supplement_v2_arXiv

# Ultrafast renormalization of the onsite Coulomb repulsion in a cuprate
superconductor

Denitsa R. Baykusheva  Department of Physics, Harvard University, Cambridge,
Massachusetts 02138, USA  Hoyoung Jang  PAL-XFEL, Pohang Accelerator
Laboratory, Pohang, Gyeongbuk 37673, Republic of Korea  Ali A. Husain
Department of Physics and Seitz Materials Research Laboratory, University of
Illinois, Urbana, IL 61801, USA  Quantum Matter Institute and Department of
Physics and Astronomy, University of British Columbia, Vancouver, BC V6T 1Z4,
Canada  Sangjun Lee  Department of Physics and Seitz Materials Research
Laboratory, University of Illinois, Urbana, IL 61801, USA  Sophia F. R.
TenHuisen  Department of Physics, Harvard University, Cambridge, Massachusetts
02138, USA  John A. Paulson School of Engineering and Applied Sciences,
Harvard University, Cambridge, Massachusetts 02138, USA  Preston Zhou
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
Sunwook Park  Department of Physics, Pohang University of Science and
Technology, Pohang 790-784, South Korea  Center for Artificial Low Dimensional
Electronic Systems, Institute for Basic Science (IBS), 77 Cheongam-Ro, Pohang
790-784, South Korea  Hoon Kim  Department of Physics, Pohang University of
Science and Technology, Pohang 790-784, South Korea  Center for Artificial Low
Dimensional Electronic Systems, Institute for Basic Science (IBS), 77
Cheongam-Ro, Pohang 790-784, South Korea  Jinkwang Kim  Department of Physics,
Pohang University of Science and Technology, Pohang 790-784, South Korea
Center for Artificial Low Dimensional Electronic Systems, Institute for Basic
Science (IBS), 77 Cheongam-Ro, Pohang 790-784, South Korea  Hyeong-Do Kim
PAL-XFEL, Pohang Accelerator Laboratory, Pohang, Gyeongbuk 37673, Republic of
Korea  Minseok Kim  PAL-XFEL, Pohang Accelerator Laboratory, Pohang, Gyeongbuk
37673, Republic of Korea  Sang-Youn Park  PAL-XFEL, Pohang Accelerator
Laboratory, Pohang, Gyeongbuk 37673, Republic of Korea  Peter Abbamonte
Department of Physics and Seitz Materials Research Laboratory, University of
Illinois, Urbana, IL 61801, USA  B. J. Kim  Department of Physics, Pohang
University of Science and Technology, Pohang 790-784, South Korea  Center for
Artificial Low Dimensional Electronic Systems, Institute for Basic Science
(IBS), 77 Cheongam-Ro, Pohang 790-784, South Korea  G. D. Gu  Condensed Matter
Physics and Materials Science, Brookhaven National Laboratory (BNL), Upton, NY
11973 USA  Yao Wang  Department of Physics and Astronomy, Clemson University,
Clemson, South Carolina 29631, USA  Matteo Mitrano  Department of Physics,
Harvard University, Cambridge, Massachusetts 02138, USA

###### Abstract

Ultrafast lasers are an increasingly important tool to control and stabilize
emergent phases in quantum materials. Among a variety of possible excitation
protocols, a particularly intriguing route is the direct light-engineering of
microscopic electronic parameters, such as the electron hopping and the local
Coulomb repulsion (Hubbard $U$). In this work, we use time-resolved x-ray
absorption spectroscopy to demonstrate the light-induced renormalization of
the Hubbard $U$ in a cuprate superconductor, La1.905Ba0.095CuO4. We show that
intense femtosecond laser pulses induce a substantial redshift of the upper
Hubbard band, while leaving the Zhang-Rice singlet energy unaffected. By
comparing the experimental data to time-dependent spectra of single- and
three-band Hubbard models, we assign this effect to a $\sim 140$ meV reduction
of the onsite Coulomb repulsion on the copper sites. Our demonstration of a
dynamical Hubbard $U$ renormalization in a copper oxide paves the way to a
novel strategy for the manipulation of superconductivity, magnetism, as well
as to the realization of other long-range-ordered phases in light-driven
quantum materials.

##  I INTRODUCTION

The electronic dynamics of strongly correlated materials is governed by a
subtle competition between itinerancy due to hopping and localization driven
by Coulomb repulsion [1, 2]. The balance between these two tendencies is
responsible for a wide variety of emergent quantum phases and its manipulation
through external perturbations is a central focus of modern condensed matter
physics [3, 4]. Ultrafast laser pulses offer an intriguing nonequilibrium
control route, particularly when fields are strong enough ($\sim 0.1$-$1$ V/Å
) to induce energy changes at the scale of the effective electronic
interactions. Electronic hopping can be controlled by transiently bending the
band structure (dynamical Franz-Keldysh effect) [5, 6, 7, 8, 9, 10], by
displacing atoms via nonlinear phonon excitation [11, 12, 13], or by coherent
light-matter dressing of the band structure (Floquet engineering) [14, 15,
16]. However, manipulating the effective Coulomb repulsion with ultrafast
lasers is less straightforward. Optical spectroscopy in organic solids hints
at the possibility of modifying the effective onsite interaction (Hubbard $U$)
by coherently driving local molecular vibrations [17, 18, 19] or enhancing
intradimer hopping [20]. Yet this route relies on the presence of molecular
orbitals coupled to local structural degrees of freedom and cannot be readily
extended to other classes of strongly correlated materials, like transition
metal oxides. In these systems, the Hubbard $U$ is a strictly atomic property
and its modification requires alternative microscopic mechanisms, such as
dynamical electronic screening [21, 22, 23, 24, 25] or Floquet-type dressing
of the Coulomb repulsion [26]. Achieving dynamical control of the Hubbard $U$
in transition metal oxides would be particularly consequential for steering
their multiple quantum phases, notably magnetism, multiferroicity, charge/spin
order, and superconductivity [27, 28].

In the specific case of the high-$T_{c}$ cuprate superconductors, the local
Coulomb repulsion has a pervasive effect on normal state properties, as well
as magnetic and superconducting phases [29, 30]. The Hubbard $U$ increases the
quasiparticle effective mass [31] and broadens the Fermi surface [32].
Furthermore, it sets the scale of the magnetic superexchange and, in a spin-
fluctuation-mediated picture, it directly determines the superconducting
pairing [33] and critical temperature [34]. Whether ultrafast lasers are able
to modify the Hubbard $U$ in cuprate superconductors is an open experimental
and theoretical problem.

Here, we demonstrate a pump-induced renormalization of the onsite Coulomb
repulsion in La2-xBaxCuO4 (LBCO), a single-layer cuprate with coexisting high-
temperature superconductivity (up to $32$ K), and charge and spin order [35,
36, 37]. We directly probe the electronic density of states of LBCO by
measuring its time-dependent x-ray absorption spectrum and show that intense
near-infrared pump pulses induce a significant transient shift of the
absorption peaks. By mapping our data onto single- and three-band Hubbard
models, we assign the shift to a renormalization of the local onsite Coulomb
repulsion on the copper orbitals. Finally, we discuss implications of these
results for the understanding of light-enhanced superconductivity, the
generation of high harmonics in correlated materials, and the realization of
coherent light-driven states.  Figure 1:  Probing effective electronic
interactions with x-ray absorption spectroscopy. (a) A LBCO ($x=9.5~{}\%$)
sample is driven by intense pump pulses at 1.55 eV, while its low-energy
electronic structure is probed by exciting core electrons into unoccupied
valence states with soft x-ray pulses. Equilibrium XAS spectrum at (b) O $K$
and (c) Cu $L$ edges, with transitions into Zhang-Rice singlets (ZRS) and
upper Hubbard band (UHB) denoted by red and purple areas, respectively.
Hatched areas indicate the main absorption edge, while the high-energy
shoulder in (c) is a ligand hole sideband which we neglect in the rest of this
work. A sketch of the density of states and the relevant x-ray transitions is
reported within each panel.

##  II Experimental Methods

Time-resolved x-ray absorption spectroscopy (trXAS) is a valuable element-
specific probe of the local electronic structure in light-driven materials
[38, 39, 40, 41, 42, 43, 44, 45], especially in insulating and/or poorly
cleavable samples. This experiment makes use of the resonant soft x-ray
scattering (RSXS) endstation of the Pohang Accelerator Laboratory x-ray free
electron laser (PAL-XFEL) [46]. We acquire trXAS spectra in fluorescence-yield
mode at both the O $K$ and Cu $L_{3}$ edges with an energy resolution of 0.046
eV and 0.34 eV, respectively. The x-ray beam, focused to $120\times
230~{}\mathrm{\mu m}^{2}$ and horizontally polarized, impinges at near-normal
incidence and is detected by an avalanche photodiode (APD) at
$2\theta=150^{\circ}$. Shot-to-shot x-ray intensity fluctuations are recorded
with a gas-monitor detector and used to normalize the XAS signal. We pump the
sample with 50-fs pulses centered at 1.55 eV, polarized in the $ab$-plane, and
focused to $600\times 600~{}\mathrm{\mu m}^{2}$ to obtain a 10 mJ/cm2 fluence
($\sim$12 MV/cm peak electric field). Since the optical penetration depth (370
nm) exceeds that of the x-ray (170-290 nm), our trXAS spectra correspond to a
homogeneously excited volume. The sample, a mm-sized single crystal of
underdoped LBCO ($x=9.5\>\%$, $T_{c}=32$ K) [37] is cleaved to expose a fresh
$ab$ surface and kept at 17 K throughout the entire experiment.

##  III Equilibrium x-ray absorption spectrum

As shown in Fig. 1a, the equilibrium electronic structure of the copper oxides
is mainly determined by the in-plane O $2p_{x,y}$ and Cu $3d_{x^{2}-y^{2}}$
orbitals [29, 47]. The $3d$ orbitals form two bands (lower and upper Hubbard
bands, LHB and UHB respectively) due to the onsite Coulomb repulsion, the LHB
being pushed below the oxygen $2p$ states. This band separation gives rise to
a charge transfer (CT) gap, in which the lowest electronic transition occurs
between oxygen (occupied) and copper (unoccupied) states. Upon hole doping,
the density of states develops an additional feature at the lower end of the
CT gap. This feature is generally attributed to a hybridized state between a
local $3d$-hole and a $2p$-hole on the four surrounding oxygens, and is known
as Zhang-Rice singlet (ZRS) [48, 49]. In this work, we measure absorption
spectra at both the O $K$ and Cu $L_{3}$ edges to map the local density of
states in the CuO2 planes and determine the transient effective interactions.

The equilibrium O $K$ and Cu $L_{3}$ edge absorption spectra are shown in Fig.
1b-c. The O $K$ spectrum features two pre-edge peaks at 528.9 and 530.5 eV
[see Supplementary Materials (SM) for further details]. The lower-energy
transition involves Zhang-Rice singlets (ZRS)
($3d^{9}\underline{L}\rightarrow\underline{1s}3d^{9}$, with $\underline{L}$
and $\underline{1s}$ being respectively the O $2p_{x,y}$ ligand and the O $1s$
core holes) [50, 49, 35]. The higher-energy peak corresponds to a transition
into the UHB ($3d^{9}\rightarrow\underline{1s}3d^{10}$), which becomes dipole-
allowed due to the mixing of $3d^{10}\underline{L}$ and $3d^{9}$
configurations in the ground state [51]. The Cu $L_{3}$ edge spectrum in Fig.
1c instead exhibits a main peak at 932.6 eV and a weaker shoulder at $\sim$934
eV, which correspond to transitions into the UHB
($3d^{9}\rightarrow\underline{2p}_{3/2}3d^{10}$) and into ligand states
($3d^{9}\underline{L}\rightarrow\underline{2p}_{3/2}3d^{10}\underline{L}$)
[51, 52] (see SM Sec. LABEL:sec:XAS_analysis_Cu for further details).

Figure 2:  Pump-induced reshaping of the x-ray absorption. XAS spectra before
($\Delta t\sim-1$ ps) and after ($\Delta t\sim 0$ ps) the pump arrival at both
O $K$ (a) and Cu $L_{3}$ (b) absorption edges. Symbols indicate equilibrium
(black), transient (red/blue), and difference spectra (grey), while solid
lines represent fits to the spectra. The main edge absorption in (a) has been
subtracted for clarity. The fit peak positions for each feature are marked by
dashed lines. The UHB is found to shift by the energy indicated in the panels
(error bars are 95 % confidence intervals), while the ZRS position is
unchanged.

##  IV Transient x-ray absorption dynamics

Having assigned the spectral features at both absorption edges, we are now
able to track light-induced changes to the LBCO electronic structure. Figure 2
reports our key experimental observation. By comparing equilibrium and
transient XAS spectra, we observe prompt photoinduced changes on both
absorption edges. At the peak of the response, the UHB peak undergoes a
substantial redshift, whereas the ZRS transition energy remains at its
equilibrium value. The UHB shift is especially prominent at the Cu $L_{3}$
edge, as its center energy decreases by 125 meV compared to a 63 meV shift at
the O $K$ edge. The integrated intensity of both peaks is unperturbed and only
the ZRS transition is found to broaden by approximately 20 %.

Figure 3:  Time dependence of the pump-induced electronic dynamics.
Equilibrium XAS spectra at O $K$ and Cu $L_{3}$ absorption edges are shown in
panels (a) and (b), respectively (shaded areas). Selected energies used to
measure the time-dependent recovery are indicated by filled and empty symbols.
Panels (c) and (e) display the temporal evolution of ZRS and UHB peaks at the
O $K$ edge, while points around the UHB peak at the the Cu $L_{3}$ edge are
reported in panels (d) and (f). Error bars on the symbols represent standard
deviations, while dash-dotted lines are fit to the data (see SM for further
details).

These photoinduced changes are prompt (close to the $\sim$80 fs pump-probe
cross correlation) and relax to equilibrium following a single exponential
behavior with a decay constant on the order of 100 fs (see Fig. 3). These
timescales are compatible with a relaxation process mediated by a dissipative
bath of optical phonons [53] and in agreement with previous ultrafast optical
spectroscopy [54, 55, 56], electron diffraction [57, 58] and photoemission
[59, 60, 61] experiments on multiple cuprate compounds.

Intriguingly, the UHB recovers more slowly than the ZRS peak. The UHB
approaches equilibrium in 320 fs (220 fs) at the O $K$ (Cu $L_{3}$) edge,
while the ZRS decays over $100$ fs (see SM). This is consistent with holon-
doublon recombination dynamics in Mott-Hubbard and CT insulators, a mechanism
also at play in other copper oxides [54, 55, 62, 63, 64], organic Mott
insulators [65, 66], and cold atoms in optical lattices [67, 68]. In our
experiment, the 1.55 eV pump excites electrons across the CT gap, thus
creating holes (holons) and double occupancies (doublons) [69]. Unlike
conduction carriers within the ZRS band, holon-doublon pairs need to dissipate
an energy approximately equal to the CT gap in order to recombine. This
process generally requires a large number of lower-energy scattering partners
(e.g. phonons, magnons, or lower energy charge fluctuations), thus making
decay events increasingly rare [68, 62, 69]. As consequence, the UHB
relaxation rate gets exponentially suppressed with the increasing CT gap [62,
69] and becomes slower than the conduction-band quasiparticle relaxation.

These data cannot be rationalized in terms of pump-induced doping or heating
effects. Hole doping in the closely related La2-xSrxCuO4 [50, 51, 52] leads to
significant intensity changes and an increased energy separation between UHB
and ZRS transitions, while we observe a decrease in the energy gap between the
two peaks. Electronic heating will redshift the entire absorption spectrum
through the creation of unoccupied states at lower energies [39, 70], with the
ZRS shifting more (129 meV) than the UHB peak (87 meV) at an estimated
$T_{e}\sim 2500$ K (see Appendix A and Sec. LABEL:sec:T-dependence of SM for
electronic temperatures between $T_{e}=0$ and $T_{e}=5300$ K). In contrast, we
experimentally observe a shift of the UHB while the ZRS remains fixed.
Finally, the excitation of optical or acoustic phonons [59, 58, 71], which may
form multiplet structures [72] and broaden the XAS spectrum beyond the natural
core hole width [73, 74], will primarily affect intensity and width (and not
the energy) of the near-edge transitions. Therefore, a more natural
explanation for our data is a dynamic renormalization of the effective
electronic interactions.

##  V Quantifying the transient Hubbard $U$

To determine which electronic interactions are affected by the pump, we
calculate the transient x-ray absorption of a three-band Hubbard model with
renormalized electronic parameters [75, 76, 77, 73, 78]. The calculation is
performed at $12.5\>\%$ hole doping in order to reduce the computational
complexity. While the superconducting and charge-ordered phases at $12.5\>\%$
hole doping are distinct from the ones of our $x=9.5\>\%$ sample [37], the XAS
spectra are similar enough in intensity and peak energy [50, 51] to justify
this simplification. The theoretical spectra closely resemble the experimental
x-ray absorption in Figs. 1-2. The O $K$ edge spectrum in Fig. 4a features one
prominent ZRS peak approximately 1 eV below the edge threshold
($E_{\mathrm{edge}}$), and additional spectral peaks around 1 eV corresponding
to the UHB. The Cu $L_{3}$ edge spectrum is instead characterized by a
prominent UHB peak around $E_{\mathrm{edge}}$ and a weaker peak 1.2 eV above
corresponding to the $3d^{9}\underline{L}$ states. All spectra are broadened
only by accounting for the core-hole lifetime, which is assumed to be 0.2 eV
at both the O $K$ and Cu $L_{3}$ edges [79, 80]. At the O $K$ edge, the
separation between ZRS and UHB centroid for $U_{d}=8.5$ eV matches the
experimental separation ($\sim 1.8$ eV), while the overall UHB intensity is
about 85 % the one of the ZRS (as expected for the 12.5 % hole-doped spectrum
[50]). To accurately determine the slope of the UHB peak shift, we only track
the sharpest feature, which is located at slightly lower energy. We
systematically tune each parameter of the three-band Hubbard Hamiltonian and
map the corresponding effect on a static XAS spectrum capturing the excited
steady state. Our observation of redshifting UHB and fixed ZRS transitions is
only reproduced by changing the onsite Coulomb repulsion $U_{d}$ (see Fig.
4a,c). All other parameters ($t_{pp}$, $U_{p}$, and $t_{pd}$) introduce an
energy shift of both peaks in tandem (see SM Sec. LABEL:sec:3band). The shift
of the sole UHB band crucially identifies the photoinduced dynamics as a
transient renormalization of the Hubbard $U$ on the copper sites. By fitting
the theoretical spectra  (see SM Sec. LABEL:sec:3band), we subsequently
convert the peak shift into an absolute change $\delta U_{d}$ of the Hubbard
$U$. The UHB peak position is found to shift linearly for small $\delta U_{d}$
(see Fig. 4b,d), while the ZRS is almost unaffected. The slope of the UHB
position as function of $U_{d}$ at the Cu $L_{3}$ edge is larger than that at
the O $K$ edge, consistent with the experiment, and the observed shifts are
reproduced by a Hubbard $U$ reduction $\delta U_{d}\sim 140$ meV.

Figure 4:  Theoretical XAS spectra of a three-band Hubbard model for variable
onsite Coulomb repulsion $U_{d}$. Panels (a) and (c) show XAS spectra at the
oxygen and copper absorption edges, respectively. The value of $U_{d}$ used in
the calculation is reported nearby each curve, while all the other electronic
parameters are kept fixed at their equilibrium values. Panels (b) and (d) show
UHB and ZRS center positions a function of $U_{d}$. The inset in panel (d) is
a cartoon of the three-band Hubbard system used in this calculation with
hopping amplitudes $t_{pd}$ and $t_{pp}$, onsite Coulomb repulsions $U_{d}$
and $U_{p}$, and core-hole interaction $U_{c}$.

##  VI A minimal theoretical description

While providing quantitative information about the electronic interactions, it
is not obvious a priori that the transient dynamics can be fully captured by a
quasiequilibrium three-band Hubbard model with tuned electronic parameters.
Therefore, we perform a full time-dependent exact diagonalization calculation
of the light-driven x-ray absorption spectrum. We downfold the three-band
Hamiltonian to an effective 2D, single-band Hubbard model [73, 48]. In this
transition, the three-band UHB maintains its original character, while the
three-band ZRS band becomes the single-band LHB. Non-bonding and Zhang-Rice
triplet states from the three-band model are ignored and the single-band Mott
gap is identified with the three-band CT gap [81, 82]. We include the pump
pulse through the standard Peierls substitution and calculate the XAS spectra
as function of pump-probe time delay, as shown in Fig. 5. The calculations are
performed on a 12D Betts cluster with two holes in order to preserve $SU(2)$
symmetry. At equilibrium, the XAS signal features two distinct transitions
into LHB and UHB, qualitatively agreeing with the experimental O $K$ edge
spectra. The peak width is determined by two contributions, the intrinsic
core-hole lifetime (set to 0.15 eV) and an extrinsic energy broadening
introduced by the finite probe pulse (see SM Sec. LABEL:sec:singlebandSM). The
UHB peak intensity is about 70 % of the LHB peak intensity (as opposed to the
equally intense peaks in Fig. 2a), while the energy separation is $U-U_{c}\sim
1$ eV [83] with $U_{c}$ being the core-hole attractive interaction. When the
pump and probe pulses overlap, the peak distance shrinks by approximately40
meV, thus indicating a transient reduction of the Hubbard $U$ by 4 %. This
Mott gap closure is quantitatively consistent with the 3 % CT gap reduction
visible in Fig. 2a. The spectra also exhibit a general broadening and an
increase of in-gap spectral weight, likely due to the creation of particle-
hole excitations. The UHB shift is found to increase nonlinearly with the pump
electric field, in agreement with our experimental findings (see Appendix B).
Our dynamical single-band calculation captures the experimental UHB peak shift
at a quantitative level and corroborates the three-band picture of a pump-
renormalized onsite Coulomb repulsion.

##  VII Possible microscopic mechanisms

We now discuss the possible microscopic origin of the observed Hubbard $U$
reduction. A first scenario involves the dynamical enhancement of dielectric
screening [21, 24, 23]. The pump pulse promotes electrons from localized
levels into highly delocalized states, which increases the screening of the
local Coulomb interaction. In a multi-band model, screening arises from both
interband and intraband excitations, while second order ($\sim t^{2}/U$)
doublon-holon and doublon-doublon interactions are the dominant screening
modes in a single-band system.

Figure 5:  Transient XAS spectrum of the single-band Hubbard model in two
dimensions. (a) Time-dependent vector potential of the pump pulse used in the
time-dependent calculations. Colored dashed lines indicate selected time
delays for the snapshots in panel (c). (b) Full trXAS spectrum simulation as
function of time delay and photon energy (c) trXAS spectra at different pump-
probe delays before (black) and during the pump (red). The peak positions of
the UHB features are marked by dashed lines. The inset depicts the single-band
Hubbard model with $t,t^{\prime}$ hopping amplitudes, and onsite Coulomb
repulsion $U$.

The presence of second-order charge and spin fluctuations is compatible with
the scattering from antiferromagnetic spin fluctuations observed in ultrafast
reflectivity experiments [56]. Intriguingly, this mechanism leads to an
average dynamical reduction of the Hubbard $U$ on both $p$ and $d$ orbitals
[21, 24, 23], while the fixed ZRS energy hints to an unperturbed Coulomb
repulsion on the $p$ orbitals. Furthermore, a transient enhancement of
dielectric screening, being proportional to the intensity instead of the
electric field, would be accompanied by oscillations of the UHB position at
twice the frequency of the laser field (beyond our current time resolution)
[21, 24].

A second scenario involves instead a Floquet-type dressing of the effective
electronic interactions [26, 84]. The vector potential of the pump dynamically
alters the Hubbard $U$ through both the average doublon number and their
mutual interactions (of order $\sim t^{2}/U$). Notably, this scheme could lead
to either an increase or a decrease of the Hubbard $U$ depending on the pump
field strength. Based on the observation that the Hubbard $U$ renormalization
lifetime is longer than the pump pulse duration (see Fig. 3), we argue that
dynamical screening may be dominant over Floquet dressing. Nonetheless, a
final assignment will require further investigation, possibly with attosecond
x-ray pulses resolving coherent dynamics at the $U$ energy scale [85].

##  VIII Conclusion

In summary, our experiment provides clear spectroscopic evidence for a
transient and reversible renormalization of the Hubbard $U$ in a cuprate
superconductor. This result has broad implications for photoinduced phase
transitions and nonlinear phenomena in strongly correlated materials.

First, it provides a possible key to interpret the recent observation of
light-enhanced superconductivity in cuprates [86, 87, 88, 89, 90], which still
lacks a comprehensive microscopic understanding. A 140-meV reduction of the
Hubbard $U$ will close the CT gap by 70 meV, or 3 % of its equilibrium value.
Since the CT gap magnitude anticorrelates with the equilibrium superconducting
$T_{c}$ in both theory and experiments [91, 92], such reduction might be
favorable to the onset of out-of-equilbrium pairing. Moreover, in a spin-
fluctuation-mediated picture, the CT gap closure will be accompanied by a
renormalization of the pairing interaction and of the spin fluctuation
spectrum [33, 93]. A lower CT gap will also increase the exchange interaction
and, hence, the characteristic spin fluctuation energy [94]. These effects
might quantitatively explain the enhancement of superconducting correlations
in photoexcited La2-xBaxCuO4 [88, 89] and could be resolved in future time-
resolved RIXS experiments [95, 96].

A dynamical renormalization of the band structure has also been discussed as
an essential ingredient to boost high-harmonic generation (HHG) efficiency in
correlated materials, e.g. NiO [21, 10]. A lower Mott gap indeed reduces the
energy cost of Zener tunneling and multiphoton ionization processes, thus
promoting electron excitation and harmonic emission under strong light
irradiation conditions.

Finally, achieving on-demand manipulation of the Coulomb repulsion in
correlated materials may enable the observation of novel nonequilibrium states
of matter, such as fragile quantum spin liquids in frustrated magnets [97, 98,
99, 100, 101, 102, 103], and $\eta$-paired superconductivity in driven Mott
insulators [104, 105, 106, 107]. In the former, a dynamical Hubbard $U$ could
modify the exchange interactions and induce a transition from an
antiferromagnet to an entangled spin liquid state. In the latter, tuning the
Hubbard $U$ in a driven-dissipative Mott insulator will dynamically alter
doublon number [105] and energy spectrum of the driven Hamiltonian [108, 104],
thus paving the way to the onset of long-range staggered superconducting
correlations.

## Acknowledgements

We thank M. Buzzi, A. Cavalleri, M. P. M. Dean, T. P. Devereaux, M. Eckstein,
D. Golež, M. Li, B. Moritz, D. Nicoletti, A. H. Reid, A. Rubio, G. Sawatzky
and M. Sentef for insightful discussions. M. M. was supported by the William
F. Milton Fund at Harvard University. D. B. acknowledges support from the
Swiss National Science Foundation through project P400P2_194343. B. J. K. was
supported by IBS-R014-A2. H. J. acknowledges support by the National Research
Foundation grant funded by the Korea government (MSIT) (Grant No.
2019R1F1A1060295). S. L., A. A. H., P. A. acknowledge support from DOE grant
DE-FG02-06ER46285. P. A. acknowledges support from the Gordon and Betty Moore
Foundation EPiQS Initiative through grant no. GBMF9452. The trXAS experiments
were performed at the SSS-RSXS endstation (proposal number: 2020-2nd-SSS-006)
of the PAL-XFEL funded by the Korea government (MSIT). Work at Brookhaven
National Laboratory was supported by the U.S. DOE, Office of Science, Office
of Basic Energy Sciences, under Contract No. DESC0012704. This research used
resources of the National Energy Research Scientific Computing Center (NERSC),
a U.S. Department of Energy Office of Science User Facility operated under
Contract No. DE-AC02-05CH11231.

##  Appendix A Calculation of possible pump-induced heating effects on the
x-ray absorption spectrum

In this Appendix, we consider whether the transient reshaping of the
experimental x-ray absorption spectrum could be due to pump-induced sample
heating. We first determine the maximum electronic temperature increase
expected under the experimental pump excitation conditions and then perform an
exact diagonalization calculation of the theoretical XAS spectrum of a three-
band Hubbard model at such electronic temperature value. For the first step,
we follow the procedure described the Supplementary Material of Ref. [90] and
assume the following form for the specific heat of LBCO:

$C_{s}(T)=\gamma T+\beta T^{3},$ (1)

with $\gamma=2.5$ $\mathrm{mJ\cdot mol^{-1}\cdot K^{-2}}$ and $\beta=0.25$
$\mathrm{mJ\cdot mol^{-1}\cdot K^{-4}}$ being the electronic and lattice
specific heat coefficients, respectively. Assuming that the pump pulse varies
on a timescale shorter than the heating-cooling dynamics, the maximum
temperature increase $\Delta T=T_{f}-T_{i}$ is obtained from the absorbed
fluence as:

$Q_{\mathrm{abs}}=\int_{T_{i}}^{T_{f}}C_{s}(T)\mathrm{d}T.$ (2)

Given the reflection coefficient of the material ($R=0.15$) and the estimated
penetration depth at 800 nm ($l_{\mathrm{p}}\approx 370$ nm), we estimate the
absorbed energy according to the relation:

$Q_{\mathrm{abs}}=\frac{F\cdot
A\cdot(1-R)}{2l_{\mathrm{p}}}N_{\mathrm{A}}V_{\mathrm{uc}},$ (3)

where $F=10$ mJ/cm2 denotes the laser fluence, $V_{\mathrm{uc}}=3.787\times
3.787\times 13.23$ $\text{\AA}^{3}$ is the volume of the unit cell, and
$N_{\mathrm{A}}$ is the Avogadro constant. The factor 2 accounts for the
presence of two LBCO formula units per unit cell. Setting the initial
temperature equal to the base temperature (17 K), we obtain a final
temperature $T_{f}=120.3$ K. In order to get an estimate for the maximal
electronic temperature increase, $\Delta
T_{e,\mathrm{max}}=T_{e,\mathrm{max}}-T_{e,0}$, we make the following
approximation of the absorbed fluence:

$Q_{\mathrm{abs}}=\int^{T_{e,\mathrm{max}}}_{T_{e,0}}C_{e}(T_{e})\mathrm{d}T_{e}.$
(4)

Here, $C_{e}(T_{e})=\gamma T_{e}$ is the electronic (temperature-dependent)
specific heat capacity. Using the fluence value estimated from Eq. (3) and
setting the initial $T_{e,0}$ equal to the base temperature, we obtain a
maximum value for $T_{e,\mathrm{max}}\approx 3240$ K. However, the pump is
attenuated while traveling across the photoexcited volume, hence a more
refined approximation for the maximum electronic temperature increase is
[109]:

$T_{e,\mathrm{max}}=\frac{1}{l_{\mathrm{p}}}\int_{0}^{l_{\mathrm{p}}}\sqrt{T_{e,0}^{2}+\frac{2Q_{\mathrm{abs}}}{\gamma}e^{-z/l_{\mathrm{p}}}}\mathrm{d}z.$
(5)

According to Eq. 5, the maximum electronic temperature increase allowed within
our experimental conditions is $T_{e,\mathrm{max}}\approx 2561$ K. Having
determined the maximum electronic temperature, we examine the corresponding
changes in the XAS spectrum. We perform finite-temperature ED calculations
within the three-band Hubbard model for temperatures ranging between $T_{e}=0$
and $T_{e}=5300$ K. In Fig. 6, we illustrate the effect of electronic heating
on the shape of the XAS signal by comparing spectra at $T_{e}=0$ and
$T_{e,\mathrm{max}}=2535.5$ K. Intuitively, electronic heating creates
unoccupied states at lower energies and leads to a shift of the entire
absorption spectrum. Both the UHB and ZRS peaks are found to shift, with the
latter redshifting more than the former. Moreover, the UHB shift at the Cu
$L_{3}$ edge is almost half the one occurring at the O $K$ edge. This is at
odds with the experimental findings in Fig. 2, where the ZRS transition does
not shift within experimental accuracy and the UHB shift at the Cu $L_{3}$
edge is almost twice larger than the one at the O $K$ edge. The discrepancy
strongly suggests that the observed XAS spectral reshaping cannot be explained
in terms of a heating-only scenario.

##  Appendix B Field dependence of the upper Hubbard band shift

In this section, we compare experimental shifts of the UHB with the
theoretically expected fluence-dependent trXAS spectra of a single-band
Hubbard model. The trXAS calculations are performed on a 12D Betts cluster
with two holes (in order to preserve $SU(2)$ symmetry) and for variable peak
electric fields between $E_{0}=1.75$ MV/cm and $E_{0}=17.5$ MV/cm. The
spectra, reported in Fig. 7a, are calculated by using the same interaction
parameters considered in Fig. 5 and the theoretical peak positions in Fig. 7b
are defined as the energy of the peak maximum. While the UHB is found to shift
appreciably, the LHB shift is much less pronounced and discernible only for
the highest field amplitudes considered here. In particular, the peak shift is
nonlinear in the field amplitude, as opposed to the predicted Hubbard $U$
reduction by dynamical screening reported in Ref. [21]. Experimental data for
the UHB peak shift at the Cu $L_{3}$ edge are also shown in Fig. 7b for two
compositions, $x=0.095$ (from 2b) and $x=0.125$ (see SM). The experimental and
theoretical trends (when rescaled by the relative size of the Mott/CT gap)
agree on a quantitative level and outline a nonlinear field dependence of the
pump-induced Hubbard $U$ reduction.

Figure 6: Theoretical XAS spectra of the three-band Hubbard model at the O $K$
edge (panel a) and the Cu $L_{3}$ edge (panel b) at zero temperature (black)
and at $T=2535.5$ K (color). Figure 7:  (a) Theoretical trXAS spectra of the
single-band Hubbard model at equilibrium ($\Delta t=-75$ fs, black) and at
$\Delta t=0$ fs for selected pump electric fields. (b) Theoretical UHB peak
shift
$\Delta\omega^{\mathrm{theor.}}=\omega(\mathrm{0\>fs})-\omega(\mathrm{-75\>fs})$
(red diamonds) as function of the pump peak field $E_{0}$ and rescaled by the
ratio between CT gap and Mott-Hubbard gap. $\Delta E_{\mathrm{CT}}=1.77$ eV is
the experimental separation of the equilibrium ZRS and UHB bands at the O $K$
edge, while $\Delta E_{\mathrm{MH}}=1.03$ eV is the separation between LHB and
UHB in the single band Hubbard model. Experimental UHB peak shift values at
the Cu $L_{3}$ edge for the two dopings are indicated by black symbols.

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