# Ultrafast control of spin-orbital separation probed with time-resolved RIXS

Aaron Müller  Department of Physics, University of Erlangen-Nürnberg, 91058
Erlangen, Germany  Francesco Grandi  Department of Physics, University of
Erlangen-Nürnberg, 91058 Erlangen, Germany  Martin Eckstein  Department of
Physics, University of Erlangen-Nürnberg, 91058 Erlangen, Germany

(August 29, 2024)

###### Abstract

Quasi-one-dimensional systems exhibit many-body effects elusive in higher
dimensions. A prime example is spin-orbital separation, which has been
measured by resonant inelastic X-ray scattering (RIXS) in Sr2CuO3. Here, we
theoretically analyze the time-resolved RIXS spectrum of Sr2CuO3 under the
action of a time-dependent electric field. We show that the external field can
reversibly modify the parameters in the effective $t-J$ model used to describe
spinon and orbiton dynamics in the material. For strong driving amplitudes, we
find that the spectrum changes qualitatively as a result of reversing the
relative spinon to orbiton velocity. The analysis shows that in general, the
spin-orbital dynamics in Mott insulators in combination with time-resolved
RIXS should provide a suitable platform to explore the reversible control of
many-body physics in the solid with strong laser fields.

Introduction –  Electrons in strongly correlated materials can be
characterized by charge, spin and orbital quantum numbers. In a prototypical
Mott insulator, the charge becomes localized by the electron-electron
repulsion, leading to highly intertwined spin-orbital physics and a variety of
ordering phenomena Tokura and Nagaosa (2000). In low-dimensional materials,
the effect of the Coulomb repulsion can be even more exotic, and the electron
can fractionalize into spin and orbital excitations (spin-orbital separation,
SOS) Wohlfeld  _et al._ (2011); Schlappa  _et al._ (2012) in analogy to spin-
charge separation (SCS) Lieb and Wu (1968); Kim  _et al._ (1996, 2006). Ultra-
short laser pulses have opened avenues to control the many-particle physics in
solids out of equilibrium Giannetti  _et al._ (2016); Basov  _et al._ (2017);
de la Torre  _et al._ (2021). A particularly versatile concept is Floquet
engineering, where the Hamiltonian of a system is dynamically modified with
time-periodic fields, as employed widely in optical lattice experiments Jotzu
_et al._ (2014); Görg  _et al._ (2019); Struck  _et al._ (2012); Görg  _et
al._ (2018). For example, a control of SCS has been proposed through Floquet
engineering of the $t-J$ model Gao  _et al._ (2020). In solids, however, a
major challenge to implement a similar dynamic control is the heating due to
photon absorption from the drive. Only few theoretical proposals have so far
been realized, such as Floquet Bloch states of weakly interacting electrons
Oka and Aoki (2009); Wang  _et al._ (2013); McIver  _et al._ (2020).

In this letter, we show that the spin and orbital degrees of freedom in Mott
insulators provide a promising platform to realize a dynamic control of many-
body physics in solids, and in particular, electron fractionalization. The
Mott gap implies a large transparency window which limits heating via multi-
photon absorption, as needed for Floquet engineering. A large gap can also
make the insulator robust against tunnelling breakdown in static fields Oka
and Aoki (2010); Eckstein  _et al._ (2010). This allows for alternative, less
explored pathways to control the low-energy spin-orbital physics with slowly
varying strong field transients. Moreover, spin-orbital excitations can be
probed using resonant inelastic X-ray scattering (RIXS) Ament  _et al._
(2011); Mitrano and Wang (2020), and recent upgrades of free-electron lasers
should provide a sufficient time ($\sim 30$ fs) and energy ($\sim 0.05$ eV)
resolution to resolve their dynamics Dunne (2020, 2018).

We particularly focus on the charge transfer insulator Sr2CuO3. The spin-
orbital excitations in this material are well described by a $t-J$ model and
display signatures of SOS, as clearly visible in static RIXS Schlappa  _et
al._ (2012). In contrast to the typical case of SCS, where the charge
excitation is much faster than the spinon, the orbiton in Sr2CuO3 is slower
than the spinon, with a speed comparable in magnitude ($J/t=2.8$). The ratio
should therefore be controllable over a wide range, eventually allowing for a
nontrivial dynamic switch between distinct regimes in which the spinon is
either faster or slower than the orbiton.

Equilibrium Hamiltonian –  The quasi-one-dimensional charge-transfer insulator
Sr2CuO3 can be described as a chain of alternating copper and oxygen atoms in
$3$d9 configuration and with filled $2$p orbitals. This makes it convenient to
work in a hole representation, with one hole per unit cell. The low point-
group symmetry of the crystal entirely lifts the degeneracy of the Cu 3d
orbitals, and the ground state is a Mott insulator with one hole in the
$3\mathrm{d}_{x^{2}-y^{2}}$ orbital (termed $a$ in the following). Without any
orbital excitation the Hamiltonian of the system is a Heisenberg model with
antiferromagnetic exchange $J$. The motion of orbital excitations on top of
this can be described by an effective $t-J$ model Wohlfeld  _et al._ (2013),
which is found well in agreement with experiment Schlappa  _et al._ (2012),

$\begin{split}\mathcal{H}_{t-J}=&-t\sum_{j,\sigma}\left(p_{j\sigma}^{\dagger}p_{j+1\sigma}+h.c.\right)-E\sum_{j}\tilde{n}_{j}+\mathcal{H}_{s}.\end{split}$
(1)

Here the fermion operator $p_{j\sigma}$ creates an orbital excitation where a
hole with spin $\sigma$ at site $j$ is transferred to another orbital $b$, and
$\tilde{n}_{j}$ counts the holes in the $a$ orbital; $\bm{S}_{j}$ is the spin
of the Cu hole, and
$\mathcal{H}_{s}=J\sum_{j}\left(\bm{S}_{j}\bm{S}_{j+1}-\frac{1}{4}\tilde{n}_{j}\tilde{n}_{j+1}\right)$
a Heisenberg Hamiltonian. An analogous Hamiltonian can be written for
different orbitons. Here we focus on $b\equiv 3\mathrm{d}_{zx}$, which turns
out to have the broadest dispersion among the $3\mathrm{d}$ orbitals. To
derive this model Wohlfeld  _et al._ (2013), the valence bands of Sr2CuO3 are
described by an ab initio charge-transfer Hamiltonian $\mathcal{H}_{c-t}$ with
nearest-neighbor hopping, local Hubbard interaction and Hund’s coupling,
charge transfer energy between the Cu and oxygen states, and a nearest-
neighbor Coulomb repulsion (for details, see Sup ). The large electron-
electron repulsion justifies a strong coupling expansion, which projects out
doubly occupied orbitals and Cu-O charge transfer excitations, and results in
a Kugel-Khomskii Hamiltonian Kugel and Khomskii (1982) for the spin and
orbital degrees of freedom of the Cu hole. Spin and orbital superexchange
proceeds via the O $2$p orbitals and is therefore obtained from a fourth-order
perturbation expansion in the p-d hopping Zaanen and Oleś (1988). By
neglecting spin flip processes in the hopping of the hole in the $b$ orbital,
and choosing a Jordan Wigner representation for the orbital pseudo-spin, the
Kugel-Khomskii Hamiltonian is then mapped on the $t-J$ model Eq. (1). In this
way, $J>0$ and $t$ in Eq. (1) relate to the spin and orbital superexchange
interaction, and $E$ is related to the crystal field splitting. When the
orbiton is present, the first part of Eq. (1) describes its motion in a Neél
antiferromagnetic background. The equilibrium parameters in Eq. (1) for
Sr2CuO3 are Wohlfeld  _et al._ (2013) $t\sim 0.085$ eV, $J\sim 0.238$ eV and
$E\sim 1.999$ eV. The large ratio $J/t\sim 2.8$ is indeed aberrant compared to
the typical value of the spin-charge $t-J$ model obtained from a single band
Hubbard model at large interaction $U$ Chao  _et al._ (1977). In the latter,
the parameter $t$ is the original electron hopping, and $J=4t^{2}/U\ll t$,
while in Eq. (1) both $t$ and $J$ originate from a fourth-order perturbation
expansion.

Figure 1: Spectral function $A(\omega,k)$ obtained form the single hole
propagator Eq. (2) in the a) slow ($J/t=0.4$) and b) fast ($J/t=2.8$) spinon
regime, computed on a chain with $L=18$ sites and Gaussian broadening. Red
dots are obtained with the spin-orbital separation ansatz Eq. (3), with
$E_{f}=E+J$ and a) $t_{f}=t$, $J_{f}=J$ and b) $t_{f}=t/2$, $J_{f}=J$.

Below we will analyze how external fields can be used to control $t$ and $J$
over a wide range, covering both $J/t>2$ (fast spinon regime) and $J/t<2$
(slow spinon regime). To prepare for this analysis, we first discuss the
equilibrium spectral function
$A(\omega,k)=-\frac{1}{\pi}\text{Im}\,G_{k}(\omega+i0^{+})$ of the momentum-
dependent single hole propagator $G_{k}$; $G_{k}$ is the Fourier transform
$G_{k}=1/L\sum_{j,j^{\prime}=0}^{L-1}e^{ik(j^{\prime}-j)}G_{j^{\prime},j}$ of
the real-space propagator

$\displaystyle
G_{j^{\prime},j}(t^{\prime},t)=-i\Bra{\Psi_{0}}\mathcal{U}_{s}(t_{0},t^{\prime})p_{j^{\prime}\sigma}^{\dagger}\mathcal{U}_{t-J}(t^{\prime},t)p_{j\sigma}\mathcal{U}_{s}(t,t_{0})\Ket{\Psi_{0}}.$
(2)

Here $\Ket{\Psi_{0}}$ is the initial ground state (no orbital excitation) at
time $t_{0}\to-\infty$, and $\mathcal{U}_{s}$ and $\mathcal{U}_{t-J}$ denote
time-evolution in the Heisenberg and $t-J$ model, respectively. The resulting
spectrum shows a distinct form in the slow (Fig. 1a) and fast (Fig. 1b) spinon
regime. As a consequence of the spin-orbital separation, insight into the form
of $A(\omega,k)$ can be acquired from a semi-phenomenological spectral
building principle Eder and Ohta (1997); Bannister and d’Ambrumenil (2000):
Assuming that the orbital excitation separates into a spinon and an orbiton,
which are treated as independent particles with dispersion
$\epsilon_{s}(k_{s})=\pi J_{f}/2\,\absolutevalue{\cos{k_{s}}}$ for $-\pi/2\leq
k_{s}\leq\pi/2$ Giamarchi (2004) and
$\epsilon_{o}(k_{o})=E_{f}-2t_{f}\,\cos{k_{o}}$, respectively, spectral weight
in $A(\omega,k)$ should be found at energies
$\epsilon(k)=\epsilon_{s}(k_{s})+\epsilon_{o}(k_{o})$, with a constraint
$k=k_{s}+k_{o}$ due to momentum conservation. Hence, one expects the peaks of
$A(\omega,k)$ at

$\epsilon(k)=E_{f}-2t_{f}\cos{k_{o}}+J_{f}\frac{\pi}{2}\absolutevalue{\cos{(k-k_{o})}}\,.$
(3)

A $1/L$-correction to the momentum of the orbiton has to be applied on account
of periodic boundary conditions Giamarchi (2004); Bohrdt  _et al._ (2018). In
Fig. 1, the result of Eq. (3) is shown by red dots. In the slow spinon regime,
the spectrum is enclosed by the lower and upper orbiton branches Suzuura and
Nagaosa (1997); Kim  _et al._ (2006), and there is an overall good agreement
between the numerical results and Eq. (3). In the fast spinon regime, the
ansatz Eq. (3) still reproduces the lower bound of the spectrum, although the
parameter $t_{f}$ needs to be renormalized with respect to the bare $t$ due to
further dressing of the orbiton Brunner  _et al._ (2000). The transition
between these two regimes can be located at the so-called supersymmetric point
$J/t=2$, where the $t-J$ model becomes exactly solvable by Bethe Ansatz Bares
_et al._ (1991).

Non-equilibrium effective Hamiltonian – To include the effect of the driving
laser, we couple the charge transfer Hamiltonian $\mathcal{H}_{c-t}$ to an
electromagnetic field using the Peierls substitution Peierls (1933); Li  _et
al._ (2020). In the length gauge within the electric-dipole approximation
Reiss (1980); Baykusheva and Wörner (2020), this is equivalent to adding a
scalar potential $\Phi_{j}(t)=-eE(t)X_{j}$ at each site (independent of the
orbital), where $e$ is the elementary charge, $E(t)$ the time-dependent
electric field, and $X_{j}=j\tilde{a}/2$ represents the position along the
chain, with the copper-copper distance $\tilde{a}\sim 0.392$ nm Rosner  _et
al._ (1997). We neglect the renormalization of the hopping due to dipole
matrix elements. Below, we parametrize the time-dependence of the electric
field as $E(t)=S(t)\cos(\Omega t)$, with an oscillatory part and an envelope
$S(t)$ with maximum amplitude $E_{0}$. The time-dependent modification of the
parameters $t$ and $J$ is then understood in two distinct limits:

In the quasi-static limit, one can derive a $t-J$ Hamiltonian at a given time
$t$ by repeating the strong coupling perturbation expansion including energy
shifts of the orbitals caused by the instantaneous scalar potential difference
$\Delta\phi=e\tilde{a}E(t)/2$ between neighboring sites. The potential
modifies the virtual intermediate state of the perturbation expansion, and
yields field-dependent parameters $t^{\phi}$, $J^{\phi}$ and $E^{\phi}$ in Eq.
(1). (For explicit expressions, see the Supplemental Material Sup .) This
quasi-static argument should apply to slowly varying fields $\Omega\ll t,J$,
up to several $10$ THz given the values of $t$ and $J$. Both $J^{\phi}$ and
$t^{\phi}$ increase with $\Delta\phi$, leading to an overall decrease of the
ratio $J/t$ (Fig. 2). The intrinsic limitation to the quasi-static control of
the spin-orbital physics is the tunnelling breakdown of the insulating state,
which would lead to a population of real charge transfer excitations. The most
rapid breakdown is expected when the gradient $\Delta\phi$ is resonant to the
nearest-neighbour charge transfer energy $E_{CT}$. Simulations in one- and
two-band Hubbard models consistently show that the carrier generation rate
becomes exponentially suppressed for fields sufficiently below $E_{CT}$ Oka
and Aoki (2010); Eckstein  _et al._ (2010); Dasari  _et al._ (2020). The
fields needed to decrease the ratio $J/t$ below the supersymmetric point are
well below this resonant amplitude $E_{CT}\sim 130$ MV/cm
($\Delta\phi_{CT}=2.6$ eV) for the present system, so that a control of the
spin-orbital physics should be possible at least transiently.

Figure 2: a) Dependence of the effective parameters $J^{\phi}$ and $t^{\phi}$
on the potential field gradient $\Delta\phi$ in eV (lower horizontal axis) and
electric field amplitude $E_{0}$ in MV$/$cm (upper horizontal axis) for the
quasi-static regime and b) the corresponding change in the characteristic
value $J/t$ of the $t-J$ model Eq. (1). The horizontal dashed line shows the
value $J/t=2$ separating slow and fast spinon regime.

In the Floquet limit, one can conveniently write a time-periodic Hamiltonian
$\mathcal{H}(t)=\mathcal{H}(t+T)$ with period $T=2\pi/\Omega$ in the Floquet
basis by moving from the Fock space $\mathcal{F}$ of the original problem to a
larger space $\mathcal{F}\otimes\mathcal{T}$, with $\mathcal{T}$ the Hilbert
space of the square-integrable periodic functions with period $T$. States in
the extended space are labelled by an additional discrete index $n$, which can
be understood as the number of photons absorbed or emitted relative to the
drive. The representation of $\mathcal{H}(t)$ on the enlarged Fock space is
called Floquet Hamiltonian $\mathcal{H}^{Fl}_{c-t}$, which defines an
effective static problem that describes the stroboscopic evolution of the
driven system. Since $\mathcal{H}^{Fl}_{c-t}$ is time-independent, one can use
a strong coupling perturbation expansion analogous to the undriven case to
derive the fourth-order spin-orbital superexchange interactions underlying the
$t-J$ model Eq. (1). The Floquet control of superexchange interactions has
been derived for various situations, including exchange interactions in the
magnetic Mentink  _et al._ (2015); Itin and Katsnelson (2015); Bukov  _et al._
(2015) and charge sector Kitamura and Aoki (2016) of the single-band Hubbard
model, antisymmetric exchange interactions which may stabilize chiral spin
liquids Claassen  _et al._ (2017), and superexchange via ligand ions Chaudhary
_et al._ (2020). To derive the superexchange interaction in the Floquet
representation in the present case, we project out both virtual charge
excitations and states in Floquet sectors $n\neq 0$ (virtual photons). The
emission or absorption of $n$ virtual photons during an electronic hopping
process shifts the energy of the intermediate states of the superexchange
process by $\pm n\Omega$. The matrix elements for such processes are
controlled by the Floquet parameter $\mathcal{E}=e\tilde{a}E_{0}/(2\Omega)$.
Explicit calculations and results for the dependence of
$t^{\mathcal{E},\Omega}$, $J^{\mathcal{E},\Omega}$ and
$E^{\mathcal{E},\Omega}$ on $\mathcal{E}$ and $\Omega$ are provided in the
Supplemental Material Sup .

In the following, Floquet theory is applied for frequencies $\Omega>t,J$, but
below the optical gap $\Delta_{gap}\sim 1.6-1.8$ eV Imada  _et al._ (1998) to
limit charge transfer excitations due to linear photon absorption. By avoiding
the Floquet resonances (resonant frequencies for which the intermediate state
energy in the superexchange process would vanish), one can further also limit
multi-photon absorption. In contrast to what is observed in the quasi-static
limit, it is possible to both decrease and increase the hopping
$t^{\mathcal{E},\Omega}$ with respect to the equilibrium value, see Fig. 3a),
while $J^{\mathcal{E},\Omega}$ can only increase in the analyzed domain, see
Fig. 3b). The ratio $J/t$ can therefore be controlled in a wide range below
the supersymmetric point $J/t=2$, see Fig. 3c). For example, for $\Omega\sim
0.95$ eV the regime $J/t<2$ is reached for a Floquet parameter
$\mathcal{E}\sim 1.5$. Divergences of $J/t$ appear upon increasing
$\mathcal{E}$ at fixed $\Omega$ due to a localization of the orbiton ($t=0$)
when the spin excitations remain mobile Gao  _et al._ (2020).

Figure 3: a), b) Dependence of the effective parameters
$t^{\mathcal{E},\Omega}$ and $J^{\mathcal{E},\Omega}$ in eV on the Floquet
parameter $\mathcal{E}$ for several values of the driving frequency $\Omega$.
c) Ratio $J^{\mathcal{E},\Omega}/t^{\mathcal{E},\Omega}$ as function of
frequency $\Omega$ and Floquet parameter $\mathcal{E}$. The white areas in the
figure correspond to regimes in which
$J^{\mathcal{E},\Omega}/t^{\mathcal{E},\Omega}$ lies outside the indicated
range. The dashed lines show the leading Floquet resonances in the given
frequency range, which involve a virtual transition to the fourth Floquet
sector ($n=4$).

Time-resolved RIXS –  To verify the dynamic control over the superexchange
interactions described in the previous section, one needs an experimental
probe able to follow the transient change of the spectrum over the whole
Brillouin zone. Time-resolved RIXS (trRIXS) seems particularly suitable for
this task. The incoming photon with energy $\omega_{i}$, momentum $k_{i}$ and
polarization $\bm{e}_{i}$ excites the hole originally in the $a$ orbital to
the $2$p core state of a copper atom. This highly unstable intermediate state
can decay within few femtoseconds to the $b$ orbital by emitting a photon with
energy $\omega_{f}$, momentum $k_{f}$ and polarization $\bm{e}_{f}$, leaving
the system with a d-d excitation. In principle, the hole can also decay into
other d orbitals, but we are only interested in the $b$ channel, as motivated
previously. Thus, measuring the energy loss ($\omega=\omega_{i}-\omega_{f}$)
and momentum loss ($k=k_{i}-k_{f}$) of the photon allows to probe the
excitation spectrum of the compound. The transient change in the spectrum due
to the action of the pump pulse can be monitored by collecting RIXS spectra
with probe pulses at several delayed times. For the short-lived intermediate
state, one can apply the ultra-short core-hole approximation Ament  _et al._
(2007). This describes the photon scattering via the intermediate state in
terms of a single RIXS operator $\mathcal{B}$, which will be the same for
time-resolved and static RIXS. The approximation reduces the computation of
the trRIXS intensity from a four-point correlation function Chen  _et al._
(2019); Eckstein and Werner (2021) to a two-point one, which can be written as
$I_{RIXS}=2(|B^{\uparrow\uparrow}_{\bm{e}_{i}\bm{e}_{f}}|^{2}+|B^{\uparrow\downarrow}_{\bm{e}_{i}\bm{e}_{f}}|^{2})A(\omega,k,\bar{t})$.
Here $B^{\uparrow\uparrow}_{\bm{e}_{i}\bm{e}_{f}}$ and
$B^{\uparrow\downarrow}_{\bm{e}_{i}\bm{e}_{f}}$ are the matrix elements of the
RIXS operator Marra (2016), and $A(\omega,k,\bar{t})$ is the convolution of
the propagator (2) with probe functions centered around a probe time
$\bar{t}$,

$A(\omega,k,\bar{t})=i\int_{-\infty}^{\infty}dt^{\prime}dt\
s(t^{\prime},\bar{t})s(t,\bar{t})e^{-i\omega(t-t^{\prime})}G_{k}(t^{\prime},t).$
(4)

The probe-pulse is thereby described semi-classically by the Gaussian envelope
$s(t,\bar{t})=\exp\left(-(t-\bar{t})^{2}/(2\sigma_{pr}^{2})\right)$ Sup . In
the following, instead of analyzing $I_{RIXS}$, we focus on
$A(\omega,k,\bar{t})$, which does not depend on the details of the
experimental apparatus used to measure the RIXS signal. The numerical
evaluation of the single hole propagator Eq. (2) is done using a Krylov time
propagation scheme with middle point approximation for the time evolution
Manmana (2005); Alvermann  _et al._ (2012); Weinberg and Bukov (2019).

Figure 4: a) Time-dependent potential field gradient $\Delta\phi(t)$ for
$\Delta\phi^{max}=1.8$ eV and $\sigma_{pu}=90$ fs and resulting change in
$J/t$. b) - e) show the time-dependent evolution of the spectral function
$A(\omega,k,\bar{t})$ for different probe envelopes with $\sigma_{pr}=20$ fs
and b) $\bar{t}=180$ fs, c) $\bar{t}=270$ fs, d) $\bar{t}=360$ fs and e)
$\bar{t}=450$ fs for a chain length $L=18$.

Dynamic Control –  As elucidated in the previous sections, it is possible to
achieve control over the parameters $t$ and $J$ of the effective $t-J$ model
Eq. (1) both in the quasi-static and the Floquet regime. Here, we explicitly
demonstrate the time-dependent modification of the spectrum for the quasi-
static limit. We choose the time-dependent profile $\Delta\phi(t)$ to be
Gaussian with variance $\sigma_{pu}=90$ fs and maximum value
$\Delta\phi^{max}=1.8$ eV, as depicted in Fig. 4a).

In Fig. 4b)-e), we show the time evolution of $A(\omega,k,\bar{t})$ during
this time-dependent perturbation. At moderately strong values of the external
field, the spectral function is not substantially changed with respect to
equilibrium, even if quantitative differences can be observed, see Fig. 4b).
However, around the maximum $\Delta\phi^{max}$ one identifies a qualitative
change in the spectrum, with the appearance of a well-defined upper orbiton
branch not present at lower values of the field, see Fig. 4c). This is a
signature of the transition from the fast to the slow spinon regime. Indeed,
as shown in Fig. 4a), the values of the ratio $J/t$ reached at
$\Delta\phi^{max}$ are well below the supersymmetric point $J/t=2$. As the
pump strength decreases, the spectrum narrows into its equilibrium shape,
which happens reversibly, see Figs. 4d) and e). This proves the possibility to
dynamically control the qualitative shape of the spectral function and hence
of the underlying collective excitations.

Conclusions – In conclusion, we showed that short pulses with realistic field
amplitudes can be used to reversibly control the spin-orbital excitations in
the quasi-one-dimensional compound Sr2CuO3. These pulses can substantially
change the relative strength of the $t$ and $J$ parameters in the effective
$t-J$ model which describes the spin and the orbital excitations in this
material. This demonstrates control over the electronic fractionalization in
solids between regimes where the spinon moves faster than the orbiton and the
contrary one. More generally, the analysis illustrates that the spin-orbital
physics in Mott insulators is a suitable platform to explore the reversible
control of many-body dynamics in solids with strong laser fields. In
particular, with the improved energy resolution available at new generation
free electron lasers spin-orbital excitations can be probed using trRIXS.
Moreover, our analysis shows that a control of the low energy physics can be
achieved by two different pathways, i.e., Floquet engineering with frequencies
$\Omega>t,J$, and a sub-cycle control with strong THz transients
($\Omega<t,J$). (We remark that these pathways can be seen as two limits of a
more general control of time-dependent exchange interactions with arbitrary
time-dependent fields Eckstein  _et al._ (2017).) There are further Mott
systems which show spin-orbital separation, and we expect a nontrivial dynamic
control in CaCu2O3 Bisogni  _et al._ (2015) or Ca2CuO3 Fumagalli  _et al._
(2020), the light-control of multi-spinon excitations in Sr2CuO3 Schlappa  _et
al._ (2018) and the control of two-spinon and two-orbital excitations in the
spin-Peierls compound TiOCl Glawion  _et al._ (2011).

Acknowledgment -  We thank Matteo Mitrano for useful discussions. This work
was supported by the ERC Starting Grant No. 716648. The authors gratefully
acknowledge the computational resources and support provided by the Regional
Computing Center Erlangen (RRZE).

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## Supplemental Material for: ’Ultrafast control of spin-orbital separation
probed with time-resolved RIXS’

The Supplemental Material is structured as follows: In Sec. .1, we discuss the
derivation of the $t-J$ model (see Eq. (1) in the main text) and its
generalization to the driven case in the quasi-static and Floquet limits. In
Sec. .2, we provide details about the derivation of the time-resolved RIXS
intensity (Eq. (4) in the main text).

###  .1 From the microscopic Hamiltonian to the effective $t-J$ model

The aim of this section is to derive a low energy Hamiltonian for the
description of the orbital excitations in Sr2CuO3 (Eq. (1) in the main text)
and to generalize it to non-equilibrium conditions in the quasi-static and
Floquet limits.

####  .1.1 Charge-transfer model

The starting point is the ab initio charge-transfer Hamiltonian
$\mathcal{H}_{c-t}=\mathcal{H}^{1}+\mathcal{H}^{2}+\mathcal{H}^{3}+\mathcal{H}^{4}$,
which describes the interactions of the coppers and oxygen atoms in the quasi-
one-dimensional compound Sr2CuO3 Wohlfeld  _et al._ (2013). The unit cell of
this system contains one Cu site and three O atoms. One oxygen lies within the
one-dimensional copper chain Cu-O-Cu-O-$\ldots$, and the other two lie above
and below the Cu atom, respectively. $\mathcal{H}^{1}$ is the tight-binding
contribution which includes a nearest-neighbor hopping between the copper and
oxygen atoms, the chemical potential and the crystal field splitting between
different orbitals

$\displaystyle\mathcal{H}^{1}=$ $\displaystyle-
t_{\sigma}\sum_{j\sigma}\left(f_{ja\sigma}^{\dagger}f_{jx\sigma}-f_{j+1,a\sigma}^{\dagger}f_{jx\sigma}+H.c.\right)-t_{\sigma
o}\sum_{j,\sigma}\left(f_{ja\sigma}^{\dagger}f_{jyo^{+}\sigma}-f_{ja\sigma}^{\dagger}f_{jyo^{-}\sigma}+H.c.\right)-t_{\pi}\sum_{j\sigma}\left(f_{jb\sigma}^{\dagger}f_{jz\sigma}-f_{j+1,b\sigma}^{\dagger}f_{jz\sigma}+H.c.\right)$
$\displaystyle+\varepsilon_{a}\sum_{j}n_{ja}+\varepsilon_{b}\sum_{j}n_{jb}+\Delta_{x}\sum_{j}\left(n_{jx}-n_{ja}\right)+\Delta_{z}\sum_{j}\left(n_{jz}-n_{jb}\right)$
$\displaystyle+\Delta_{yo}\sum_{j}\left(n_{jyo^{+}}-n_{ja}\right)+\Delta_{yo}\sum_{j}\left(n_{jyo^{-}}-n_{ja}\right)+\Delta_{xo}\sum_{j}\left(n_{jxo^{+}}-n_{jb}\right)+\Delta_{xo}\sum_{j}\left(n_{jxo^{-}}-n_{jb}\right)\,.$
(.1.1)

$\mathcal{H}^{2}$ is the onsite Coulomb repulsion on the copper sites, which
includes intra- and interband Hubbard repulsion, Hund’s interaction and pair-
hopping terms

$\displaystyle\mathcal{H}^{2}=$
$\displaystyle\sum_{j,\sigma,\alpha<\beta}\left(U-2J^{\alpha\beta}_{H}\right)n_{j\alpha\sigma}n_{j\beta\bar{\sigma}}+\sum_{j,\sigma,\alpha<\beta}\left(U-3J^{\alpha\beta}_{H}\right)n_{j\alpha\sigma}n_{j\beta\sigma}+U\sum_{j,\alpha}n_{j\alpha\uparrow}n_{j\alpha\downarrow}$
$\displaystyle-\sum_{j,\sigma,\alpha<\beta}J^{\alpha\beta}_{H}f_{j\alpha\sigma}^{\dagger}f_{j\alpha\bar{\sigma}}f_{j\beta\bar{\sigma}}^{\dagger}f_{j\beta\sigma}+\sum_{j,\sigma,\alpha<\beta}J^{\alpha\beta}_{H}f_{j\alpha\uparrow}^{\dagger}f_{j\alpha\downarrow}^{\dagger}f_{j\beta\downarrow}f_{j\beta\uparrow}\,.$
(.1.2)

Similarly, $\mathcal{H}^{3}$ describes the onsite Coulomb repulsion on the
oxygen sites

$\displaystyle\mathcal{H}^{3}=$
$\displaystyle\sum_{j,\sigma,\mu<\nu}\left(U_{p}-2J^{\mu\nu}_{H}\right)n_{j\mu\sigma}n_{j\nu\bar{\sigma}}+\sum_{j,\sigma,\mu<\nu}\left(U_{p}-3J^{\mu\nu}_{H}\right)n_{j\mu\sigma}n_{j\nu\sigma}+U_{p}\sum_{j,\mu}n_{j\mu\uparrow}n_{j\mu\downarrow}$
$\displaystyle-\sum_{j,\sigma,\mu<\nu}J^{\mu\nu}_{H}f_{j\mu\sigma}^{\dagger}f_{j\mu\bar{\sigma}}f_{j\nu\bar{\sigma}}^{\dagger}f_{j\nu\sigma}+\sum_{j,\sigma,\mu<\nu}J^{\mu\nu}_{H}f_{j\mu\uparrow}^{\dagger}f_{j\mu\downarrow}^{\dagger}f_{j\nu\downarrow}f_{j\nu\uparrow}\,.$
(.1.3)

Finally, $\mathcal{H}^{4}$ is the nearest-neighbor Coulomb repulsion

$\displaystyle\mathcal{H}^{4}=V_{dp}\sum_{j\mu\alpha}n_{j\alpha}\left(n_{j\mu}+n_{j\mu^{+}}+n_{j\mu^{-}}+n_{j+1,\mu}\right)\,.$
(.1.4)

The two 3d Cu orbitals considered in the previous expressions are labeled as
$a\equiv 3\mathrm{d}_{x^{2}-y^{2}}$ and $b\equiv 3\mathrm{d}_{zx}$, the 2p O
orbitals within the chain are $x\equiv 2\mathrm{p}_{x}$ and $z\equiv
2\mathrm{p}_{z}$, and the 2p O orbitals above (below) the chain are
$xo^{+}\equiv 2\mathrm{p}_{x}$ ($xo^{-}\equiv 2\mathrm{p}_{x}$) and
$yo^{+}\equiv 2\mathrm{p}_{y}$ ($yo^{-}\equiv 2\mathrm{p}_{y}$). The indices
$\alpha,\beta\in\\{a,b\\}$ describe the copper orbitals and
$\mu,\nu\in\\{x,z\\}$ the oxygen orbitals. $f_{j\kappa\sigma}^{\dagger}$
creates a hole at site $j$ in orbital $\kappa\in\\{a,b,x,z\\}$ with spin
$\sigma\in\\{\uparrow,\downarrow\\}$ ($\bar{\sigma}=-\sigma$), while the
number operators are $n_{j\kappa}=n_{j\kappa\uparrow}+n_{j\kappa\downarrow}$
with $n_{j\kappa\sigma}=f_{j\kappa\sigma}^{\dagger}f_{j\kappa\sigma}$. In the
hole representation and for the system in the ground state, there is a single
hole in the unit cell which occupies the $a$ orbital. The other 3d Cu orbitals
are not considered here as their characteristic dispersion relations are
weaker. The relevant values of the parameters in Eqs. (.1.1) - (.1.4) are then
$t_{\sigma}=1.5$ eV, $t_{\pi}=0.83$ eV, $t_{\sigma o}=1.8$ eV, $t_{\pi o}=1.0$
eV, $\Delta_{x}=3.0$ eV, $\Delta_{z}=2.2$ eV, $\Delta_{xo}=2.5$ eV,
$\Delta_{yo}=3.5$ eV, $U=8.8$ eV, $U_{p}=4.4$ eV, $J^{b}_{H}=J^{ab}_{H}=1.2$
eV, $J^{p}_{H}=0.83$ eV, $V_{dp}=1.0$ eV, $\varepsilon_{a}=0.0$ eV and
$\varepsilon_{b}=-0.5$ eV, in accordance with Wohlfeld  _et al._ (2013);
Neudert  _et al._ (2000).

####  .1.2 $t-J$ model

The charge-transfer Hamiltonian $\mathcal{H}_{c-t}$ provided in Sec. .1.1
belongs to the regime $t/U\ll 1$ and $t/\Delta<1$, with $t$ any of the
hoppings and $\Delta$ any of the crystal fields appearing in Eqs. (.1.1) -
(.1.4), making the ground state a Mott insulator with a hole localized on the
$a$ orbital on each copper site. Similarly, when a single orbital excitation
is present, the hole is localized on the $b$ orbital. As outlined in Wohlfeld
_et al._ (2013), the second- and the fourth-orders of the perturbative
expansion in the hoppings lead to the expression

$\displaystyle\bar{\mathcal{H}}=\mathcal{H}^{II}+\mathcal{H}^{IV}_{a}+\mathcal{H}^{IV}_{b}\,.$
(.1.5)

The second-order contribution has two important consequences: Firstly, it
leads to a renormalization of the on-site energies of the orbitals; Secondly,
it results in a renormalization of the hoppings within the chain, due to
hybridization of the Cu orbitals with the oxygen atoms above and below the
copper-oxygen chain. The first effect is given by

$\displaystyle\mathcal{H}^{II}=\varepsilon_{a}\sum_{j}\tilde{n}_{ja}+\bar{\varepsilon}_{b}\sum_{j}\tilde{n}_{jb}\;,$
(.1.6)

where the tilde over the occupation operators implies that double occupations
are prohibited in this strong coupling expansion and

$\displaystyle\bar{\varepsilon}_{b}=\varepsilon_{b}+\frac{2t_{\sigma
o}^{2}}{\Delta_{yo}}+\frac{2t_{\sigma}^{2}}{\Delta_{x}}-\frac{2t_{\pi}^{2}}{\Delta_{z}}\;.$
(.1.7)

Eq. (.1.6) is an effective crystal field between $a$ and $b$ orbitals. By
introducing the pseudospin operators

$\displaystyle\sigma^{z}_{j}=\frac{1}{2}(\tilde{n}_{jb}-\tilde{n}_{ja}),\quad\sigma^{+}_{j}=\tilde{f}_{jb}^{\dagger}\tilde{f}_{ja},\quad\sigma^{-}_{j}=\tilde{f}_{ja}^{\dagger}\tilde{f}_{jb}\;,$
(.1.8)

one can rewrite Eq. (.1.6) as

$\displaystyle\mathcal{H}^{II}=\varepsilon_{a}\sum_{j}\left(\frac{1}{2}-\sigma^{z}_{j}\right)+\bar{\varepsilon}_{b}\sum_{j}\left(\frac{1}{2}+\sigma^{z}_{j}\right)\;.$
(.1.9)

The renormalization of the hopping affects only
$\bar{t}_{\sigma}=\lambda_{a}t_{\sigma}$ with

$\displaystyle\lambda_{a}=\frac{\Delta_{yo}-e_{a}}{\sqrt{2t_{\sigma
o}^{2}+(\Delta_{yo}-e_{a})^{2}}}\,,\qquad
e_{a}=\frac{1}{2}\left(\Delta_{yo}-\sqrt{\Delta_{yo}^{2}+8t_{\sigma
o}^{2}}\right)\,.$ (.1.10)

For additional details about this derivation, we refer the reader to Wohlfeld
_et al._ (2013).

The first fourth-order contribution to Eq. (.1.5) describes the superexchange
process of two holes in the $a$ orbitals at two distinct copper atoms

$\displaystyle\mathcal{H}^{IV}_{a}=J\sum_{j}\mathcal{P}_{j,j+1}\left(\bm{S}_{j}\bm{S}_{j+1}-\frac{1}{4}\tilde{n}_{j}\tilde{n}_{j+1}\right)\,,$
(.1.11)

where $\mathcal{H}^{IV}_{a}\equiv\mathcal{H}_{s}$ defined in the main text.
Here, we introduced the spin operators

$\displaystyle
S^{z}_{j}=\frac{1}{2}(\tilde{n}_{j\uparrow}-\tilde{n}_{j\downarrow}),\quad
S^{+}_{j}=\tilde{f}_{j\uparrow}^{\dagger}\tilde{f}_{j\downarrow},\quad
S^{-}_{j}=\tilde{f}_{j\downarrow}^{\dagger}\tilde{f}_{j\uparrow}\,,$ (.1.12)

and we defined the superexchange interaction $J=J^{1}(1+2R)>0$, with

$\displaystyle
J^{1}=\frac{4\bar{t}_{\sigma}^{4}}{\left(\Delta_{x}+V_{dp}\right)^{2}U}\,,\qquad
R=\frac{U}{2\Delta_{x}+U_{p}}\,.$ (.1.13)

The operator $\mathcal{P}_{j,j+1}=(1/2-\sigma_{i}^{z})(1/2-\sigma_{i+1}^{z})$
projects out any orbital excitation along the bond $(j,j+1)$. The Hamiltonian
Eq. (.1.11) describes a Heisenberg antiferromagnetic interaction between
nearest copper sites.

The second fourth-order contribution to Eq. (.1.5) describes the superexchange
process in the presence of an orbital excitation, i.e. a single hole occupying
the $b$ orbital. It reads

$\displaystyle\mathcal{H}^{IV}_{b}=$
$\displaystyle\sum_{j}\left(\bm{S}_{j}\bm{S}_{j+1}+\frac{3}{4}\right)\left\\{\left[\frac{r^{b1}}{2}\left(J^{1}+J^{b2}\right)+R^{b1}\sum_{\mu,\nu\in\\{x,z\\}}J^{b12}_{\mu\nu}\right]\left[\sigma_{j}^{z}\sigma_{j+1}^{z}-\frac{1}{4}\right]+\left[r^{b1}J^{b12}_{xz}+\frac{R^{b1}}{2}\sum_{\mu,\nu\in\\{x,z\\}}J^{b12}_{\mu\nu}\right]\left[\sigma_{j}^{+}\sigma_{j+1}^{-}+\sigma_{j}^{-}\sigma_{j+1}^{+}\right]\right\\}$
$\displaystyle+$
$\displaystyle\sum_{j}\left(\frac{1}{4}-\bm{S}_{j}\bm{S}_{j+1}\right)\left\\{\left[\frac{r^{b2}}{2}\left(J^{1}+J^{b2}\right)+R^{b2}\sum_{\mu,\nu\in\\{x,z\\}}J^{b12}_{\mu\nu}\right]\left[\sigma_{j}^{z}\sigma_{j+1}^{z}-\frac{1}{4}\right]-\left[r^{b2}J^{b12}_{xz}+\frac{R^{b2}}{2}\sum_{\mu,\nu\in\\{x,z\\}}J^{b12}_{\mu\nu}\right]\left[\sigma_{j}^{+}\sigma_{j+1}^{-}+\sigma_{j}^{-}\sigma_{j+1}^{+}\right]\right\\}$
(.1.14)

with

$\displaystyle
J^{b2}=\frac{4t_{\pi}^{4}}{\left(\Delta_{z}+V_{dp}\right)^{2}U}\,,$
$\displaystyle r^{b1}=\frac{U}{U-3J_{H}^{b}}\,,\quad$ $\displaystyle
R^{b1}=\frac{U}{\Delta_{x}+\Delta_{z}+U_{p}-3J^{p}_{H}}\,,$ (.1.15)
$\displaystyle
J_{\mu\nu}^{b12}=\frac{2t_{\pi}^{2}\bar{t}_{\sigma}^{2}}{\left(\Delta_{\mu}+V_{dp}\right)U\left(\Delta_{\nu}+V_{dp}\right)}\,,\quad$
$\displaystyle r^{b2}=\frac{U}{U-J_{H}^{b}}\,,$ $\displaystyle
R^{b2}=\frac{U}{\Delta_{x}+\Delta_{z}+U_{p}-J^{p}_{H}}\,.$ (.1.16)

Eq. (.1.2) is a spin-orbital Hamiltonian of the Kugel-Khomskii type.

The Hamiltonian Eq. (.1.5) describes the low-energy spin-orbital dynamics of
the quasi-one-dimensional compound Sr2CuO3. Given the dimensionality of the
system, spin and pseudospin operators Eqs. (.1.12) and (.1.8) can be
conveniently rewritten through the Jordan-Wigner transformation

$\displaystyle
S_{j}^{+}=\exp\left(-i\pi\sum_{l=1}^{j-1}n_{l\alpha}\right)\alpha_{j}^{\dagger}\,,\quad
S_{j}^{-}=\alpha_{j}\exp\left(i\pi\sum_{l=1}^{j-1}n_{l\alpha}\right),\quad
S_{j}^{z}=n_{j\alpha}-\frac{1}{2}\,,$ (.1.17)
$\displaystyle\sigma_{j}^{+}=\exp\left(-i\pi\sum_{l=1}^{j-1}n_{l\beta}\right)\beta_{j}^{\dagger}\,,\quad\sigma_{j}^{-}=\beta_{j}\exp\left(i\pi\sum_{l=1}^{j-1}n_{l\beta}\right),\quad\sigma_{j}^{z}=n_{j\beta}-\frac{1}{2}\,,$
(.1.18)

where $\alpha_{j}$ ($\beta_{j}$) is a fermionic operator annihilating a spinon
(pseudospinon) on site $j$ and $n_{j\alpha}$ ($n_{j\beta}$) is the spinon
(pseudospinon) number operator. As discussed in Wohlfeld  _et al._ (2013), a
spinon and a pseudospinon cannot be on the same site, leading to the
constraint $n_{j\alpha}+n_{j\beta}\leq 1$ for each site $j$, and it is assumed
that the hopping of the pseudospinon does not change the spin of the hole in
the $b$ orbital.

Finally, by defining the fermionic operators

$\displaystyle\tilde{p}_{j\uparrow}=\beta_{j}^{\dagger}\,,\qquad\tilde{p}_{j\uparrow}=\beta_{j}^{\dagger}\alpha_{j}\exp\left(i\pi\sum_{l=1}^{j-1}n_{l\alpha}\right),$
(.1.19)

which act in a Hilbert space without double occupations and introducing back
the spin operators Eq. (.1.17), the Hamiltonian Eq. (.1.5) takes the form

$\bar{\mathcal{H}}\equiv\mathcal{H}_{t-J}=-t\sum_{j,\sigma}\left(p_{j\sigma}^{\dagger}p_{j+1\sigma}+h.c.\right)-E\sum_{j}\tilde{n}_{j}+J\sum_{j}\left(\bm{S}_{j}\bm{S}_{j+1}-\frac{1}{4}\tilde{n}_{j}\tilde{n}_{j+1}\right),$
(.1.20)

with

$\displaystyle J$ $\displaystyle=J^{1}(1+2R)\,,$ (.1.21) $\displaystyle t$
$\displaystyle=\frac{1}{4}\left[2(3r^{b1}+r^{b2})J^{b12}_{xz}+(3R^{b1}+R^{b2})\sum_{\mu,\nu\in\\{x,z\\}}J^{b12}_{\mu\nu}\right],$
(.1.22) $\displaystyle E$
$\displaystyle=\bar{\varepsilon}_{b}-\left[\frac{r^{b1}}{2}\left(J^{1}+J^{b2}\right)+R^{b1}\sum_{\mu,\nu\in\\{x,z\\}}J^{b12}_{\mu\nu}\right]\;.$
(.1.23)

Eq. (.1.20) is equivalent to Eq. (1) of the main text.

####  .1.3 Quasi-static limit

In this section, we analyze how the parameters of the effective $t-J$ model
Eq. (.1.21) change under the action of an electric field included as a scalar
potential $\phi_{j\kappa}$, which we assume slowly varying, i.e. $\Omega=0$.
The charge-transfer Hamiltonian, in this case, has an additional contribution

$\mathcal{H}_{c-t}^{\phi}=\mathcal{H}_{c-t}+\sum_{j,\kappa}\phi_{j\kappa}n_{j\kappa}\,.$
(.1.24)

Considering the same derivation sketched in Sec. .1.2, one arrives at an
expression for the $t-J$ model akin to Eq. (.1.20), with new parameters

$\displaystyle J^{\phi}$
$\displaystyle=\sum_{s\in\\{+,-\\}}J_{s}^{1}r_{s}^{a}+R^{a}\left(\sqrt{J_{+}^{1}}+\sqrt{J_{-}^{1}}\right)^{2}\,,$
(.1.25) $\displaystyle t^{\phi}$
$\displaystyle=\sum_{s\in\\{+,-\\}}\left[\frac{3r^{b1}_{s}+r^{b2}_{s}}{8}\left(J^{b12}_{xz,s}+J^{b12}_{zx,s}\right)+\frac{3R^{b1}+R^{b2}}{8}\sum_{\mu,\nu\in\\{x,z\\}}J^{b12}_{\mu\nu,s}\right],$
(.1.26) $\displaystyle E^{\phi}$
$\displaystyle=\bar{\varepsilon}_{b}^{\phi}-\sum_{s\in\\{+,-\\}}\left[\frac{r^{b1}_{s}}{2}\left(J^{1}_{s}+J^{b2}_{s}\right)+R^{b1}\sum_{\mu,\nu\in\\{x,z\\}}J^{b12}_{\mu\nu,s}\right]$
(.1.27)

with

$\displaystyle
J_{\pm}^{1}=\frac{2\bar{t}_{\sigma}^{4}}{\left(\Delta_{x}+V_{dp}\pm\Delta\phi\right)^{2}U}\,,$
$\displaystyle r_{\pm}^{a}=\frac{U}{U\pm 2\Delta\phi}\,,$ $\displaystyle
R^{a}=\frac{U}{2\Delta_{x}+U_{p}}\,,$ (.1.28) $\displaystyle
J_{\pm}^{b2}=\frac{2t_{\pi}^{4}}{\left(\Delta_{z}+V_{dp}\pm\Delta\phi\right)^{2}U}\,,$
$\displaystyle r_{\pm}^{b1}=\frac{U}{U-3J_{H}^{b}\pm 2\Delta\phi}\,,\quad$
$\displaystyle R^{b1}=\frac{U}{\Delta_{x}+\Delta_{z}+U_{p}-3J^{p}_{H}}\,,$
(.1.29) $\displaystyle
J_{\mu\nu,\pm}^{b12}=\frac{t_{\pi}^{2}\bar{t}_{\sigma}^{2}}{\left(\Delta_{\mu}+V_{dp}\pm\Delta\phi\right)U\left(\Delta_{\nu}+V_{dp}\pm\Delta\phi\right)}\,,\quad$
$\displaystyle r_{\pm}^{b2}=\frac{U}{U-J_{H}^{b}\pm 2\Delta\phi}\,,$
$\displaystyle R^{b2}=\frac{U}{\Delta_{x}+\Delta_{z}+U_{p}-J^{p}_{H}}\,,$
(.1.30)

and

$\displaystyle\bar{\varepsilon}_{b}^{\phi}=\varepsilon_{b}+\frac{2t_{\sigma
o}^{2}}{\Delta_{yo}}+t_{\sigma}^{2}\left(\frac{1}{\Delta_{x}+\Delta\phi}+\frac{1}{\Delta_{x}-\Delta\phi}\right)-t_{\pi}^{2}\left(\frac{1}{\Delta_{z}+\Delta\phi}+\frac{1}{\Delta_{z}-\Delta\phi}\right)\;.$
(.1.31)

The time-dependency of the problem is provided by the substitution
$\Delta\phi\rightarrow\Delta\phi(t)=\phi_{j\mu}(t)-\phi_{j\alpha}(t)\equiv\phi_{j+1\alpha}(t)-\phi_{j\mu}(t)$
for $\alpha\in\\{a,b\\}$ and $\mu\in\\{x,z\\}$.

In the main text, we did not take into account the renormalization of the
crystal field Eq. (.1.31) provided by the scalar potential, considering
instead its expression in equilibrium Eq. (.1.7), as it only shifts the
energies.

####  .1.4 Floquet limit

The effect of a driving laser on the charge-transfer Hamiltonian is included
via the Peierls substitution, which modifies the hopping between two nearest
neighbor sites $i$ and $j$ as $(t_{\sigma})_{ij}\rightarrow
t_{\sigma}\exp(ieaA_{ij}(t)/2)$ and $(t_{\pi})_{ij}\rightarrow
t_{\pi}\exp(ieaA_{ij}(t)/2)$. Here $A_{ij}(t)$ is a time-dependent vector
potential which we assume to be directed along the chain direction, so it does
not influence the hoppings $t_{\sigma o}$ and $t_{\pi o}$. Considering an
electric field $E(t)=E_{0}\sin(\Omega t)$, the corresponding vector potential
reads $A_{ij}(t)=-E_{0}(i-j)\cos(\Omega t)/\Omega$. This way, the charge-
transfer Hamiltonian $\mathcal{H}_{c-t}^{\text{Fl}}(t)$ becomes periodic in
time with period $T=2\pi/\Omega$
($\mathcal{H}_{c-t}^{\text{Fl}}(t)=\mathcal{H}_{c-t}^{\text{Fl}}(t+T)$). The
solution of the Schrödinger equation $|\psi(t)\rangle$ can be rewritten, by
using the Floquet theorem

$\displaystyle\Ket{\psi(t)}=e^{i\epsilon_{\alpha}t}\Ket{\psi_{\alpha}(t)}\,,$
(.1.32)

with $|\psi_{\alpha}(t)\rangle=|\psi_{\alpha}(t+T)\rangle$ time periodic and
$\epsilon_{\alpha}$ a Floquet quasi-energy defined up to integer multiples of
$\Omega$. $|\psi_{\alpha}(t)\rangle$ can be expanded in Fourier series as
$|\psi_{\alpha}(t)\rangle=\sum_{n=-\infty}^{\infty}\exp(-i\Omega
nt)\,|\psi_{\alpha,n}\rangle$, where $|\psi_{\alpha,n}\rangle$ is the
component of the wave function in the n-th Floquet sector. In this basis, the
Schrödinger equation reads

$\displaystyle(\epsilon_{\alpha}+n\Omega)|\psi_{\alpha,n}\rangle=\sum_{n^{\prime}}(\mathcal{H}_{c-t}^{\text{Fl}})_{n-n^{\prime}}|\psi_{\alpha,n^{\prime}}\rangle$
(.1.33)

with $(\mathcal{H}_{c-t}^{\text{Fl}})_{n}$ the n-th Fourier component of the
Hamiltonian $(\mathcal{H}_{c-t}^{\text{Fl}})_{n}=1/T\int_{0}^{T}\
dt\exp(i\Omega nt)\,\mathcal{H}_{c-t}^{\text{Fl}}(t)$. Thereby, only the
tight-binding contribution $\mathcal{H}^{1}$ of the charge-transfer
Hamiltonian is changed

$\displaystyle(\mathcal{H}^{1})_{n}=$ $\displaystyle-
t_{\sigma}J_{n}(\mathcal{E})\sum_{j\sigma}\left(f_{ja\sigma}^{\dagger}f_{jx\sigma}-(-1)^{n}f_{j+1,a\sigma}^{\dagger}f_{jx\sigma}+(-1)^{n}f_{jx\sigma}^{\dagger}f_{ja\sigma}-f_{jx\sigma}^{\dagger}f_{j+1,a\sigma}\right)$
$\displaystyle-
t_{\pi}J_{n}(\mathcal{E})\sum_{j\sigma}\left(f_{jb\sigma}^{\dagger}f_{jz\sigma}-(-1)^{n}f_{j+1,b\sigma}^{\dagger}f_{jz\sigma}+(-1)^{n}f_{jz\sigma}^{\dagger}f_{jb\sigma}-f_{jz\sigma}^{\dagger}f_{j+1,b\sigma}\right)$
$\displaystyle-t_{\sigma
o}\sum_{j,\sigma}\left(f_{ja\sigma}^{\dagger}f_{jyo^{+}\sigma}-f_{ja\sigma}^{\dagger}f_{jyo^{-}\sigma}+H.c.\right)+\varepsilon_{a}\sum_{j}n_{ja}+\varepsilon_{b}\sum_{j}n_{jb}$
$\displaystyle+\Delta_{x}\sum_{j}\left(n_{jx}-n_{ja}\right)+\Delta_{z}\sum_{j}\left(n_{jz}-n_{jb}\right)+\Delta_{yo}\sum_{j}\left(n_{jyo^{+}}-n_{ja}\right)+\Delta_{yo}\sum_{j}\left(n_{jyo^{-}}-n_{ja}\right)\,,$
(.1.34)

where $J_{n}(\mathcal{E})$ is the n-th Bessel function of the first kind and
$\mathcal{E}=e\tilde{a}E_{0}/(2\Omega)$ is the Floquet parameter. Thus, the
hopping allows the system to change Floquet sectors by the emission or the
absorption of photons. Starting from the effective time-independent Floquet
Hamiltonian, one can follow a similar procedure as presented in the previous
sections, leading to a $t-J$ model with parameters

$\displaystyle J^{\mathcal{E},\Omega}$
$\displaystyle=\sum_{n,l,k}J^{1}_{nlk}(1+R^{a}_{nlk}),$ (.1.35) $\displaystyle
t^{\mathcal{E},\Omega}$
$\displaystyle=\sum_{n,l,k}\left[\frac{3r^{b1}_{l}+r_{l}^{b2}}{8}\left(J^{b12}_{xz,nlk}+J^{b12}_{zx,nlk}\right)+\frac{3R^{b1}_{nlk}+R^{b2}_{nlk}}{8}\left((-1)^{l}J^{b12}_{xx,nlk}+(-1)^{l}J^{b12}_{zz,nlk}+J^{b12}_{xz,nlk}+J^{b12}_{zx,nlk}\right)\right]\,,$
(.1.36) $\displaystyle E^{\mathcal{E},\Omega}$
$\displaystyle=\bar{\varepsilon}_{b}^{\mathcal{E},\Omega}-\sum_{n,l,k}\left[\frac{r^{b1}_{l}}{2}\left(J^{1}_{nlk}+J^{b2}_{nlk}\right)+R^{b1}_{nlk}\left(J^{b12}_{xx,nlk}+J^{b12}_{zz,nlk}+(-1)^{l}J^{b12}_{xz,nlk}+(-1)^{l}J^{b12}_{zx,nlk}\right)\right]\;,$
(.1.37)

where

$\displaystyle
J^{1}_{nlk}=\frac{4\bar{t}_{\sigma}^{4}J_{n}(\mathcal{E})J_{l-n}(\mathcal{E})J_{l-k}(\mathcal{E})J_{k}(\mathcal{E})}{(\Delta_{x}+V_{dp}+n\omega)(U+l\omega)(\Delta_{x}+V_{dp}+k\omega)}\,,\quad$
$\displaystyle
R^{a}_{nlk}=(-1)^{n+k}\frac{U+l\omega}{2\Delta_{x}+U_{p}+l\omega}\left(1+(-1)^{l}\right)\,,$
(.1.38) $\displaystyle
J^{b2}_{nlk}=\frac{4t_{\pi}^{4}J_{n}(\mathcal{E})J_{l-n}(\mathcal{E})J_{l-k}(\mathcal{E})J_{k}(\mathcal{E})}{(\Delta_{z}+V_{dp}+n\omega)(U+l\omega)(\Delta_{z}+V_{dp}+k\omega)}\,,$
$\displaystyle r^{b1}_{l}=\frac{U+l\omega}{U-3J_{H}+l\omega}\,,$
$\displaystyle
R^{b1}_{nlk}=(-1)^{n+k}\frac{U+l\omega}{\Delta_{x}+\Delta_{z}+U_{p}-3J_{H}^{p}+l\omega}\,,$
(.1.39) $\displaystyle
J^{b12}_{\mu\nu,nlk}=\frac{2t_{\pi}^{2}\bar{t}_{\sigma}^{2}J_{n}(\mathcal{E})J_{l-n}(\mathcal{E})J_{l-k}(\mathcal{E})J_{k}(\mathcal{E})}{(\Delta_{\mu}+V_{dp}+n\omega)(U+l\omega)(\Delta_{\nu}+V_{dp}+k\omega)}\;,$
$\displaystyle r^{b2}_{l}=\frac{U+l\omega}{U-J_{H}+l\omega}\,,$ $\displaystyle
R^{b2}_{nlk}=(-1)^{n+k}\frac{U+l\omega}{\Delta_{x}+\Delta_{z}+U_{p}-J_{H}^{p}+l\omega}$
(.1.40)

and

$\displaystyle\bar{\varepsilon}_{b}^{\mathcal{E},\Omega}=\varepsilon_{b}+\frac{2t_{\sigma
o}^{2}}{\Delta_{yo}}+t_{\sigma}^{2}\sum_{n=-\infty}^{+\infty}\frac{J_{n}^{2}(\mathcal{E})}{\Delta_{x}+n\Omega}-t_{\pi}^{2}\sum_{n=-\infty}^{+\infty}\frac{J_{n}^{2}(\mathcal{E})}{\Delta_{z}+n\Omega}\;.$
(.1.41)

###  .2 Theory of time-resolved RIXS

trRIXS is a photon-in photon-out technique which can be described as a second
order X-ray scattering process. Before the arrival of the pump, we assume the
system to be in the ground state $\Ket{\Psi_{0}}$ of $\mathcal{H}_{c-t}$, i.e.
a state in which the holes occupy the $a$ orbitals of the coppers. The
incoming photon with energy $\omega_{i}$, momentum $k_{i}$ and polarization
$\bm{e}_{i}$ can excite the $a$ hole to the $2$p core state of a copper atom.
Within the dipole approximation, this process can be described by the dipole
transition operator $\mathcal{D}_{j\bm{e}_{i}}(t)$. The highly unstable
intermediate state with the core-hole decays within few femtoseconds in one of
the $3\mathrm{d}$ orbital by emitting a photon with energy $\omega_{f}$,
momentum $k_{f}$ and polarization $\bm{e}_{f}$ (described by
$\mathcal{D}^{\dagger}_{j\bm{e}_{f}}(t)$). The overall process is called
direct RIXS, which is the dominant scattering channel since the transition
$2\mathrm{p}\leftrightarrow 3\mathrm{d}$ is dipole allowed Marra (2016).

If we turn on the pump, the system is no longer in equilibrium and we describe
it with

$\displaystyle\bar{\mathcal{H}}_{0}(t)=\mathcal{H}_{0}(t)+\sum_{\begin{subarray}{c}k_{i},\bm{e}_{i}\\\
k_{f},\bm{e}_{f}\end{subarray}}\Big{(}\omega_{i}a^{\dagger}_{k_{i},\bm{e}_{i}}a_{k_{i},\bm{e}_{i}}+\omega_{f}a^{\dagger}_{k_{f},\bm{e}_{f}}a_{k_{f},\bm{e}_{f}}\Big{)}\,,$
(.2.1)

where $\mathcal{H}_{0}(t)$ is the time-dependent $\mathcal{H}_{c-t}$ and the
last contribution is the free photon Hamiltonian $\mathcal{H}_{ph}$.

In the interaction (Dirac) picture the states evolve as
$\bar{\mathcal{U}}_{0}(t,t_{0})|\Psi_{0}\rangle\otimes|\Psi_{ph}\rangle$,
where
$\bar{\mathcal{U}}_{0}(t,t_{0})=\mathcal{T}_{D}\exp\left(-i\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\bar{\mathcal{H}}_{0}(t^{\prime})\right)$
is the time evolution operator with Dyson’s time-ordering operator
$\mathcal{T}_{D}$. For the later use we analogously define time evolution
operators for the photonic and electronic subsystem
$\mathcal{U}_{ph}(t,t_{0})=\exp\left(-i\mathcal{H}_{ph}(t-t_{0})\right)$ and
$\mathcal{U}_{0}(t,t_{0})=\mathcal{T}_{D}\exp\left(-i\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\mathcal{H}_{0}(t^{\prime})\right)$.
The photonic states in the weak-probe limit can be approximated as
$|\Psi_{ph}\rangle\approx a^{\dagger}_{{k}_{i}\bm{e}_{i}}\ket{0}$. The probe
is generally much weaker than the pump, thus we can treat its contribution as
a perturbation $\mathcal{H}^{\prime}(t)$ of the full Hamiltonian
$\mathcal{H}(t)=\bar{\mathcal{H}}_{0}(t)+\mathcal{H}^{\prime}(t)$. By
expanding to second order the time evolution operator reads

$\displaystyle\mathcal{U}(t,t_{0})$
$\displaystyle=\mathcal{T}_{D}\exp\left(-i\int_{t_{0}}^{t}\mathrm{d}t^{\prime}\mathcal{H}(t^{\prime})\right)$
(.2.2)
$\displaystyle\approx\bar{\mathcal{U}}_{0}(t,t_{0})-i\int_{t_{0}}^{t}\mathrm{d}t_{1}\bar{\mathcal{U}}_{0}(t,t_{1})\mathcal{H}^{\prime}(t_{1})\bar{\mathcal{U}}_{0}(t_{1},t_{0})-\int_{t_{0}}^{t}\mathrm{d}t_{2}\int_{t_{0}}^{t_{2}}\mathrm{d}t_{1}\bar{\mathcal{U}}_{0}(t,t_{2})\mathcal{H}^{\prime}(t_{2})\bar{\mathcal{U}}_{0}(t_{2},t_{1})\mathcal{H}^{\prime}(t_{1})\bar{\mathcal{U}}_{0}(t_{1},t_{0})\,.$
(.2.3)

In the experiment one can measure the flux of photons with given momentum,
polarization and energy

$\displaystyle I_{k_{f}\bm{e}_{f}}$
$\displaystyle=\lim_{t\rightarrow\infty}\Braket{\bar{\mathcal{U}}_{0}^{\dagger}(t,t_{0})a_{k_{f}\bm{e}_{f}}^{\dagger}a_{k_{f}\bm{e}_{f}}\bar{\mathcal{U}}_{0}(t,t_{0})}$
$\displaystyle\approx
I_{k_{f}\bm{e}_{f}}^{(0)}+I_{k_{f}\bm{e}_{f}}^{(2)}+I_{k_{f}\bm{e}_{f}}^{(4)}\;,$
(.2.4)

which can be calculated order by order. Note that in Eq. (.2.4), just the even
terms survive since they are the only ones that guarantee the same amount of
photon creation and annihilation operators in the expectation value. The
zeroth order corresponds to elastic reflection of the photon and the second
order to time-resolved X-ray absorption spectroscopy (trXAS). The fourth order
contains the actual RIXS spectrum, where we have to include an incoming and
outgoing process on each branch of the Keldysh contour. Therefore, we
decompose the probe Hamiltonian into
$\mathcal{H}^{\prime}(t)=\mathcal{H}_{in}(t)+\mathcal{H}_{out}(t)+H.c.$ and
define them as

$\displaystyle\mathcal{H}_{in}(t)=s(t)\frac{1}{\sqrt{L}}\sum_{j{q}\bm{e}}\mathcal{D}_{j\bm{e}}(t)e^{iqj}a_{q\bm{e}},\qquad\mathcal{H}_{out}(t)=\frac{1}{\sqrt{L}}\sum_{j{q}\bm{e}}a^{\dagger}_{q\bm{e}}e^{-iqj}\mathcal{D}^{\dagger}_{j\bm{e}}(t)\;,$
(.2.5)

where $s(t)\equiv s(t,\bar{t})$ is the probe envelope function defined in the
main text. In direct RIXS the core electron is excited directly into the
valence band. Thus, the local dipole transition operator is
$\mathcal{D}_{j\bm{e}}(t)=\sum_{\sigma\alpha\rho}M^{\bm{e}}_{\alpha\rho}(t)f^{\dagger}_{j\alpha\sigma}f_{j\rho\sigma}$,
where $f_{j\alpha\sigma}$ ($f_{j\rho\sigma}$) annihilates an electron at site
$j$ with spin $\sigma$ in valence $\alpha$ (core $\rho$) orbital.
$M^{\bm{e}}_{\alpha\rho}(t)$ denotes the associated matrix element with photon
polarization $\bm{e}$, which, in non-equilibrium, can be time-dependent Chen
_et al._ (2019). If we only collect the relevant terms for trRIXS we get

$\displaystyle I_{RIXS}\coloneqq I_{k_{f}\bm{e}_{f}}^{(4)}$
$\displaystyle=\int_{t_{0}}^{\infty}\mathrm{d}t^{\prime}_{2}\int_{t_{0}}^{t^{\prime}_{2}}\mathrm{d}t^{\prime}_{1}\int_{t_{0}}^{\infty}\mathrm{d}t_{2}\int_{t_{0}}^{t_{2}}\mathrm{d}t_{1}\Big{\langle}\bar{\mathcal{U}}_{0}(t_{0},t^{\prime}_{1})\mathcal{H}^{\dagger}_{in}(t^{\prime}_{1})\bar{\mathcal{U}}_{0}(t^{\prime}_{1},t^{\prime}_{2})\mathcal{H}^{\dagger}_{out}(t^{\prime}_{2})\bar{\mathcal{U}}_{0}(t^{\prime}_{2},\infty)a_{k_{f}\bm{e}_{f}}^{\dagger}$
$\displaystyle\hphantom{==}\times
a_{k_{f}\bm{e}_{f}}\bar{\mathcal{U}}_{0}(\infty,t_{2})\mathcal{H}_{out}(t_{2})\bar{\mathcal{U}}_{0}(t_{2},t_{1})\mathcal{H}_{in}(t_{1})\bar{\mathcal{U}}_{0}(t_{1},t_{0})\Big{\rangle}$
(.2.6) $\displaystyle=\frac{1}{L^{2}}\sum_{\begin{subarray}{c}mj\\\
m^{\prime}j^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}q^{\prime}_{2}q^{\prime}_{1}q_{2}q_{1}\\\
\bm{e}^{\prime}_{2}\bm{e}^{\prime}_{1}\bm{e}_{2}\bm{e}_{1}\end{subarray}}\int_{t_{0}}^{\infty}\mathrm{d}t^{\prime}_{2}\int_{t_{0}}^{t^{\prime}_{2}}\mathrm{d}t^{\prime}_{1}\int_{t_{0}}^{\infty}\mathrm{d}t_{2}\int_{t_{0}}^{t_{2}}\mathrm{d}t_{1}s(t^{\prime}_{1})s(t_{1})$
$\displaystyle\hphantom{==}\times\Big{\langle}\bar{\mathcal{U}}_{0}(t_{0},t^{\prime}_{1})a^{\dagger}_{q^{\prime}_{1}e^{\prime}_{1}}e^{-iq^{\prime}_{1}m^{\prime}}\mathcal{D}_{{m^{\prime}}\bm{e}^{\prime}_{1}}^{\dagger}(t^{\prime}_{1})\bar{\mathcal{U}}_{0}(t^{\prime}_{1},t^{\prime}_{2})\mathcal{D}_{j^{\prime}\bm{e}^{\prime}_{2}}(t^{\prime}_{2})e^{iq^{\prime}_{2}j^{\prime}}a_{q^{\prime}_{2}\bm{e}^{\prime}_{2}}\bar{\mathcal{U}}_{0}(t^{\prime}_{2},\infty)a_{k_{f}\bm{e}_{f}}^{\dagger}$
$\displaystyle\hphantom{==}\times
a_{k_{f}\bm{e}_{f}}\bar{\mathcal{U}}_{0}(\infty,t_{2})a^{\dagger}_{q_{2}\bm{e}_{2}}e^{-iq_{2}m}\mathcal{D}_{{m}\bm{e}_{2}}^{\dagger}(t_{2})\bar{\mathcal{U}}_{0}(t_{2},t_{1})\mathcal{D}_{j\bm{e}_{1}}(t_{1})e^{iq_{1}j}a_{q_{1}\bm{e}_{1}}\bar{\mathcal{U}}_{0}(t_{1},t_{0})\Big{\rangle}\,.$
(.2.7)

By separating the average value into the electronic and photonic part, we
realize that for the photonic part it has to hold

$\displaystyle\Bra{0}a_{k_{i}\bm{e}_{i}}\mathcal{U}_{ph}(t_{0},t^{\prime}_{1})a^{\dagger}_{q^{\prime}_{1}\bm{e}^{\prime}_{1}}\mathcal{U}_{ph}(t^{\prime}_{1},t^{\prime}_{2})a_{q^{\prime}_{2}\bm{e}^{\prime}_{2}}\mathcal{U}_{ph}(t^{\prime}_{2},\infty)a_{k_{f}\bm{e}_{f}}^{\dagger}a_{k_{f}\bm{e}_{f}}\mathcal{U}_{ph}(\infty,t_{2})a^{\dagger}_{q_{2}\bm{e}_{2}}\mathcal{U}_{ph}(t_{2},t_{1})a_{q_{1}\bm{e}_{1}}\mathcal{U}_{ph}(t_{1},t_{0})a_{k_{i}\bm{e}_{i}}^{\dagger}\Ket{0}=$
$\displaystyle=\delta_{k_{i}q^{\prime}_{1}}\delta_{\bm{e}_{i}\bm{e}^{\prime}_{1}}\delta_{k_{i}q_{1}}\delta_{\bm{e}_{i}\bm{e}_{1}}\delta_{k_{f}q^{\prime}_{2}}\delta_{\bm{e}_{f}\bm{e}^{\prime}_{2}}\delta_{k_{f}q_{2}}\delta_{\bm{e}_{f}\bm{e}_{2}}e^{i\omega_{i}(t^{\prime}_{1}-t_{1})-i\omega_{f}(t^{\prime}_{2}-t_{2})}\,.$
(.2.8)

We assume that in the intermediate state the core hole does not move, i.e.
$m=j$ and $m^{\prime}=j^{\prime}$. Substituting that, Eq. (.2) and
$k=k_{f}-k_{i}$ into Eq. (.2.7) yields

$\displaystyle I_{RIXS}=$
$\displaystyle\frac{1}{L}\sum_{jj^{\prime}}\int_{t_{0}}^{\infty}\mathrm{d}t^{\prime}_{2}\int_{t_{0}}^{t^{\prime}_{2}}\mathrm{d}t^{\prime}_{1}\int_{t_{0}}^{\infty}\mathrm{d}t_{2}\int_{t_{0}}^{t_{2}}\mathrm{d}t_{1}s(t^{\prime}_{1})s(t_{1})e^{-ik(j-j^{\prime})}e^{i\omega_{i}(t^{\prime}_{1}-t_{1})-i\omega_{f}(t^{\prime}_{2}-t_{2})}$
$\displaystyle\times\Big{\langle}\mathcal{U}_{0}(t_{0},t^{\prime}_{1})\mathcal{D}_{j^{\prime}\bm{e}_{i}}^{\dagger}(t^{\prime}_{1})\mathcal{U}_{0}(t^{\prime}_{1},t^{\prime}_{2})\mathcal{D}_{j^{\prime}\bm{e}_{f}}(t^{\prime}_{2})\mathcal{U}_{0}(t^{\prime}_{2},t_{2})\mathcal{D}_{j\bm{e}_{f}}^{\dagger}(t_{2})\mathcal{U}_{0}(t_{2},t_{1})\mathcal{D}_{j\bm{e}_{i}}(t_{1})\mathcal{U}_{0}(t_{1},t_{0})\Big{\rangle}\,.$
(.2.9)

By introducing the local RIXS operator

$\displaystyle\mathcal{B}_{j\bm{e}_{i}\bm{e_{f}}}\coloneqq\mathcal{D}_{j\bm{e}_{f}}^{\dagger}(t_{1})\frac{1}{(\omega_{f}-\mathcal{H}_{0}+i\Gamma)}\mathcal{D}_{j\bm{e}_{i}}(t_{1})$
(.2.10)

with $1/\Gamma$ the core-hole lifetime Marra  _et al._ (2012), we can
implement the ultra-short core-hole lifetime approximation Ament  _et al._
(2007), which reduces Eq. (.2.9) to a two point correlation function

$\displaystyle
I_{RIXS}\approx\frac{1}{L}\sum_{j^{\prime}=0}^{L-1}\int_{t_{0}}^{\infty}\mathrm{d}t^{\prime}_{1}\int_{t_{0}}^{\infty}\mathrm{d}t_{1}s(t^{\prime}_{1})s(t_{1})e^{-i\omega(t_{1}-t^{\prime}_{1})}e^{ikj^{\prime}}\Bra{\Psi_{0}}\mathcal{U}_{0}(t_{0},t^{\prime}_{1})\mathcal{B}_{j^{\prime}\bm{e}_{i}\bm{e}_{f}}^{\dagger}\mathcal{U}_{0}(t^{\prime}_{1},t_{1})\mathcal{B}_{0\bm{e}_{i}\bm{e}_{f}}\mathcal{U}_{0}(t_{1},t_{0})\Ket{\Psi_{0}}\;,$
(.2.11)

where $\omega\coloneqq\omega_{i}-\omega_{f}$. The time evolution operator
$\mathcal{U}_{0}(t^{\prime},t)$ now acts in the low energy sector, i.e. there
are no charge excitations. It is, thus, determined by the $t-J$ or Heisenberg
Hamiltonian in the quasi-static or Floquet limit. Since we work with periodic
boundary conditions (PBC), in Eq. (.2.11) we assume, without any loss of
generality, the action of the first $\mathcal{B}_{j\bm{e}_{i}\bm{e_{f}}}$ on
site $j=0$. The local RIXS operators can than be expanded in the basis of the
local hole spin and orbital as

$\displaystyle\mathcal{B}_{j\bm{e}_{i}\bm{e}_{f}}=\sum_{\sigma\sigma^{\prime}}B^{\sigma\sigma^{\prime}}_{\bm{e}_{i}\bm{e}_{f}}f_{jb\sigma}^{\dagger}f_{ja\sigma^{\prime}}\;.$
(.2.12)

The RIXS matrix elements $B^{\sigma\sigma^{\prime}}_{\bm{e}_{i}\bm{e}_{f}}$
can be calculated following Marra (2016).

By considering that the derived $t-J$ Hamiltonian conserves the spin of the
hole, we can write Eq. (.2.11) as

$\displaystyle I_{RIXS}=$
$\displaystyle\frac{1}{L}\int_{t_{0}}^{\infty}\mathrm{d}t^{\prime}_{1}\int_{t_{0}}^{\infty}\mathrm{d}t_{1}s(t^{\prime}_{1})s(t_{1})e^{-i\omega(t_{1}-t^{\prime}_{1})}\Big{[}\Bra{\Psi_{0}}\mathcal{U}_{0}(t_{0},t^{\prime}_{1})\sum_{j^{\prime}}e^{ikj^{\prime}}\sum_{\sigma^{\prime}}B^{\uparrow\sigma^{\prime}*}_{\bm{e}_{i}\bm{e}_{f}}f_{j^{\prime}a\sigma^{\prime}}^{\dagger}f_{j^{\prime}b\uparrow}\mathcal{U}_{0}(t^{\prime}_{1},t_{1})\sum_{\sigma}B^{\uparrow\sigma}_{\bm{e}_{i}\bm{e}_{f}}f_{0b\uparrow}^{\dagger}f_{0a\sigma^{\prime}}\mathcal{U}_{0}(t_{1},t_{0})\Ket{\Psi_{0}}$
$\displaystyle+\Bra{\Psi_{0}}\mathcal{U}_{0}(t_{0},t^{\prime}_{1})\sum_{j^{\prime}}e^{ikj^{\prime}}\sum_{\sigma^{\prime}}B^{\downarrow\sigma*}_{\bm{e}_{i}\bm{e}_{f}}f_{j^{\prime}a\sigma}^{\dagger}f_{j^{\prime}b\downarrow}\mathcal{U}_{0}(t^{\prime}_{1},t_{1})\sum_{\sigma}B^{\downarrow\sigma}_{\bm{e}_{i}\bm{e}_{f}}f_{0b\downarrow}^{\dagger}f_{0a\sigma}\mathcal{U}_{0}(t_{1},t_{0})\Ket{\Psi_{0}}\Big{]}.$
(.2.13)

If we now apply the transformations explained in App..1 to obtain the
effective $t-J$ Hamiltonian, we get for one of the local RIXS operators

$\displaystyle\sum_{\sigma}B^{\uparrow\sigma}_{\bm{e}_{i}\bm{e}_{f}}f_{0b\uparrow}^{\dagger}f_{0a\sigma}=B^{\uparrow\uparrow}_{\bm{e}_{i}\bm{e}_{f}}p_{0\uparrow}+B^{\uparrow\downarrow}_{\bm{e}_{i}\bm{e}_{f}}p_{0\downarrow}\,,$
(.2.14)

whereby the RIXS matrix elements generally obey
$B^{\uparrow\uparrow}_{\bm{e}_{i}\bm{e}_{f}}B^{\downarrow\uparrow*}_{\bm{e}_{i}\bm{e}_{f}}=-B^{\downarrow\downarrow*}_{\bm{e}_{i}\bm{e}_{f}}B^{\uparrow\downarrow}_{\bm{e}_{i}\bm{e}_{f}}$.
Using these properties together with the SU(2) invariance of
$\mathcal{H}_{t-J}$, the RIXS intensity reduces to

$\displaystyle I_{RIXS}\left(\omega,k,\bar{t}\right)=$
$\displaystyle\frac{2}{L}\left(\absolutevalue{B^{\uparrow\uparrow}_{\bm{e}_{i}\bm{e}_{f}}}^{2}+\absolutevalue{B^{\uparrow\downarrow}_{\bm{e}_{i}\bm{e}_{f}}}^{2}\right)\int_{t_{0}}^{\infty}\mathrm{d}t^{\prime}_{1}\int_{t_{0}}^{\infty}\mathrm{d}t_{1}s(t^{\prime}_{1},\bar{t})s(t_{1},\bar{t})e^{-i\omega(t_{1}-t^{\prime}_{1})}$
$\displaystyle\times\Bra{\Psi_{0}}\mathcal{U}_{s}(t_{0},t^{\prime}_{1})\sum_{j^{\prime}}e^{ikj^{\prime}}p_{j^{\prime}\uparrow}^{\dagger}\mathcal{U}_{t-J}(t^{\prime}_{1},t_{1})p_{0\uparrow}\mathcal{U}_{s}(t_{1},t_{0})\Ket{\Psi_{0}}$
(.2.15)

with the according time evolution operators of $\mathcal{H}_{s}$ and
$\mathcal{H}_{t-J}$. At this point we can recognize Eq. (4) of the main text.
The RIXS matrix elements in the ultra-short core-hole lifetime approximation
can be calculated as

$\displaystyle
B^{\uparrow\uparrow}_{\bm{e}_{i}\bm{e}_{f}}=\frac{e^{*}_{f,z}}{\Gamma}\left(e_{i,y}-ie_{i,x}\right),\qquad
B^{\uparrow\downarrow}_{\bm{e}_{i}\bm{e}_{f}}=\frac{e^{*}_{f,x}}{\Gamma}\left(ie_{i,x}-e_{i,y}\right).$
(.2.16)

The polarisation of the incoming and outgoing photons depends on the
experimental setup, thus, we do not specify them here.