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Samuel Beaulieu, Shuo Dong, Viktor Christiansson, Philipp Werner, Tommaso Pincelli, Jonas D. Ziegler, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Alexey Chernikov, Martin Wolf, Laurenz Rettig, Ralph Ernstorfer, Michael Schüler

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[Berry curvature signatures in chiroptical excitonic transitions](https://mdr.nims.go.jp/datasets/be1042dd-6428-4057-b8f1-e790ff9b16a6)

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Berry curvature signatures in chiroptical excitonic transitionsBeaulieu et al., Sci. Adv. 10, eadk3897 (2024)     28 June 2024S c i e n c e  A d v a n c e s  |  R e s e ar  c h  A r t i c l e1 of 9C O N D E N S E D  M AT T E R  P H Y S I C SBerry curvature signatures in chiroptical excitonic transitionsSamuel Beaulieu1*†, Shuo Dong2,3*†, Viktor Christiansson4, Philipp Werner4, Tommaso Pincelli3,5, Jonas D. Ziegler6, Takashi Taniguchi7, Kenji Watanabe8, Alexey Chernikov6, Martin Wolf3,  Laurenz Rettig3, Ralph Ernstorfer3,5, Michael Schüler9,10*The topology of the electronic band structure of solids can be described by its Berry curvature distribution across the Brillouin zone. We theoretically introduce and experimentally demonstrate a general methodology based on the measurement of energy- and momentum-resolved optical transition rates, allowing to reveal signatures of Berry curvature texture in reciprocal space. By performing time- and angle-resolved photoemission spectroscopy of atomically thin WSe2 using polarization-modulated excitations, we demonstrate that excitons become an asset in extracting the quantum geometrical properties of solids. We also investigate the resilience of our measurement protocol against ultrafast scattering processes following direct chiroptical transitions.INTRODUCTIONElectron transport and dynamics in periodically ordered solids are governed by intrinsic quantum mechanical properties, such as the electronic band structure and the interaction between electrons, phonons, and other quasiparticles. The quantum geometrical prop-erties of the Bloch wave function, manifesting as Berry curvature (property that reflects handedness of Bloch electrons), band topol-ogy, Fermi-liquid transport properties (1), current-noise character-istics (2), or the geometric origin of superfluidity in flat-band systems (3), play a fundamental role in all of these microscopic mechanisms. More generally, the quantum geometry of Bloch electrons is of capi-tal importance, as it provides key insights into the intricate interplay between quantum mechanics and materials’ electronic properties. Recently, the link between the quantum geometry and light-matter interaction has entered the stage, providing insights into physical mechanisms underlying peculiar optoelectronic responses of topo-logical materials (4–8).However, a momentum-resolved measurement of Bloch elec-trons’ quantum geometry still remains a grand challenge. A direct approach, exploiting the close link between the quantum geometry and light-matter interaction, has been introduced in the context of cold atoms, where paradigmatic model systems can directly be implemented. Because inter-band transition dipole matrix elements are equivalent to the Berry connection (9), the rate of transitions from occupied to unoccupied bands upon resonant monochromatic irradiation has been shown to be a direct measure of the underlying quantum geometry (10). In particular, the circular dichroism in the absorption is a fingerprint of a Chern insulating state, which has been demonstrated out-of-equilibrium (11), for fractional quantum Hall systems (12), and in optical lattices (13). Yet, applying this ap-proach to diagnose a material’s quantum geometry is not straight-forward. For systems with locally nonzero but globally vanishing Berry curvature, the total (momentum-integrated) optical oscillator strength does not provide any specific information on the quantum geometrical properties. However, it has been predicted that quan-tum geometric information can, in principle, be extracted from di-chroism in the momentum-resolved optical oscillator strength (4, 14, 15). Even if the connection between k-resolved optical oscillator strengths of inter-band transitions and Berry curvature is already established (4), an associated experimental measurement protocol for extracting local quantum geometric information of materials is still missing. Tackling this problem requires going beyond standard optical spectroscopic probes, as they lack momentum resolution.This is where angle-resolved photoemission spectroscopy (ARPES) (16, 17) has entered the stage. Signatures of local Berry curvature in solids can be extracted by using circularly polarized ionizing radia-tion (18–23). The basic principle of this approach is based on the close relation between Berry curvature and orbital angular momen-tum (OAM). Intuitively, it has been shown that OAM is linked with a self-rotation of the initial state, which is reflected in the dipole se-lection rules in the ARPES matrix elements—circular dichroism can emerge because of propensity rules in photoemission for electrons co- or counter-rotating with circularly polarized light. However, de-spite its feasibility, extracting information on the Berry curvature from photoemission transition dipole matrix elements is not straight-forward in practice due to the influence of the experimental geometry (24) and effects of complicated photoelectron final states (25, 26).The extension to the time domain using time-resolved ARPES (trARPES), a powerful technique to measure out-of-equilibrium band structures and excited states of crystalline solids, allows, in principle, to directly measure the momentum-resolved optical inter-band transition rate. For example, momentum-resolved linear dichroism in bilayer MoS2 in trARPES has been shown to reveal intralayer single-particle hopping (27), and band-resolved photo-currents have been observed in the topological insulator Bi2Se3 (28). 1Université de Bordeaux - CNRS - CEA, CELIA, UMR5107, F33405 Talence, France. 2Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China. 3Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany. 4Department of Phys-ics, University of Fribourg, 1700 Fribourg, Switzerland. 5Institut für Optik und Atomare Physik, Technische Universität Berlin, Strasse des 17 Juni 135, 10623 Berlin, Germany. 6Institute of Applied Physics and Würzburg-Dresden Cluster of Excel-lence ct.qmat, Technische Universität Dresden, 01062 Dresden, Germany. 7Re-search Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 8Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 9Laboratory for Materials Simulations, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland. 10Department of Physics, University of Fribourg, CH-1700 Fribourg, Switzerland.*Corresponding author. Email: samuel.​beaulieu@​u-bordeaux.​fr (S.B.); dongshuo@​iphy.​ac.​cn (S.D.); michael.​schueler@​psi.​ch (M.S.)†These authors contributed equally to this work.Copyright © 2024 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC). Downloaded from https://www.science.org at National Institute for Materials Science on June 28, 2024mailto:samuel.​beaulieu@​u-bordeaux.​frmailto:dongshuo@​iphy.​ac.​cnmailto:dongshuo@​iphy.​ac.​cnmailto:michael.​schueler@​psi.​chhttp://crossmark.crossref.org/dialog/?doi=10.1126%2Fsciadv.adk3897&domain=pdf&date_stamp=2024-06-28Beaulieu et al., Sci. Adv. 10, eadk3897 (2024)     28 June 2024S c i e n c e  A d v a n c e s  |  R e s e ar  c h  A r t i c l e2 of 9Extending this approach to chiral (circular) excitations allows to translate the cold-atom concept of the dichroism of the depletion rate into pump dichroism of the population of the unoccupied states. However, the rich ultrafast dynamics within the photoexcited material, leading to the redistribution of optically prepared excited states both in energy and momentum, can blur the direct relation-ship between measured photoemission intensities and momentum-resolved optical oscillator strength. In particular, electron-electron and electron-phonon scattering can smear out the initial energy-momentum distribution of pump-induced excited states on the femtosecond timescale. In addition, many-body excitations such as excitons or correlated in-gap states are often the dominating excita-tion channel, which, at first sight, seems to obscure the link between optical transition rates and quantum geometry.In this work, we, in turn, use the many-body excitations to ex-tract the quantum geometrical properties of solids. In particular, by exploiting the optical selection rules for chiral valley-excitons, we map out the Berry curvature texture of the prototypical atomically thin transition metal dichalcogenide (TMDC) WSe2. We show that the measurement of the momentum-resolved chiroptical oscillator strength, using optical pump polarization-modulation in trARPES, allows us to access the electronic wave function’s quantum geometry texture in materials.RESULTSMonolayer WSe2 (ML-WSe2) has broken inversion symmetry and strong spin-orbit coupling, leading to locked spin, orbital, and valley degrees of freedom (29). These symmetry considerations imply pe-culiar valley-selective optical selection rules, leading to strong circu-lar dichroism (30–32), a property that is at the heart of our approach. These material systems are also characterized by specific OAM and Berry curvature texture in reciprocal space. In addition, monolayer WSe2 has a direct bandgap at the two inequivalent K and K′ valleys. Because of the reduced screening resulting from its atomically thin nature, its excitons have large binding energies and dominate their optical responses, even at room temperature. As a result, strongly bound (hundreds of milli–electron volts) bright excitons composed of electrons and holes in the vicinity of K/K′ in the top valence and bottom conduction band are formed (known as A-excitons) upon resonant photoexcitation. These strongly bound excitons are stable against momentum scattering for relatively long timescales. In con-trast, typical band-to-band single-particle excitations at higher en-ergy are subject to electron-electron and electron-phonon scattering on the femtosecond timescale. The key concept of our approach is summarized in Fig. 1A: The Berry curvature of the valence and con-duction bands is tied to OAM. Therefore, excitons as bound states of electrons and holes become chiral excitations, whose population is determined by whether the chirality of the pump aligns with their intrinsic chirality. In turn, the exciton population (as measured from trARPES) is characteristic of the Berry curvature of the under-lying valence and conduction band. While the chirality of excitons has been discussed in terms of winding numbers (33) and from first principles (34), its use for the reconstruction of the Berry curvature texture is an unexplored territory.ExperimentsIn our trARPES setup, bright K/K′ excitons are resonantly prepared at room temperature by a resonant near-infrared (NIR) pump pulse [760 nm, ℏωIR = 1.63 eV, ∼45-fs full-width-at-half-maximum (FWHM) duration]. Electrons with momenta corresponding to first Brillouin zone (BZ; Fig. 2A) are ejected from the sample [ML-WSe2 on thin hBN flake on a slightly Nb-doped rutile TiO2 (100) substrate; for more details, see Materials and Methods] through the photoelec-tric effect induced by linearly p-polarized extreme ultraviolet (XUV) pulses (57 nm, ℏωpr = 21.7 eV and ∼20-fs FWHM duration). Mea-surements are performed at the pump-probe overlap (Δt = 0) to max-imize the signal emerging from bright excitons (Fig.  2B) while simultaneously minimizing the contribution of ultrafast scatter-ing processes following photoexcitation. We recorded two-color (NIR +  XUV) ARPES spectra while continuously rotating the quarter-wave plate (QWP) angle θ, leading to a pump polarization-modulation from left-hand circularly polarized (LCP) to right-hand circularly polarized (RCP), passing through linearly s-polarized (Fig. 2D, top). This continuous polarization-modulated photoemission ABFig. 1. Illustration of Berry curvature texture and exciton population along with the schematic of the experimental measurement protocol. (A) Single par-ticle top valence band and bottom conduction band of WSe2 close to the K valley. Because of the Berry curvature (represented by color shading), the electrons and holes created upon photoexcitation have intrinsic OAM. The optical transition rate is modulated by the chirality of excitons, which manifests in the population Pλ of the excitons, and of the pump pulse and serves as a probe of the Berry curvature. The experiment effectively measures the exciton density ∑λ Pλ∣Yλ(k)∣2, with Pλ de-noting the population and Yλ(k) standing for the envelope function of the exci-tons. (B) Sketch of the experimental setup, featuring a polarization-modulated infrared pump and linearly polarized extreme ultraviolet (XUV) probe pulses. Pho-toelectrons are collected by a time-of-flight momentum microscope detector.Downloaded from https://www.science.org at National Institute for Materials Science on June 28, 2024Beaulieu et al., Sci. Adv. 10, eadk3897 (2024)     28 June 2024S c i e n c e  A d v a n c e s  |  R e s e ar  c h  A r t i c l e3 of 9measurement protocol is analogous to a lock-in detection scheme. Using Fourier analysis, this measurement scheme allows us to isolate signals that are modulated at the helicity-swap frequency, efficiently rejecting all other frequency components coming from, e.g., linear dichroism, experimental geometry, or artifacts (imperfection of the waveplate, misalignments, etc.). The photoemission data are ac-quired using a time-of-flight momentum microscope, which allows to detect each photoelectron as a single event, as a function of NIR quarter-waveplate angle (θ), resulting in four-dimensional (4D) pho-toemission intensity data, I(kx, ky, E, θ). More information about the experimental setup can be found in Materials and Methods.A typical ARPES signal along K-Γ-​K′ high-symmetry direction (pump polarization-integrated) is shown in Fig. 2C. Bright excitons directly manifest themselves in Fig. 2C as strongly localized (in energy-momentum space) pump-induced signals (E − EVBM ∼ ℏωIR) at the BZ boundaries in the trARPES spectra (35, 36). In addition, photoemission intensity at Γ, which can be attributed to laser-assisted photoemission (LAPE) (37), and signatures of momentum-indirect dark excitons at the Σ valleys are also visible in Fig. 2C.In Fig. 2D, we show the modulation of the photoemission signal from bright excitons at K (K′), momentum and energy-integrated for the three equivalent valleys, as a function of the NIR QWP angle. Note that, before summing the signal emerging from the three equivalent K and K′ valleys, we made sure that the modulation in each equivalent valley followed the same trend. Signals originating from excitons located around both K and K′ valleys are strongly modulated, with a dominating oscillation component with a 180° period (helicity-swap period). The π-phase shift between the modu-lations of the K and K′ excitons indicates that these quasiparticles are created upon the absorption of light with opposite chirality, RCP and LCP, respectively. The π2-phase with an identical population of K and K′ valley excitons reflects equal excitation with a linearly polarized pump. These results already indicate that the phase of the exciton pop-ulation modulation encodes some information related to their intrin-sic valley pseudospin degree of freedom.From the full θ-dependent intensity, we can perform a Fourier analysis of the experimentally measures signals in Fig. 2D. Besides a non-oscillating background (encoded in the n = 0 component), the n = 1 Fourier component is dominant at both K and K′ (Fig. 2E), consistent with the modulation of the light chirality. The n = 2 Fourier coefficient is originating mainly from linear dichroism, i.e., the modulation between s- and p-components of pump pulses. Be-cause we recorded 4D ARPES data I(kx, ky, E, θ), we have access to the polarization-modulated (θ) photoemission signal for each en-ergy (E) and momenta (kx, ky) coordinates. We can thus perform the energy- and momentum-resolved Fourier analysis, i.e., compute the Fourier components for each voxelThis procedure yields complex quantities containing the full in-formation on the excitation with linearly polarized photons [encoded in I2(kx, ky, E)] and circular dichroism [encoded in I1(kx, ky, E)]. I0(kx, ky, E) and the imaginary part of I1(kx, ky, E) computed from the experimental data are shown in Fig. 3 (C and D), respectively.While the dominant components of Im[I1(kx, ky, E)] are strong signals at BZ corners with alternating signs between K and K′ valleys, suggesting qualitatively some similarity with the OAM and Berry curvature texture, the detailed understanding of the origin of these features requires some theoretical analysis, which is done in the following sections.Theory of exciton signaturesWe treat excitons in the electron-hole basis, expanding the many- body stateIn(kx , ky ,E) =12π ∫π−πdθ e−2inθI(kx , ky ,E, θ) (1)∣Ψexcpλ⟩ =�kαβY λαβ(p, k)c†k+pαckβ ∣Ψ0 ⟩ (2)ABC D EFig. 2. Optical polarization-modulated pump-probe photoemission in monolayer WSe2. (A) Sketch of the Brillouin zone (BZ) of WSe2 with the high-symmetry points. (B) Sketch of the overlapping pump and probe pulses. (C) Optical polarization-averaged trARPES signal along kx (K-Λ-Γ-Λ′-​K′). The intensity has been multiplied by 1000 for unoccupied states. (D) The ellipticity factor (Stokes parameter S3) of the pump pulse, which is controlled by the continuous rotation of quarter-wave-plate (QWP) angle θ (top), along with the ellipticity-resolved photoemission intensity of excited states around the K and K′ points. (E) The absolute value of the Fourier coefficients associ-ated with the polarization-modulated photoemission intensities from excitonic states in (D). The highlighted coefficient (n = 1) is associated with the helicity-swap fre-quency, i.e., captures the effects of circular polarization.Downloaded from https://www.science.org at National Institute for Materials Science on June 28, 2024Beaulieu et al., Sci. Adv. 10, eadk3897 (2024)     28 June 2024S c i e n c e  A d v a n c e s  |  R e s e ar  c h  A r t i c l e4 of 9Here, p denotes the center-of-mass momentum of the exciton (different states labeled by λ), while c†k+pα (ckβ) creates an electron (a hole) in the conduction (valence) band α (β) with corresponding momentum; ∣Ψ0〉 is the ground state. The envelope function Yαβ(p, k), in which its Fourier transform limited size can be experimentally measured (35, 38, 39), describes the localization of the excitons. For excitons in TMDCs, Yαβ(p, k) is strongly localized around k = K/K′ for bright excitons, while for the dark excitons, k is localized around K/K′ (Λ) for holes (electrons).In the linear-response regime, the population Pexc of the bright excitons is obtained from Fermi’s Golden rule (assuming atom-ic units)where Eλexc is the energy of the two A-excitons relative to the ground state, while eIR(θ) denotes the polarization of the NIR pump pulse. The dipole matrix element of the excitons is given by Mλ, while S(ω) stands for the Fourier transform of the envelope of the pump pulse [all other constant prefactors have been absorbed into S(ω)]. Com-bining the wave function (Eq. 2) and the exciton population (Eq. 3) with the trARPES formalism (40, 41) and assuming that the exciton population stays constant over the duration of the probe pulse, one findsHere, εβ(k) denotes the energy of the valence bands, ωpr denotes the photon energy of the probe pulse, and E denotes the energy of the final states, all entering a Gaussian function g(ω) whose width is determined by the duration of the probe pulse. We also include the dark excitons (p ≠ 0) in Eq. 4, as they get populated on a sub-100-fs timescale due to electron-phonon scattering (42). Neglecting photo-emission matrix elements, the experimental intensity I(kx, ky, E, θ) is obtained from Eq. 4 by summing over all exciton momenta p in the first BZ.Apart from enabling a direct comparison with the experimental results, our theory allows us to trace the dependence of the trARPES intensity on the QWP angle θ back to the exciton population. For the bright excitons, the Fourier components (Eq. 1) are thus deter-mined by In(kx , ky ,E) ∝ ∫ π−πdθ∕ (2π) e−2inθ ∣ eIR(θ) ⋅Mλ ∣2 . Working out the pump polarization eIR(θ) in the given experimental geome-try, the n = 1 Fourier component is given bywhere α denotes the angle of incidence. The combination of matrix elements in Eq. 5 is directly proportional to the circular dichroismHere, PLCPexc ( PRCPexc ) is the exciton population that would be gener-ated by a pump with LCP (RCP) polarization in normal incidence. The component In=2 is related to linear dichroism. In summary, sweeping over the QWP angle θ and Fourier transforming the ARPES signal provide direct access to energy- and momentum-resolved chiroptical (pump) circular dichroism in normal incidence, while the experimental geometry enters only as a prefactor.Impact of Berry curvature on excitonsTo trace the impact of the quantum geometry on the pump-induced exciton population, we analyze the dipole transition matrix element Mλ of the bright excitons in Eq. 3. The light-matter coupling is ex-pressed through the coupling of the pump electric field Ep(t) and the polarization operator P̂ : Ĥlm = −E(t) ⋅ P̂ . For inter-band transitions, the matrix elements of P̂ in the basis of Bloch states ∣ψkα〉 are given by Aαα′(k) = 〈ψkα∣r∣ψkα′〉. With the modern theory of polarization Pλexc(θ) = S2(ωIR − Eλexc) ∣ eIR(θ) ⋅Mλ ∣2 (3)Ipλ(kx , ky ,E, θ)∝ g(εβ(k−p)+Eλexc(p)+ωpr−E)×Pλexc(p, θ)∑β∣Y λαβ(p, k) ∣2(4)Im[In=1(kx , ky ,E)] ∝cosα2Im[(Mλx)∗Mλy] (5)Im[In=1(kx , ky ,E)]∝ −cosα4(PLCPexc−PRCPexc) (6)ABCDEFGHFig. 3. Berry curvature, spin texture, and Fourier components of the dichroic signal of excitons. (A) Berry curvature of monolayer WSe2 along indicated high-symmetry points. a.u., atomic units. (B) Spin expectation value texture 〈Sz〉 along the same high-symmetry points. The arrows illustrate the pump excitation and the rel-evant exciton scattering processes in the electron and hole picture. (C) Polarization-averaged photoemission intensity (equivalent to the n = 0 Fourier component), energy-integrated over the excited state’s region. (D) Imaginary part of the n = 1 Fourier component In=1(kx, ky, E) [energy-integrated as in (C)]. (E and F) Theoretical predic-tions (without any scattering) of the n = 0 and n = 1 Fourier components of the intensity corresponding to (C) and (D). (G and H) Theoretical predictions where the interval-ley scattering has been included.Downloaded from https://www.science.org at National Institute for Materials Science on June 28, 2024Beaulieu et al., Sci. Adv. 10, eadk3897 (2024)     28 June 2024S c i e n c e  A d v a n c e s  |  R e s e ar  c h  A r t i c l e5 of 9(9), we can identify the matrix elements Aαα′(k) with the Berry connections i〈ukα∣∇kukα′〉 (∣ukα〉 is the cell-periodic part of the Bloch wave function). Combining this with the exciton wave function (Eq. 2), the exciton transition matrix element becomes Mλ =∑kαβ Yλαβ(k)Aαβ(k) . Inserting into Fermi’s Golden rule (Eq. 3) and exploiting the localization in momentum space, we obtain the leading contribution to the circular dichroism PCDexc= PLCPexc− PRCPexcHere, VBZ is the area of the BZ. For TMDCs, the quantum geom-etry in the vicinity of the K/K′ valleys is determined by the top va-lence (β) and the bottom conduction (α) band (43). As a consequence, the Berry connections can be related to the Berry curvature, yieldingThe distinct Berry curvature texture in monolayer TMDCs (see Fig. 3A) thus determines the exciton population induced by circu-larly polarization light, giving rise to valley polarization. On the basis of this close connection, we can track the signatures of the quantum geometry: The dichroic exciton population and the exci-ton envelope function [which can be determined independently; (35)] directly correspond to the Berry curvature texture in the case of two relevant bands (for more bands, the correspondence stays intact qualitatively). In particular, the strongly localized nature of Y λαβ(k) (35) effectively limits the BZ integral in Eq. 8 to either the K or K′ valley. While absolute numbers can only be extracted using accurate theory input, the positive-negative texture of the dichroic exciton population is directly proportional to the texture of the Ber-ry curvature.We are now ready to analyze the Fourier transform of the mea-sured polarization-modulated photoemission intensities (Eq. 1), in an energy- and momentum-resolved fashion. In particular, the n = 1 component reflects the circular dichroism (Eq. 6), which should di-rectly reflect the Berry curvature texture (Eq. 8). The imaginary part Im[In=1(kx, ky, E)], energy-integrated over the spectral region where the excitons peaks occur (Fig. 3D), shows clear dichroic features at the K/K′ valleys. The alternating positive-negative pattern matches exactly the behavior of the in the conduction band Berry curvature (Fig. 3A).The Fourier component Im[In=1(kx, ky, E)] obtained with our theoretical calculations (Fig. 3F) is in very good agreement with the experiment. We obtain the identical positive-negative pattern that, within the theory, can exactly be traced back to the momentum de-pendence of the Berry curvature (see Eq. 8). The width of the peaks is governed by the exciton envelope function Y λαβ(k) , which is esti-mated as Δk ≈ 0.1 Å−1 FWHM. The peak width Δk determines the resolution of the Berry curvature, as Ωα(k) broadened by the mod-ules square of the exciton envelope function enters the dichroic signal (Eq. 8). Ultrafast scattering processes further reduce the resolution.Role of ultrafast scattering processesApart from the bright excitons manifesting in the trARPES signal at K/K′, the experimental data clearly feature additional excited states signals around the Λ/Λ′ valleys. Despite being clearly weaker than at the K/K′ valleys, these features are characterized by the same alternating sign pattern between adjacent Λ/Λ′ valleys. The origin of the population at the Λ/Λ′ valleys is well-known: It origi-nates from K-Λ intervalley scattering, leading to the formation of momentum-forbidden dark excitons, with electron and hole resid-ing at the Λ and K valleys, respectively. Because of their momentum-indirect nature, these excitons cannot be prepared by a direct (vertical) optical transition. Understanding the origin of the Λ/Λ′ valleys Im[In=1(kx, ky, E)] texture thus requires more sophisticated modeling, including ultrafast scattering processes following photo-excitation.Electron-phonon and electron-electron scattering limit the life-time of the bright excitons. Two mechanisms are dominant on tens of femtosecond timescale: (i) electrons scattering to the Λ valleys and (ii) electrons scattering from K to K′ (or K′ to K) (44, 45). The spin polarization and the Berry curvature are locked, and adjacent K/K′ valleys are characterized by opposite spin and Berry curva-ture textures (see Fig. 3B). These properties strongly influence the ultrafast exciton dynamics in 2D systems (35, 36, 42, 46). To com-pare experiment and theory directly, we solved a quantum-master equationHere, ρ(t) is the many-body density matrix in the space of the ground state (index ν = 0) and the bright (ν = 1 and 2, correspond-ing to p = 0) and dark (ν > 2, corresponding to p ≠ 0) excitons. We can thus identify Pλexc(p, t) = ρνν(t) for ν > 0. The scattering opera-tors Dn[ρ] (n labels the scattering channels) are constructed such that they incorporate (i) K↔K′ scattering (rate γn = T−1K→K� ), (ii) K→Λ scattering (rate γn = T−1K→Λ ), and (iii) general dephasing of the off-diagonal components (rate γn = T−1deph ). Note that the excitons localized at the K/K′ valleys are degenerate; nevertheless, the K↔K′ scattering can be phenomenologically captured by the quantum-master equation (Eq. 9), as it is equivalent to field-free pure dephas-ing. The underlying microscopic process can either be mediated by intervalley electron-hole exchange (44, 45) or be assisted by pho-nons (47). The diagonal components of the time-dependent exciton Hamiltonian are given by the exciton energies Eν = Eλexc(p) , while the off-diagonal elements Hν0(t) = −EIR(t) · Mλ (for ν denoting the bright excitons) describe the light-matter coupling. Substituting the exciton population obtained from solving the master equation (Eq.  9) (averaging over the duration of the probe pulse) into the trARPES expression (Eq. 4) yields an excellent match with the ex-perimental exciton (polarization-averaged) intensity (Fig. 3, C and G) for TK→K′ = 120 fs and TK→Λ = 80 fs. The only major difference is the intensity peak around the Γ point observed in the experiments, which is attributed to LAPE (37). Similarly, the agreement between experiment and theory is improved for the n = 1 Fourier component (Fig. 3, D and H).Notably, despite being significantly weaker, the dichroism encoded in the n = 1 Fourier component from the Λ valleys has the same sign as the dichroism at the closest K or K′ valley. While the Berry curva-ture texture in the Λ valleys is roughly similar to the corresponding K/K′ valley, it has a pronounced momentum dependence (weaker for smaller parallel momenta), which is not observed in the experi-ments nor in the theory. The dichroism is determined by the pump-induced population, i.e., by the inter-band vertical optical transitions. With LCP (RCP) polarization, the spin-polarized electrons forming PCDexc= −S2(ωIR − Eλexc) × ∫dkVBZIm[Axαβ(k)Ayβα(k)] ∣Y λαβ(k) ∣2 (7)PCDexc= −12S2(ωIR − Eλexc) ∫dkVBZΩα(k) ∣Yλαβ(k) ∣2 (8)ddt�(t) = −i[H(t), �(t)] +∑nγnDn[�(t)] (9)Downloaded from https://www.science.org at National Institute for Materials Science on June 28, 2024Beaulieu et al., Sci. Adv. 10, eadk3897 (2024)     28 June 2024S c i e n c e  A d v a n c e s  |  R e s e ar  c h  A r t i c l e6 of 9the bright excitons at K (K′) scatter to Λ valleys with the same spin, while spin-flip processes have a low probability (see Fig. 3B) (46). Therefore, the valley selectivity of the pump-induced bright exciton population is preserved by the K→Λ (K′→Λ) scattering process, due to the constraint on scattering pathways imposed by the spin texture. This “memory” effect is also present in our calculations (Fig. 3H), confirming this physical mechanism.In contrast, post-optical transition ultrafast intervalley scattering involving spin-flip processes would reduce the measured dichroism. In particular, K↔K′ (or vice-versa) scattering would give rise to electron populations in the minority valley, thus leading to a weaker polarization modulation of the valley-resolved population. While it is very challenging to control them experimentally, our theoretical approach allows us to investigate the role of scattering processes by tuning their characteristic times TK→K′ and TK→Λ (Fig. 4).We first investigate the situation where only K → K′ scattering chan-nel is activated (i.e., K → Λ is forbidden, TK→Λ = ∞; see Fig. 4A). In this case, the population of the excitons localized at K/K′ approach the same value rapidly, thus reducing the dichroic signal. Note that even for scattering times as fast as TK→K′ = 10 fs, which has been used for the simulation in Fig. 3A, the dichroism is not fully suppressed. Ultrafast scattering processes thus blur the direct correspondence between the momentum-resolved optical transition rate and the Berry curvature.In Fig. 4B, we investigate another extreme scenario with ultrafast K→Λ scattering (TK→Λ = 10 fs) and forbidden K → K′ channel (TK→K′ = ∞). In this case, the dichroic trARPES signal from the Λ valleys dominates. Similar to Fig. 4A, the quantum geometric tex-ture still leaves its imprint onto the dichroic Im[In=1(kx, ky)] signal, despite the rapid population transfer.DISCUSSIONOur joint experimental and theoretical work introduces a robust scheme to extract local quantum geometric properties of the electronic structure of materials using momentum-resolved many-body optical transition rates, here exemplified for a TMDC monolayer (WSe2). We exploit the direct relationship between chiroptical selection rules for bright excitons and their Berry curvature to design a viable mea-surement protocol to access its texture in reciprocal space. Using continuous pump polarization modulation in trARPES in an anal-ogous fashion to the lock-in detection scheme, we isolate signals modulated at the helicity-swap frequency. This measurement scheme allows for extracting a pure optical circular dichroism signal, efficiently removing all contamination coming from linear pump contribu-tions, experimental geometry, or other experimental artifacts. This Fourier analysis protocol is particularly interesting for ARPES mea-surements, which are performed at off-normal angles of incidence, leading to nontrivial experimental geometric effects competing with intrinsic signals of interest.Our theoretical model allowed us to investigate the resilience of our dichroic signal toward ultrafast scattering following optical transitions. Ultrafast reorganization of populations in energy and momentum space may blur the one-to-one correspondence be-tween momentum-resolved optical transition rate and Berry curva-ture. However, even in the scenario where the scattering time is shorter than the pulse duration, our calculations demonstrate that the quantum geometric texture still leaves its imprint onto the di-chroic Im[In=1(kx, ky)] signal. With sub-50-fs temporal resolution routinely available in trARPES setups, this measurement scheme can be applied to a wide range of material systems. To demonstrate the more general applicability of our measurement scheme, we have also simulated the dichroic response Im[In=1(kx, ky)] for the surface of the topological topological insulator Bi2Te3 under comparable conditions (see note S4). Similar to the case of fast intervalley scat-tering, the dichroic signal contains clear signatures of the Berry cur-vature even for short scattering times. Hence, our scheme can also detect strongly localized Berry curvatuer as long as (i) inter-band transitions can be induced in the relevant region, and (ii), in the case of excitonic transitions, the exciton wave function has signifcant overlap with the relevant region in momentum space. The surface sensitivity furthermore enables the extraction of chiroptical signa-tures for the emerging field of Berry curvature engineering at sur-faces (48–50).It is also very interesting to compare our approach to measure-ment protocols developed in other communities to access quantum geometric properties of various types of systems. Our measurement protocol can be compared to some work from cold atoms commu-nity (10). Tran and co-workers (10) predicted that the total absorp-tion rate of Bloch bands of a quantum lattice system can satisfy a quantization law imposed by global (momentum-integrated) topo-logical invariant, i.e., Chern number. Our work can be seen as a momentum-resolved analog of this approach. In TMDCs, the Berry curvature is alternating between adjacent valleys, such that its inte-gral over the BZ (Chern number) is vanishing. Thus, adding mo-mentum resolution to this approach is not an incremental step forward but is fundamental to studying the Berry curvature texture in materials. Moreover, the experimental measurement of quantum geometric properties of polaritonic systems is based on conceptually similar ideas to our methodology (51–53). It relies on the measure-ment of polarization-, momentum-, and energy-resolved photoluminescence. Because photoluminescence is a radiative process, only a limited fraction of excitons, namely, those whose in-plane wave vectors lie within the radiative light cone, can be measured (54). This is a fundamental limitation of this technique, which prevents it A BFig. 4. Impact of ultrafast scattering on the dichroism. (A) Time-dependent population of the exciton states upon pumping with LCP light in normal incidence, along with the envelope of the pump pulse and the probe pulse (top), and corre-sponding energy-integrated Fourier signal Im[In=1(kx, ky)] for TK→K′ = 10 fs, TK→Λ = ∞. (B) Same as (A), but for TK→K′ = ∞ and TK→Λ = 10 fs. The color scale is consistent with Fig. 3 (F and H).Downloaded from https://www.science.org at National Institute for Materials Science on June 28, 2024Beaulieu et al., Sci. Adv. 10, eadk3897 (2024)     28 June 2024S c i e n c e  A d v a n c e s  |  R e s e ar  c h  A r t i c l e7 of 9from being used to access the full Brillouin texture of Berry curvature. However, extending the ideas developed in polarization-, momentum-, and energy-resolved photoluminescence from polaritonic systems to polarization-, momentum-, and energy-resolved photoemission from material systems to access quantum geometrical properties is for sure a successful roadmap for the future.It is also interesting to mention that a simple extension of our scheme would be compatible with the recent proposal to experi-mentally measure the quantum metric (14), i.e., the real part of the quantum geometric tensor (Berry curvature is the imaginary part of the quantum geometric tensor). A light-matter interaction–based protocol to measure the quantum metric would be highly desirable, as this momentum-resolved quantity has been predicted to be of capital importance in the emergence of a broad range of physical phenomena, e.g., anomalous Hall effect (55), orbital magnetic sus-ceptibility (56), exciton Lamb shift (57), and superconductivity (58). Beyond single-particle physics, the interplay of quantum geometry and many-body excitations (excitons in our case) is an interesting avenue, where novel phenomena have been predicted (59).Moreover, being intrinsically compatible with ultrafast time-resolved measurements, extending our scheme to a three-pulses trARPES approach would allow measuring ultrafast light-induced modifica-tion of local quantum geometric properties of solids undergoing dynamics.MATERIALS AND METHODSExperimentsThe optical setup underlying our time- and angle-resolved photo-emission spectroscopy experiments is based on a homebuilt optical parametric chirped-pulse amplifier (OPCPA). The OPCPA is deliv-ering up to 30 μJ per pulse (15 W, 800 nm, 30 fs) at 500-kHz repeti-tion rate (60). In the probe arm, the second harmonic of the OPCPA output (400 nm) is used to drive high-order harmonic generation by tightly focusing (15-μm FWHM) laser pulses onto a thin and dense Argon gas jet. The nonlinear interaction between the laser pulses and the Argon atoms leads to the generation of a comb of odd har-monics of the driving laser, extending up to the 11th order. A single harmonic (seventh order, 21.7 eV) is isolated by reflection off a focus-ing multilayer XUV mirror and transmission through a 400-nm-thick Sn metallic filter. A photon flux of up to 2 × 1011 photons/s at the sample position is obtained (110-meV FWHM) (61). As a pump beam, we used s-polarized NIR pulses (760 nm, ℏωIR = 1.63 eV, ∼45-fs FWHM duration) to resonantly prepare bright A-excitons in ML-WSe2 sample. We use a QWP located before the pump and probe recombination chamber to control the polarization state of the pump pulse. The NIR pump and XUV probe pulses are noncollinear recombined and focused onto the sample lying in the photoemis-sion end-station. The photoemission data are acquired using a time-of-flight momentum microscope (METIS1000, SPECS GmbH), allowing to detect each photoelectron as a single event, as a function of NIR quarter-waveplate angle (θ). The resulting 4D photoemission intensity data have the coordinates I(kx, ky, E, θ).Concerning the preparation of the atomically thin TMDC sam-ple, first, thin hBN flakes are mechanically exfoliated on polydimeth-ylsiloxane (PDMS) and transferred onto a 0.5 wt % Nb-doped rutile TiO2 (100) substrate. Subsequently, monolayer WSe2 is exfoliated from bulk crystals (HQ Graphene) on PDMS and stamped on top of the previously transferred hBN flake. The sample is then annealed in a high vacuum at 180°C for at least 2 hours at each step. The hBN serves as an atomically smooth buffer layer to prevent the corru-gation of substrate surface roughness (62), and the slightly conduc-tive substrate TiO2 reduces the space charging effect from trARPES measurements (63).First-principle calculationsWe performed density functional theory calculations with the full-electron code FLEUR (64) within the Perdew-Burke-Ernzerhof approximation (65) to the exchange-correlation functional and sub-sequently constructed projective Wannier functions ϕj(r) using the Wannier90 code (66). We included the W-d and the Se-p orbitals. As the next step, we performed a one-shot G0W0 calculation (67) to obtain the self-energy Σα(k, ω), from which the quasiparticle ener-gies εα(k) are computed. The resulting quasiparticle Hamiltonian is expressed in the Wannier basis, yielding an 11-orbital model repro-ducing the G0W0 bands with high accuracy.As the next step, we performed constrained random-phase ap-proximation calculations (68) to obtain the Coulomb matrix ele-ments in the Wannier basis using the SPEX code (69). Because of reduction to the bands spanned by the Wannier functions, the Cou-lomb interaction attains a frequency dependence. However, as the energy scale of the screening effects is much bigger than the band-gap, we approximate the interaction as static (ω = 0). Furthermore, we only keep the density-density matrix elements due to the localized nature of the Wannier functions. Thus, we obtain the interaction Hamiltonianwhere n̂Rj = ĉ†RjĉRj is the density operator for the lattice site R and orbital j. The Coulomb interactions Ujj′(R − R′) are presented in fig. S2, along with full details of the calculations.Wannier modelWith the G0W0-Wannier Hamiltonian and the Coulomb interac-tions, we have a flexible and accurate model for the electronic struc-ture, including excitons. To obtain the exciton envelope function, we solved the Wannier equation (70, 71) for a selected pair of valence (β) and conduction (α) bandsThe effective interaction Wαβ(k, k′, p) is the inter-band screened interaction. As the precise dielectric environment of the substrate is hard to characterize, we used the effective continuum model from (72, 73). The model dielectric function ϵ(q) is parameterized by the dielectric constant at ω → ∞, ϵ∞, the substrate dielectric function ϵsub, and the effective thickness of the WSe2 layer deff. We fixed deff = 6.48 Å as in (73) while adjusting ϵ∞ and ϵsub to match the exciton binding energies observed in the experiments. The resulting absorp-tion spectrum (see fig. S3) is in good agreement with first-principle calculations for WSe2 on hBN substrate.Once the exciton envelope functions Y λαβ(pk) (we take the lowest states λ only) have been determined, optical matrix elements are computed as Mλexc= δp,0∑kαβ Yλαβ(p, k)Aαβ(k) . The Berry connec-tions Aαβ(k) are calculated from the Wannier Hamiltonian as in (74).Ĥ int =12∑R,R�∑jj�Ujj� (R − R�)n̂Rjn̂R�j� (10)[εα(k+p)−εβ(k)−Eλexc(p)]Y λαβ(p, k)−∑qWαβ(k+p, k+q, q)Y λαβ(p, q)=0(11)Downloaded from https://www.science.org at National Institute for Materials Science on June 28, 2024Beaulieu et al., Sci. Adv. 10, eadk3897 (2024)     28 June 2024S c i e n c e  A d v a n c e s  |  R e s e ar  c h  A r t i c l e8 of 9Time-dependent dynamicsTo simulate the population dynamics, we derived the quantum-master equation (Eq. 9) from the Lindblad formalism. Thus, the scattering operators are constructed aswhere {, } denotes the anti-commutator. The Lindblad operators are constructed as projectors as follows: (i) Ln =∣Ψexc0λ⟩ ⟨Ψexcpλ�∣ for the scattering process from K/K′ (corresponding to ν = 1,2) to the dark exciton states with corresponding momentum p (ν′ > 2), (ii) Ln =∣Ψexc0λ⟩ ⟨Ψexc0λ�∣ for the K↔K′ process with ν = 1,2, ν′ = 2,1, and (ii) Ln =∣Ψ0 ⟩ ⟨Ψ0 ∣ +∑pλ ∣Ψexcpλ⟩ ⟨Ψexcpλ∣ to capture the dephasing of off-diagonal components of the density matrix. We fix Tdeph = 40 fs for all calculations.Inserting the scattering operators (Eq. 12), the optical transition matrix elements Mλ, and the pump pulse with parameters consistent with the experiments into the master equation (Eq. 9) yields the time-dependent density matrix ρνν(t), from which the trARPES spectra presented in Fig. 3 and 4 are computed.Strictly speaking, the Lindblad equation can only be applied to the dissipation from an excited state to a low-energy state that is much slower than the pump-driven Rabi oscillations between the two states. 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Freudenstein, J. Reimann, D. Afanasiev, K. A. Kokh, O. E. Tereshchenko, J. Güdde, M. A. Sentef, U. Höfer, R. Huber, Build-up and dephasing of Floquet-Bloch bands on subcycle timescales. Nature 616, 696–701 (2023).Acknowledgments Funding: This work was funded by the Max Planck Society, the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant no. ERC452 2015-CoG-682843), H2020-FETOPEN-2018-2019-2020-01 (OPTOLogic—grant agreement no. 899794), the German Research Foundation (DFG) within the Emmy Noether program (grant no. RE 3977/1), the priority program SPP2244 (projects 443366970 and 443405595), the SFB/TRR 227 “Ultrafast Spin Dynamics” (projects A09 and B07), and the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter (ct.qmat) (EXC 2147, project ID 390858490). This research was also supported by the NCCR MARVEL, a National Centre of Competence in Research, funded by the Swiss National Science Foundation (grant number 205602). K.W. and T.T. acknowledge support from the JSPS KAKENHI (grant numbers 20H00354, 21H05233, and 23H02052) and World Premier International Research Center Initiative (WPI), MEXT, Japan. T.P. acknowledges funding from the Alexander von Humboldt Foundation. M.S. acknowledges support from SNSF Ambizione grant no. PZ00P2 193527. P.W. acknowledges support from ERC Consolidator grant no. 724103. S.B. acknowledges support from ERC Starting Grant ERC-2022-STG no. 101076639. Funded by the European Union. Views and opinions expressed are, however, those of the authors only and do not necessarily reflect those of the European Union. Neither the European Union nor the granting authority can be held responsible for them. Author contributions: S.B. and M.S. conceived the idea. S.D. performed the experiments. S.B. and M.S. analyzed the experimental data. R.E., L.R., and M.W. were responsible for developing the experimental infrastructures. S.B., S.D., and T.P. participated in maintaining and running the experimental apparatus. J.D.Z. and A.C. prepared the ML sample with the hBN substrate provided by T.T. and K.W. M.S. developed the theory with inputs from V.C. and P.W. S.B. and M.S. wrote the first draft of the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Raw trARPES data along with Python scripts for post-processing and visualization as well as the Python scripts producing the figures in the main text can be found at Materials Cloud Archive 2023.128 (2023): https://doi.org/10.24435/materialscloud:zq-tj DOI: 10.24435/materialscloud:zq-tj.Submitted 21 August 2023 Accepted 24 May 2024 Published 28 June 2024 10.1126/sciadv.adk3897Downloaded from https://www.science.org at National Institute for Materials Science on June 28, 2024https://doi.org/10.24435/materialscloud:zq-tjhttps://doi.org/10.24435/materialscloud:zq-tjhttp://dx.doi.org/10.24435/materialscloud:zq-tj Berry curvature signatures in chiroptical excitonic transitions INTRODUCTION RESULTS Experiments Theory of exciton signatures Impact of Berry curvature on excitons Role of ultrafast scattering processes DISCUSSION MATERIALS AND METHODS Experiments First-principle calculations Wannier model Time-dependent dynamics Supplementary Materials This PDF file includes: REFERENCES AND NOTES Acknowledgments