, the nonequilibrium
entanglement entropy increases at the same time of the light-driven QFI,
further supporting the notion that the nonequilibrium state exhibits light-
enhanced entanglement. It is, however, worth noting that, unlike the QFI, the
entanglement entropy is not experimentally accessible in spectroscopic
measurements of quantum materials.

Figure 10: Comparison between transient entanglement entropy and QFI density
for a light-driven extended-Hubbard model. Model parameters ($U=8t_{h}$ and
$V=-t_{h}$) and pump conditions ($A_{0}=1$ and $\Omega=10t_{h}$) are the same
as those in Fig. 4 of the main text.

##  Appendix E Upper Bound on the QFI based on the Single-Particle Fermionic
Modes

In this appendix, we follow Ref. 70 and derive the QFI bounds based on single-
particle fermionic modes (SPFM). For the 1D extended Hubbard model of $N$
sites (with periodic boundary conditions) considered in the main text, the
operator ${\rho}^{\mathrm{s}}_{q}$ in Eq. (15) obtained as a linear
superposition of local operators $S_{i}^{z}$ is non-hermitian at all momenta
(${\rho}^{\mathrm{s}\dagger}_{q}={\rho}^{\mathrm{s}}_{-q}$ except $q=0,\pi$).
The corresponding Hermitian operator can be chosen as
${O}_{q}=({\rho}_{q}^{\mathrm{s}}+{\rho}_{-q}^{\mathrm{s}})/2N$, such that

${O}_{q}=\frac{1}{N}\sum_{j,\sigma}s_{\sigma}\cos(jq)\,{c}_{j\sigma}^{\dagger}{c}_{j\sigma}.$
(23)

where $s_{\uparrow}=+,s_{\downarrow}=-$. The selection of local density
operators in ${O}_{q}$ avoids ambiguities related to fermion anticommutation
and sign. The fluctuation of this ${O}_{q}$ operator is related to the
(instantaneous) QFI of the spin operator ${\rho}^{\mathrm{s}}_{q}$ through

$4\langle{O}_{q}^{2}\rangle=\frac{1}{N}\langle\,({\rho}^{\mathrm{s}}_{q})^{2}+({\rho}^{\mathrm{s}}_{-q})^{2}\rangle+\frac{f_{\mathrm{Q}}(q,t)}{2}.$
(24)

For the simulated translational-symmetric system in Sec. IV, the time-
dependent wavefunction $|\psi(t)\rangle$ has conserved momentum quantum
number. This implies that $\langle({\rho}^{\mathrm{s}}_{q})^{2}\rangle\neq 0$
only when $q=0,\pi$. At the same wavevectors $q=0,\pi$,
${\rho}^{\mathrm{s}}_{q}$ is Hermitian, therefore Eq. (24) leads to

$f_{\mathrm{Q}}(q,t)=\frac{8\langle{O}_{q}^{2}\rangle}{(1+\delta_{q0}+\delta_{q\pi})}.$
(25)

To estimate the upper bound on $f_{\mathrm{Q}}(q,t)$, we consider the upper
bound on $\langle{O}^{2}_{q}\rangle$ as derived in Ref. 70 for a fermionic
many-body $k$-producible pure quantum state. For the creation (annihilation)
operators ${c}_{n\sigma}^{\dagger}$ (${c}_{n\sigma}$), all SPFMs are spanned
by $n=1,2,\cdots N$ and spin $\sigma=\uparrow,\downarrow$. If
$P_{m}=[(n_{1}^{m},\sigma_{1}^{m}),(n_{2}^{m},\sigma_{2}^{m}),\cdots,(n_{N_{m}}^{m},\sigma_{N_{m}}^{m})]$
denotes a unique partition that contains $N_{m}$ SPFMs, then
$\mathcal{P}=P_{1}\cup P_{2}\cup\cdots\cup P_{N_{\mathrm{parts}}}$ (where
$N_{\mathrm{parts}}\leq 2N$ is the number of partitions) forms the complete
set of SPFMs. With this notation, Ref. 70 shows that if
$|\Psi(t)\rangle_{k-\textrm{prod}}$ has a fixed number of electrons $N_{e}$,
then the lower bound on
$\langle{O}_{q}^{2}\rangle={}_{k-\textrm{prod}}\langle\Psi(t)|{O}_{q}^{2}|\Psi(t)\rangle_{k-\textrm{prod}}$
can be determined by choosing a partition
$\bar{\mathcal{P}}=\bar{P}_{1}\cup\bar{P}_{2}\cup\cdots\bar{P}_{N_{\mathrm{parts}}}$
which maximizes the right-hand side of the following inequality

$\displaystyle\langle{O}_{q}^{2}\rangle$ $\displaystyle\leq$
$\displaystyle\frac{1}{4}\sum_{m=1}^{N_{\mathrm{parts}}}\left[\sum_{\beta\in
P_{m}^{\mathrm{high}}}A_{\beta}(q)-\sum_{\beta\in
P_{m}^{\mathrm{low}}}A_{\beta}(q)\right]^{2},$ (26)

where $A_{\beta}(q)\in\\{\sigma\cos(qn)|\sigma=\pm 1,n=1,2,\cdots,N\\}$ such
that $A_{1}(q)\geq A_{2}(q)\geq\cdots\geq A_{2N}(q)$ and
$\bar{P}_{m}=P^{\mathrm{high}}_{m}\cup P^{\mathrm{mid}}_{m}\cup
P^{\mathrm{low}}_{m}$ represents a decomposition of the partition
$\bar{P}_{m}$. The right-hand side in Eq. (26) can be numerically evaluated
via the algorithm presented in Ref. 70, assuming conserved particle number
$N_{e}$. Thus, the upper bound of QFI based on SPFM can be obtained by
substituting Eq. (26) into Eq. (25), which is the fermionic bound mentioned at
the end of Sec. V.

We note that Eq. (26) does not automatically impose SU(2), translational, and
point-group symmetries, which are challenging to implement in sizeable
fermionic systems. Therefore, this bound is usually not strict for doped
fermionic systems and for 2-producible SPFM states we find it to be $\sim
1.5$, i.e. much higher than the value obtained for the spin modes in the main
text Fig. 6. Crucially, since the QFI bounds represent a sufficient but not
necessary condition for the presence of entanglement, the overall scale of the
QFI cannot exclude the presence of other highly-entangled states and the
entanglement depth should be estimated via the highest available $k$ witnessed
by different QFIs. A more accurate and basis-independent definition of
entanglement in fermionic systems requires considering indistinguishable
particles subject to anticommutation relations, and the superselection rule
[108, 109]. Entanglement measures satisfying these conditions, such as the the
Slater rank [100, 99], usually require high-order correlation functions beyond
the ones accessible with spectra in the form of the main text Eq. (2).
Therefore, distilling entanglement in the fermionic basis from spectroscopic
measurements remains an open and interesting question.

## References

  * Keimer and Moore [2017] B. Keimer and J. Moore, The Physics of Quantum Materials, Nat. Phys. 13, 1045 (2017).
  * Matthias  _et al._ [1963] B. T. Matthias, T. H. Geballe, and V. B. Compton, Superconductivity, Rev. Mod. Phys. 35, 1 (1963).
  * Wen [2017] X.-G. Wen, Colloquium: Zoo of Quantum-Topological Phases of Matter, Rev. Mod. Phys. 89, 041004 (2017).
  * Balents [2010] L. Balents, Spin Liquids in Frustrated Magnets, Nature 464, 199 (2010).
  * Broholm  _et al._ [2020] C. Broholm, R. Cava, S. Kivelson, D. Nocera, M. Norman, and T. Senthil, Quantum Spin Liquids, Science 367, eaay0668 (2020).
  * Amico  _et al._ [2008] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entanglement in Many-body Systems, Rev. Mod. Phys. 80, 517 (2008).
  * Horodecki  _et al._ [2009] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum Entanglement, Rev. Mod. Phys. 81, 865 (2009).
  * Imada  _et al._ [1998] M. Imada, A. Fujimori, and Y. Tokura, Metal-Insulator Transitions, Rev. Mod. Phys. 70, 1039 (1998).
  * Kondo  _et al._ [2015] T. Kondo, W. Malaeb, Y. Ishida, T. Sasagawa, H. Sakamoto, T. Takeuchi, T. Tohyama, and S. Shin, Point Nodes Persisting Far Beyond Tc in Bi2212, Nat. Commun. 6, 7699 (2015).
  * Faeth  _et al._ [2021] B. D. Faeth, S. Yang, J. K. Kawasaki, J. N. Nelson, P. Mishra, L. Chen, D. G. Schlom, and K. M. Shen, Incoherent Cooper Pairing and Pseudogap Behavior in Single-Layer FeSe/SrTiO3, Phys. Rev. X 11, 021054 (2021).
  * Xu  _et al._ [2021] Y. Xu, H. Rong, Q. Wang, D. Wu, Y. Hu, Y. Cai, Q. Gao, H. Yan, C. Li, C. Yin, H. Chen, J. Huang, Z. Zhu, Y. Huang, G. Liu, Z. Xu, Z. Lin, and X. J. Zhou, Spectroscopic Evidence of Superconductivity Pairing at 83 K in Single-Layer FeSe/SrTiO3 Films, Nat. Commun. 12, 2840 (2021).
  * He  _et al._ [2021] Y. He, S.-D. Chen, Z.-X. Li, D. Zhao, D. Song, Y. Yoshida, H. Eisaki, T. Wu, X.-H. Chen, D.-H. Lu, C. Meingast, T. P. Devereaux, R. J. Birgeneau, M. Hashimoto, D.-H. Lee, and Z.-X. Shen, Superconducting Fluctuations in Overdoped $Bi_{2}Sr_{2}CaCu_{2}O_{8+\delta}$, Phys. Rev. X 11, 031068 (2021).
  * Chen  _et al._ [2022] C. Chen, X. Chen, W. Tang, Z. Li, S. Wang, S. Ding, C. Jozwiak, A. Bostwick, E. Rotenberg, M. Hashimoto, D. Lu, J. P.C. Ruff, S. G. Louie, R. Birgenau, Y. Chen, Y. Wang, and Y. He, Lattice Fluctuation Induced Pseudogap in Quasi-one-dimensional Ta2NiSe5, arXiv:2203.06817 (2022).
  * Vedral [2008] V. Vedral, Quantifying Entanglement in Macroscopic Systems, Nature 453, 1004 (2008).
  * Gühne and Tóth [2009] O. Gühne and G. Tóth, Entanglement Detection, Phys. Rep. 474, 1 (2009).
  * de Leon  _et al._ [2021] N. P. de Leon, K. M. Itoh, D. Kim, K. K. Mehta, T. E. Northup, H. Paik, B. Palmer, N. Samarth, S. Sangtawesin, and D. Steuerman, Materials challenges and opportunities for quantum computing hardware, Science 372, eabb2823 (2021).
  * Häffner  _et al._ [2005] H. Häffner, W. Hänsel, C. Roos, J. Benhelm, D. Chek-al Kar, M. Chwalla, T. Körber, U. Rapol, M. Riebe, P. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, Scalable Multiparticle Entanglement of Trapped Ions, Nature 438, 643 (2005).
  * Esteve  _et al._ [2008] J. Esteve, C. Gross, A. Weller, S. Giovanazzi, and M. K. Oberthaler, Squeezing and Entanglement in a Bose–Einstein Condensate, Nature 455, 1216 (2008).
  * Gross  _et al._ [2010] C. Gross, T. Zibold, E. Nicklas, J. Esteve, and M. K. Oberthaler, Nonlinear Atom Interferometer Surpasses Classical Precision Limit, Nature 464, 1165 (2010).
  * Van Enk and Beenakker [2012] S. Van Enk and C. Beenakker, Measuring Tr$\rho$n on Single Copies of $\rho$ Using Random Measurements, Phys. Rev. Lett. 108, 110503 (2012).
  * Islam  _et al._ [2015] R. Islam, R. Ma, P. M. Preiss, M. E. Tai, A. Lukin, M. Rispoli, and M. Greiner, Measuring Entanglement Entropy in a Quantum Many-body System, Nature 528, 77 (2015).
  * Kaufman  _et al._ [2016] A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Quantum Thermalization Through Entanglement in an Isolated Many-body System, Science 353, 794 (2016).
  * Tubman [2016] N. M. Tubman, Measuring Quantum Entanglement, Machine Learning and Wave Function Tomography: Bridging Theory and Experiment with the Quantum Gas Microscope, arXiv:1609.08142 (2016).
  * Linke  _et al._ [2018] N. M. Linke, S. Johri, C. Figgatt, K. A. Landsman, A. Y. Matsuura, and C. Monroe, Measuring the Rényi Entropy of a Two-Site Fermi-Hubbard Model on a Trapped Ion Quantum Computer, Phys. Rev. A 98, 052334 (2018).
  * Brydges  _et al._ [2019] T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, and C. F. Roos, Probing Rényi Entanglement Entropy via Randomized Measurements, Science 364, 260 (2019).
  * Schweigler  _et al._ [2017] T. Schweigler, V. Kasper, S. Erne, I. Mazets, B. Rauer, F. Cataldini, T. Langen, T. Gasenzer, J. Berges, and J. Schmiedmayer, Experimental Characterization of a Quantum Many-body System via Higher-order Correlations, Nature 545, 323 (2017).
  * Hilker  _et al._ [2017] T. A. Hilker, G. Salomon, F. Grusdt, A. Omran, M. Boll, E. Demler, I. Bloch, and C. Gross, Revealing Hidden Antiferromagnetic Correlations in Doped Hubbard Chains via String Correlators, Science 357, 484 (2017).
  * Salomon  _et al._ [2019] G. Salomon, J. Koepsell, J. Vijayan, T. A. Hilker, J. Nespolo, L. Pollet, I. Bloch, and C. Gross, Direct Observation of Incommensurate Magnetism in Hubbard Chains, Nature 565, 56 (2019).
  * Koepsell  _et al._ [2019] J. Koepsell, J. Vijayan, P. Sompet, F. Grusdt, T. A. Hilker, E. Demler, G. Salomon, I. Bloch, and C. Gross, Imaging Magnetic Polarons in the Doped Fermi-Hubbard Model, Nature 572, 358 (2019).
  * Vijayan  _et al._ [2020] J. Vijayan, P. Sompet, G. Salomon, J. Koepsell, S. Hirthe, A. Bohrdt, F. Grusdt, I. Bloch, and C. Gross, Time-Resolved Observation of Spin-Charge Deconfinement in Fermionic Hubbard Chains, Science 367, 186 (2020).
  * Prüfer  _et al._ [2020] M. Prüfer, T. V. Zache, P. Kunkel, S. Lannig, A. Bonnin, H. Strobel, J. Berges, and M. K. Oberthaler, Experimental Extraction of the Quantum Effective Action for a Non-Equilibrium Many-Body System, Nat. Phys. 16, 1012 (2020).
  * Zache  _et al._ [2020] T. V. Zache, T. Schweigler, S. Erne, J. Schmiedmayer, and J. Berges, Extracting the Field Theory Description of a Quantum Many-Body System from Experimental Data, Phys. Rev. X 10, 011020 (2020).
  * Koepsell  _et al._ [2020] J. Koepsell, D. Bourgund, P. Sompet, S. Hirthe, A. Bohrdt, Y. Wang, F. Grusdt, E. Demler, G. Salomon, C. Gross, and I. Bloch, Microscopic Evolution of Doped Mott Insulators from Polaronic Metal to Fermi Liquid, Science 374, 82 (2020).
  * Wootters [1981] W. K. Wootters, Statistical Distance and Hilbert Space, Phys. Rev. D 23, 357 (1981).
  * Carvalho  _et al._ [2004] A. R. Carvalho, F. Mintert, and A. Buchleitner, Decoherence and Multipartite Entanglement, Phys. Rev. Lett. 93, 230501 (2004).
  * Mintert  _et al._ [2005] F. Mintert, M. Kuś, and A. Buchleitner, Concurrence of Mixed Multipartite Quantum States, Phys. Rev. Lett. 95, 260502 (2005).
  * Aolita and Mintert [2006] L. Aolita and F. Mintert, Measuring Multipartite Concurrence with a Single Factorizable Observable, Phys. Rev. Lett. 97, 050501 (2006).
  * Sørensen and Mølmer [2001] A. S. Sørensen and K. Mølmer, Entanglement and Extreme Spin Squeezing, Phys. Rev. Lett. 86, 4431 (2001).
  * Gühne  _et al._ [2005] O. Gühne, G. Tóth, and H. J. Briegel, Multipartite Entanglement in Spin Chains, New J. Phys. 7, 229 (2005).
  * Acín  _et al._ [2001] A. Acín, D. Bruß, M. Lewenstein, and A. Sanpera, Classification of Mixed Three-qubit States, Phys. Rev. Lett. 87, 040401 (2001).
  * Friis  _et al._ [2019] N. Friis, G. Vitagliano, M. Malik, and M. Huber, Entanglement Certification from Theory to Experiment, Nat. Rev. Phys. 1, 72 (2019).
  * Terhal [2000] B. M. Terhal, Bell Inequalities and the Separability Criterion, Phys. Lett. A 271, 319 (2000).
  * Coffman  _et al._ [2000] V. Coffman, J. Kundu, and W. K. Wootters, Distributed Entanglement, Phys. Rev. A 61, 052306 (2000).
  * Amico  _et al._ [2004] L. Amico, A. Osterloh, F. Plastina, R. Fazio, and G. M. Palma, Dynamics of Entanglement in One-Dimensional Spin Systems, Phys. Rev. A 69, 022304 (2004).
  * Roscilde  _et al._ [2004] T. Roscilde, P. Verrucchi, A. Fubini, S. Haas, and V. Tognetti, Studying Quantum Spin Systems Through Entanglement Estimators, Phys. Rev. Lett. 93, 167203 (2004).
  * Brukner  _et al._ [2006] Č. Brukner, V. Vedral, and A. Zeilinger, Crucial Role of Quantum Entanglement in Bulk Properties of Solids, Phys. Rev. A 73, 012110 (2006).
  * Pezzé and Smerzi [2009] L. Pezzé and A. Smerzi, Entanglement, Nonlinear Dynamics, and the Heisenberg Limit, Phys. Rev. Lett. 102, 100401 (2009).
  * Hyllus  _et al._ [2012] P. Hyllus, W. Laskowski, R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, L. Pezzé, and A. Smerzi, Fisher Information and Multiparticle Entanglement, Phys. Rev. A 85, 022321 (2012).
  * Tóth [2012] G. Tóth, Multipartite Entanglement and High-precision Metrology, Phys. Rev. A 85, 022322 (2012).
  * Hauke  _et al._ [2016] P. Hauke, M. Heyl, L. Tagliacozzo, and P. Zoller, Measuring Multipartite Entanglement Through Dynamic Susceptibilities, Nat. Phys. 12, 778 (2016).
  * Helstrom [1969] C. W. Helstrom, Quantum Detection and Estimation Theory, J. Stat. Phys. 1, 231 (1969).
  * Braunstein and Caves [1994] S. L. Braunstein and C. M. Caves, Statistical Distance and the Geometry of Quantum States, Phys. Rev. Lett. 72, 3439 (1994).
  * Braunstein  _et al._ [1996] S. L. Braunstein, C. M. Caves, and G. J. Milburn, Generalized Uncertainty Relations: Theory, Examples, and Lorentz Invariance, Ann. Phys. 247, 135 (1996).
  * Mathew  _et al._ [2020] G. Mathew, S. L. Silva, A. Jain, A. Mohan, D. Adroja, V. Sakai, C. Tomy, A. Banerjee, R. Goreti, R. Singh, _et al._ , Experimental Realization of Multipartite Entanglement via Quantum Fisher Information in a Uniform Antiferromagnetic Quantum Spin Chain, Phys. Rev. Research 2, 043329 (2020).
  * Laurell  _et al._ [2021] P. Laurell, A. Scheie, C. J. Mukherjee, M. M. Koza, M. Enderle, Z. Tylczynski, S. Okamoto, R. Coldea, D. A. Tennant, and G. Alvarez, Quantifying and Controlling Entanglement in the Quantum Magnet Cs2 CoCl4, Phys. Rev. Lett. 127, 037201 (2021).
  * Scheie  _et al._ [2021] A. Scheie, P. Laurell, A. Samarakoon, B. Lake, S. Nagler, G. Granroth, S. Okamoto, G. Alvarez, and D. Tennant, Witnessing Entanglement in Quantum Magnets Using Neutron Scattering, Phys. Rev. B 103, 224434 (2021).
  * Zhang and Averitt [2014] J. Zhang and R. Averitt, Dynamics and Control in Complex Transition Metal Oxides, Annu. Rev. Mater. Res. 44, 19 (2014).
  * Basov  _et al._ [2017] D. Basov, R. Averitt, and D. Hsieh, Towards Properties on Demand in Quantum Materials, Nat. Mater. 16, 1077 (2017).
  * de la Torre  _et al._ [2021] A. de la Torre, D. M. Kennes, M. Claassen, S. Gerber, J. W. McIver, and M. A. Sentef, Colloquium: Nonthermal Pathways to Ultrafast Control in Quantum Materials, Rev. Mod. Phys. 93, 041002 (2021).
  * Alba and Calabrese [2017] V. Alba and P. Calabrese, Entanglement and Thermodynamics after a Quantum Quench in Integrable Systems, Proc. Natl. Acad. Sci. U.S.A. 114, 7947 (2017).
  * Nahum  _et al._ [2017] A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum Entanglement Growth under Random Unitary Dynamics, Phys. Rev. X 7, 031016 (2017).
  * Dean  _et al._ [2016] M. P. Dean, Y. Cao, X. Liu, S. Wall, D. Zhu, R. Mankowsky, V. Thampy, X. Chen, J. G. Vale, D. Casa, J. Kim, A. Said, P. Juhas, R. Alonso-Mori, J. Glownia, A. Robert, J. Robinson, M. Sikorski, S. Song, M. Kozina, H. Lemke, L. Patthey, S. Owada, T. Katayama, M. Yabashi, Y. Tanaka, T. Togashi, J. Liu, C. R. Serrao, B. J. Kim, L. Huber, C.-L. Chang, D. F. McMorrow, M. Först, and J. P. Hill, Ultrafast Energy-and Momentum-resolved Dynamics of Magnetic Correlations in the Photo-doped Mott Insulator $Sr_{2}IrO_{4}$, Nat. Mater. 15, 601 (2016).
  * Cao  _et al._ [2019] Y. Cao, D. Mazzone, D. Meyers, J. Hill, X. Liu, S. Wall, and M. Dean, Ultrafast Dynamics of Spin and Orbital Correlations in Quantum Materials: an Energy-and Momentum-resolved Perspective, Philos. Trans. R. Soc. A 377, 20170480 (2019).
  * Mitrano and Wang [2020] M. Mitrano and Y. Wang, Probing Light-driven Quantum Materials with Ultrafast Resonant Inelastic X-ray Scattering, Commun. Phys. 3, 184 (2020).
  * Mitrano  _et al._ [2019a] M. Mitrano, S. Lee, A. A. Husain, L. Delacretaz, M. Zhu, G. de la Peña Munoz, S. X.-L. Sun, Y. I. Joe, A. H. Reid, S. F. Wandel, G. Coslovich, W. Schlotter, T. VanDriel, J. Schneeloch, G. Gu, S. Hartnoll, N. Goldenfeld, and P. Abbamonte, Ultrafast Time-Resolved X-ray Scattering Reveals Diffusive Charge Order Dynamics in La2–xBaxCuO4, Sci. Adv. 5, eaax3346 (2019a).
  * Mazzone  _et al._ [2021] D. G. Mazzone, D. Meyers, Y. Cao, J. G. Vale, C. D. Dashwood, Y. Shi, A. J. A. James, N. J. Robinson, J. Lin, V. Thampy, Y. Tanaka, A. S. Johnson, H. Miao, R. Wang, T. A. Assefa, J. Kim, D. Casa, R. Mankowsky, D. Zhu, R. Alonso-Mori, S. Song, H. Yavas, T. Katayama, M. Yabashi, Y. Kubota, S. Owada, J. Liu, J. Yang, R. M. Konik, I. K. Robinson, J. P. Hill, D. F. McMorrow, M. Först, S. Wall, X. Liu, and M. P. M. Dean, Laser-Induced Transient Magnons in Sr3Ir2O7 Throughout the Brillouin Zone, Proc. Natl. Acad. Sci. U.S.A. 118 (2021).
  * Mitrano  _et al._ [2019b] M. Mitrano, S. Lee, A. A. Husain, M. Zhu, G. de la Peña Munoz, S. X.-L. Sun, Y. I. Joe, A. H. Reid, S. F. Wandel, G. Coslovich, W. Schlotter, T. vanDriel, J. Schneeloch, G. Gu, N. Goldenfeld, and P. Abbamonte, Evidence for Photoinduced Sliding of the Charge-order Condensate in La 1.875 Ba 0.125 CuO4, Phys. Rev. B 100, 205125 (2019b).
  * Parchenko  _et al._ [2020] S. Parchenko, E. Paris, D. McNally, E. Abreu, M. Dantz, E. M. Bothschafter, A. H. Reid, W. F. Schlotter, M.-F. Lin, S. F. Wandel, G. Coslovich, S. Zohar, G. L. Dakovski, J. J. Turner, S. Moeller, Y. Tseng, M. Radovic, C. Saathe, M. Agaaker, J. E. Nordgren, S. L. Johnson, T. Schmitt, and U. Staub, Orbital Dynamics during an Ultrafast Insulator to Metal Transition, Phys. Rev. Research 2, 023110 (2020).
  * Pang and Jordan [2017] S. Pang and A. N. Jordan, Optimal Adaptive Control for Quantum Metrology with Time-dependent Hamiltonians, Nat. Commun. 8, 14695 (2017).
  * de Almeida and Hauke [2021] R. C. de Almeida and P. Hauke, From Entanglement Certification with Quench Dynamics to Multipartite Entanglement of Interacting Fermions, Phys. Rev. Research 3, L032051 (2021).
  * Chen  _et al._ [2021] Z. Chen, Y. Wang, S. N. Rebec, T. Jia, M. Hashimoto, D. Lu, B. Moritz, R. G. Moore, T. P. Devereaux, and Z.-X. Shen, Anomalously Strong Near-neighbor Attraction in Doped 1D Cuprate Chains, Science 373, 1235 (2021).
  * Wang  _et al._ [2017] Y. Wang, M. Claassen, B. Moritz, and T. Devereaux, Producing Coherent Excitations in Pumped Mott Antiferromagnetic Insulators, Phys. Rev. B 96, 235142 (2017).
  * Freericks  _et al._ [2009] J. Freericks, H. Krishnamurthy, and T. Pruschke, Theoretical Description of Time-resolved Photoemission Spectroscopy: Application to Pump-probe Experiments, Phys. Rev. Lett. 102, 136401 (2009).
  * Lovesey [1984] S. W. Lovesey,  _Theory of Neutron Scattering from Condensed Matter_ (Oxford University Press, 1984).
  * Ament  _et al._ [2009] L. J. Ament, G. Ghiringhelli, M. M. Sala, L. Braicovich, and J. van den Brink, Theoretical Demonstration of How the Dispersion of Magnetic Excitations in Cuprate Compounds Can Be Determined Using Resonant Inelastic X-Ray Scattering, Phys. Rev. Lett. 103, 117003 (2009).
  * Wang  _et al._ [2021a] Y. Wang, Y. Chen, T. P. Devereaux, B. Moritz, and M. Mitrano, X-Ray Scattering from Light-Driven Spin Fluctuations in a Doped Mott Insulator, Commun. Phys. 4, 212 (2021a).
  * Jia  _et al._ [2016] C. Jia, K. Wohlfeld, Y. Wang, B. Moritz, and T. P. Devereaux, Using RIXS to Uncover Elementary Charge and Spin Excitations, Phy. Rev. X 6, 021020 (2016).
  * Schlappa  _et al._ [2012] J. Schlappa, K. Wohlfeld, K. J. Zhou, M. Mourigal, M. W. Haverkort, V. N. Strocov, L. Hozoi, C. Monney, S. Nishimoto, S. Singh, A. Revcolevschi, J.-S. Caux, L. Patthey, H. M. Rønnow, J. van den Brink, and T. Schmitt, Spin–orbital separation in the quasi-one-dimensional mott insulator sr2cuo3, Nature 485, 82 (2012).
  * Wang  _et al._ [2021b] Y. Wang, Z. Chen, T. Shi, B. Moritz, and T. P. Devereaux, Phonon-Mediated Long-Range Attractive Interaction in One-Dimensional Cuprates, Phys. Rev. Lett. 127, 197003 (2021b).
  * Lehoucq  _et al._ [1998] R. B. Lehoucq, D. C. Sorensen, and C. Yang,  _ARPACK Users' Guide: Solution of Large-scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods_ (SIAM, 1998).
  * Jia  _et al._ [2018] C. Jia, Y. Wang, C. Mendl, B. Moritz, and T. Devereaux, Paradeisos: a Perfect Hashing Algorithm for Many-body Eigenvalue Problems, Comput. Phys. Commun. 224, 81 (2018).
  * Manmana  _et al._ [2007] S. R. Manmana, S. Wessel, R. M. Noack, and A. Muramatsu, Strongly Correlated Fermions After a Quantum Quench, Phys. Rev. Lett. 98, 210405 (2007).
  * Balzer  _et al._ [2011] M. Balzer, N. Gdaniec, and M. Potthoff, Krylov-space Approach to the Equilibrium and Nonequilibrium Single-particle Green’s Function, J. Phys. Condens. Matter 24, 035603 (2011).
  * Lin  _et al._ [1995] H. Lin, E. Gagliano, D. Campbell, E. Fradkin, and J. Gubernatis, The phase diagram of the one-dimensional extended hubbard model, in  _The Hubbard Model_ (Springer, 1995) pp. 315–326.
  * Qu  _et al._ [2021] D.-W. Qu, B.-B. Chen, H.-C. Jiang, Y. Wang, and W. Li, Spin-Triplet Pairing Induced by Near-Neighbor Attraction in the Extended Hubbard Model, arXiv:2110.00564 (2021).
  * Pärschke  _et al._ [2019] E. M. Pärschke, Y. Wang, B. Moritz, T. P. Devereaux, C.-C. Chen, and K. Wohlfeld, Numerical Investigation of Spin Excitations in a Doped Spin Chain, Phys. Rev. B 99, 205102 (2019).
  * Jia  _et al._ [2014] C. J. Jia, E. A. Nowadnick, K. Wohlfeld, Y. F. Kung, C. C. Chen, S. Johnston, T. Tohyama, B. Moritz, and T. P. Devereaux, Persistent Spin Excitations in Doped Antiferromagnets Revealed by Resonant Inelastic Light Scattering, Nat. Commun. 5, 3314 (2014).
  * Mentink  _et al._ [2015] J. Mentink, K. Balzer, and M. Eckstein, Ultrafast and Reversible Control of the Exchange Interaction in Mott Insulators, Nat. Commun. 6, 1 (2015).
  * Chen  _et al._ [2019] Y. Chen, Y. Wang, C. Jia, B. Moritz, A. M. Shvaika, J. K. Freericks, and T. P. Devereaux, Theory for Time-resolved Resonant Inelastic X-ray Scattering, Phys. Rev. B 99, 104306 (2019).
  * Chen  _et al._ [2020] Y. Chen, Y. Wang, M. Claassen, B. Moritz, and T. P. Devereaux, Observing Photo-induced Chiral Edge States of Graphene Nanoribbons in Pump-probe Spectroscopies, npj Quantum Mater. 5, 84 (2020).
  * Strobel  _et al._ [2014] H. Strobel, W. Muessel, D. Linnemann, T. Zibold, D. B. Hume, L. Pezzè, A. Smerzi, and M. K. Oberthaler, Fisher Information and Entanglement of Non-Gaussian Spin States, Science 345, 424 (2014).
  * Lorenzana  _et al._ [2005] J. Lorenzana, G. Seibold, and R. Coldea, Sum rules and missing spectral weight in magnetic neutron scattering in the cuprates, Physical Review B 72, 224511 (2005).
  * Laurell  _et al._ [2022] P. Laurell, A. Scheie, D. A. Tennant, S. Okamoto, G. Alvarez, and E. Dagotto, Magnetic Excitations, Nonclassicality, and Quantum Wake Spin Dynamics in the Hubbard Chain, Phys. Rev. B 106, 085110 (2022).
  * Lin and Hirsch [1986] H. Q. Lin and J. E. Hirsch, Condensation Transition in the One-dimensional Extended Hubbard Model, Phys. Rev. B 33, 8155 (1986).
  * Lin  _et al._ [1997] H. Lin, E. Gagliano, and D. Campbell, Phase separation in the 1-D extended hubbard model, Physica C: Superconductivity 282, 1875 (1997).
  * Xiang  _et al._ [2019] Y.-Y. Xiang, X.-J. Liu, Y.-H. Yuan, J. Cao, and C.-M. Tang, Doping Dependence of the Phase Diagram in One-dimensional Extended Hubbard Model: a Functional Renormalization Group Study, J. Phys.: Condens. Matter 31, 125601 (2019).
  * Shinjo  _et al._ [2019] K. Shinjo, K. Sasaki, S. Hase, S. Sota, S. Ejima, S. Yunoki, and T. Tohyama, Machine Learning Phase Diagram in the Half-Filled One-Dimensional Extended Hubbard Model, Journal of the Physical Society of Japan 88, 065001 (2019).
  * Suresh  _et al._ [2022] A. Suresh, R. Soares, P. Mondal, J. Pires, J. Lopes, A. Ferreira, A. Feiguin, P. Plecháč, and B. Nikolić, Electron-mediated entanglement of two distant macroscopic ferromagnets within a nonequilibrium spintronic device, arXiv:2210.06634 (2022).
  * Eckert  _et al._ [2002] K. Eckert, J. Schliemann, D. Bruß, and M. Lewenstein, Quantum Correlations in Systems of Indistinguishable Particles, Ann. Phys. 299, 88 (2002).
  * Schliemann  _et al._ [2001a] J. Schliemann, J. I. Cirac, M. Kuś, M. Lewenstein, and D. Loss, Quantum Correlations in Two-fermion Systems, Phys. Rev. A 64, 022303 (2001a).
  * Schliemann  _et al._ [2001b] J. Schliemann, D. Loss, and A. MacDonald, Double-occupancy Errors, Adiabaticity, and Entanglement of Spin Qubits in Quantum Dots, Phys. Rev. B 63, 085311 (2001b).
  * Kraus  _et al._ [2009] C. V. Kraus, M. M. Wolf, J. I. Cirac, and G. Giedke, Pairing in Fermionic Systems: A Quantum-Information Perspective, Phys. Rev. A 79, 012306 (2009).
  * Braicovich  _et al._ [2010] L. Braicovich, J. van den Brink, V. Bisogni, M. M. Sala, L. J. P. Ament, N. B. Brookes, G. M. De Luca, M. Salluzzo, T. Schmitt, V. N. Strocov, and G. Ghiringhelli, Magnetic excitations and phase separation in the underdoped ${\mathrm{la}}_{2-x}{\mathrm{sr}}_{x}{\mathrm{cuo}}_{4}$ superconductor measured by resonant inelastic x-ray scattering, Phys. Rev. Lett. 104, 077002 (2010).
  * Haverkort [2010] M. W. Haverkort, Theory of resonant inelastic x-ray scattering by collective magnetic excitations, Phys. Rev. Lett. 105, 167404 (2010).
  * Robarts  _et al._ [2021] H. C. Robarts, M. García-Fernández, J. Li, A. Nag, A. C. Walters, N. E. Headings, S. M. Hayden, and K.-J. Zhou, Dynamical spin susceptibility in ${\mathrm{la}}_{2}{\mathrm{cuo}}_{4}$ studied by resonant inelastic x-ray scattering, Phys. Rev. B 103, 224427 (2021).
  * Tsutsui  _et al._ [2000] K. Tsutsui, H. Kondo, T. Tohyama, and S. Maekawa, Resonant Inelastic X-ray Scattering Spectrum in High-Tc Cuprates, Physica B: Condensed Matter 284, 457 (2000).
  * Kourtis  _et al._ [2012] S. Kourtis, J. van den Brink, and M. Daghofer, Exact Diagonalization Results for Resonant Inelastic X-ray Scattering Spectra of One-dimensional Mott Insulators, Phys. Rev. B 85, 064423 (2012).
  * Wiseman and Vaccaro [2003] H. M. Wiseman and J. A. Vaccaro, Entanglement of Indistinguishable Particles Shared Between Two Parties, Physical review letters 91, 097902 (2003).
  * Bañuls  _et al._ [2007] M.-C. Bañuls, J. I. Cirac, and M. M. Wolf, Entanglement in Fermionic Systems, Phys. Rev. A 76, 022311 (2007).