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B. Evrard, H. S. Adlong, A. A. Ghita, T. Uto, L. Ciorciaro, [K. Watanabe](https://orcid.org/0000-0003-3701-8119), [T. Taniguchi](https://orcid.org/0000-0002-1467-3105), M. Kroner, A. İmamoğlu

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[ac Stark Spectroscopy of Interactions between Moiré Excitons and Polarons](https://mdr.nims.go.jp/datasets/6a7c4599-b190-4f39-afd2-0a346886aa2d)

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ac Stark Spectroscopy of Interactions between Moiré Excitons and Polaronsac Stark Spectroscopy of Interactions between Moiré Excitons and PolaronsB. Evrard ,1 H. S. Adlong ,1,2 A. A. Ghita ,1 T. Uto ,1 L. Ciorciaro ,1 K. Watanabe ,3T. Taniguchi ,3 M. Kroner ,1 and A. İmamoğlu 11Institute for Quantum Electronics, ETH Zürich, Zürich, Switzerland2Institute for Theoretical Physics, ETH Zürich, Zürich, Switzerland3Research Center for Electronic and Optical Materials, NIMS, 1-1 Namiki, Tsukuba 305-0044, Japan(Received 26 February 2024; revised 4 November 2024; accepted 4 March 2025; published 1 April 2025)We use nonlinear pump-probe spectroscopy to study optical excitations in a charge-tunable MoSe2=WS2moiré heterostructure. An intense red-detuned laser pulse creates a photonic dressing of the material byintroducing a large virtual population of excitons or exciton polarons in a deep moiré potential. Bymeasuring the resulting ac Stark effect with a weak resonant laser pulse, we gain access to the nature andmutual interactions of the elementary optical excitations. At charge neutrality, our measurements reveal thatdifferent exciton resonances, associated with confinement of their center-of-mass motion in the moirépotential, have a significant spatial overlap. The resulting short-range interactions manifest themselves as adensity-dependent blueshift for same-valley excitons and bound biexciton states for opposite-valleyexcitons. The attractive polaron resonance that appears upon injection of electrons into the heterostructureshows a contrasting behavior: Here, we observe an electron-density-independent light shift and a clearpump-power-dependent saturation. These features are equivalent to that of an ensemble of independenttwo-level emitters and indicate a breakdown of the Fermi-polaron picture for optical excitations of electronssubject to a strong moiré potential. Our work establishes an experimental approach to elucidate theelementary optical excitations of semiconductor moiré heterostructures, providing a solid ground for thespectroscopy of correlated electronic and excitonic states in such materials.DOI: 10.1103/PhysRevX.15.021002 Subject Areas: Condensed Matter Physics,Semiconductor Physics,Strongly Correlated MaterialsI. INTRODUCTIONSemiconductor moiré materials have emerged as a richplayground for the exploration of strongly correlatedelectrons [1–3]. In twisted bilayers of transition metaldichalcogenides (TMDs), linear spectroscopy has enabledthe observation of a wealth of many-body states, rangingfrom correlated Mott-Wigner states [4–6], through kineticmagnetism [7] to fractional Chern insulators [8,9]. Indeed,attractive and repulsive exciton polarons, resulting from thedynamical dressing of excitons by itinerant charges [10–15], provide built-in sensors for both the charge andmagnetic order of electrons in TMDs [16,17]. Goingbeyond a mere diagnostic purpose, optical excitations ofmoiré materials have also been proposed as building blocksof correlated bosonic systems [18–21]. Consequently, acomplete characterization of moiré exciton and exciton-polaron resonances is essential for correctly interpretingspectroscopic signatures of a broad set of correlated statesin semiconductor moiré materials [22–27].Here, we use nonlinear ac Stark spectroscopy to measurethe interactions and possible bound states arising betweendifferent elementary optical excitations of a MoSe2=WS2heterostructure. Our approach relies on an intense red-detuned pump beam to generate a significant density ofvirtual excitations and on a weak broadband probe pulse tomonitor the subsequent modification of the optical spectrum.By detuning the pump away from any resonances, wealleviate real-absorption-induced modification of the elec-tronic state and look at the coherent scattering response of thesystem. In TMD monolayers, repulsive Coulomb exchangeinteractions between itinerant excitations generated by thepump and the probe result in a blueshift when both lasers arecocircularly polarized [28–35]. For cross-circular polariza-tion, the sign of the light shift depends on the pumpfrequency: This is a manifestation of an underlyingFeshbach resonance, occurring when the pump detuningfrom the exciton resonance equals the biexciton bindingenergy [36–40]. Our prior measurements also revealed astriking electron density dependence of the light shift whichallowed us to determine a dramatic enhancement ofPublished by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.PHYSICAL REVIEW X 15, 021002 (2025)2160-3308=25=15(2)=021002(17) 021002-1 Published by the American Physical Societyhttps://orcid.org/0000-0003-1949-0241https://orcid.org/0000-0002-7782-3975https://orcid.org/0000-0003-0156-7685https://orcid.org/0009-0001-6987-2507https://orcid.org/0000-0001-7750-6411https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0001-8154-5990https://orcid.org/0000-0002-0641-1631https://ror.org/05a28rw58https://ror.org/05a28rw58https://ror.org/026v1ze26https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevX.15.021002&domain=pdf&date_stamp=2025-04-01https://doi.org/10.1103/PhysRevX.15.021002https://doi.org/10.1103/PhysRevX.15.021002https://doi.org/10.1103/PhysRevX.15.021002https://doi.org/10.1103/PhysRevX.15.021002https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/interactions between attractive polarons (APs) [34]. In thiswork, we employ ac Stark spectroscopy on a TMD bilayerfeaturing a deep moiré potential to reveal three new features:(i) At charge neutrality where two bright moiré excitonsdominate the spectrum, we observe a blueshift of a moiréexcitonmode induced by the interactionwith the othermode.This interspecies ac Stark effect, with no equivalent in amonolayer TMD, reveals the extent of spatial overlapbetween the two orthonormal modes. (ii) We identify moirébiexciton Feshbach resonances associated with bound statesof both the same and different moiré exciton modes [41]. Toexplain the nature of these biexciton states, we develop atheoretical model for the scattering of excitons in a moirépotential and find good qualitative agreement with theexperiment. (iii)We find that the electrondensity dependenceof the ac Stark shift, as well as the pump-power-dependentsaturation, of theAP resonance is qualitatively different fromthat of a monolayer. Our findings correspond to those of anensemble of independent two-level emitters, indicating abreakdown of the Fermi-polaron picture for optical excita-tions of electrons subject to a strong moiré potential.Our measurements are carried out in a ≃0° stackedMoSe2=WS2 heterostructure, exhibiting a type I bandalignment where the lowest (highest) energy moiré con-duction (valence) band resides in MoSe2 [Figs. 1(a) and 1(b)]. The electron-density-dependent reflection contrastexhibits four bright resonances, which we identify asMX1, MX2, AP, and MX3 to be consistent with the notationused in an earlier publication [7] [Fig. 1(c)]. A strong pumplaser with a finite red detuning from a given resonancegenerates a large virtual population of the correspondingmoiré exciton modes that exists only during theapproximately 0.2 ps duration of the pump pulse. Weestimate a maximum virtual exciton population in MX1of approximately 1012 cm−2 for a pump detuning ofδ1 ≈ 20 meV (for more details, see Appendix E). A weak,broadband probe pulse then measures the energy shift of allexcitonic species concurrently [Fig. 1(d)]. Throughout thiswork, we set the pump laser detuning from the excitonicresonances to be much smaller than the exciton bindingenergy: In this limit, the dominant contribution to the lightshift of the resonances originates from exciton-excitoninteractions [28–34]. We determine both intra- and inter-species interaction strengths between the moiré exciton orpolaron modes as a function of the electron density bymeasuring the light shift in this small detuning limit.II. INTERACTIONS BETWEENMOIRÉ EXCITONSWe begin by performing nonlinear ac Stark spectros-copy at charge neutrality in order to understand how themoiré potential modifies exciton-exciton interactions andbiexcitons. Since the exciton binding energy (Bohr radius)is much larger (smaller) than all other relevant energy(length) scales, the role of the moiré potential is to inducea periodic potential for the center-of-mass motion of the1s exciton [42]. When the moiré potential is weak, wewould observe umklapp resonances, blueshifted from thek ¼ 0 1s exciton mode by ≃ðℏkMÞ2=ð2mexÞ, where kM isthe reciprocal moiré lattice wave vector and mex is theexciton effective mass. In the opposite limit, we expectresonances corresponding to exciton modes localizedaround the local minima of the moiré potential. Incharge-neutral MoSe2=WS2 heterostructures, two brightmoiré exciton resonances MX1 and MX2 are observed inthe normalized reflection spectrum ΔR ¼ ðR − RbgÞ=Rbg,where R and Rbg denote the reflection spectrum from thesample and background, respectively [see Fig. 1(c)]. Thesplitting between these two peaks of approximately41 meV is significantly larger than the expected umklappFIG. 1. Overview of the sample and experiment. (a) Schematic of the device. The TMDs MoSe2 and WS2 form the moiré material thatwe investigate. It is encapsulated in two approximately 35-nm-thick h-BN flakes and dual gated with graphite electrodes to enable anindependent control of the chemical potential and electric field. The latter is kept close to zero, such that only MoSe2 is doped at lowdensities, given the band alignment (b). The evolution of the reflection spectrum of MoSe2 as a function of the electronic density(c) reveals several moiré exciton and polaron resonances, which we investigate. We rely on a pump-probe scheme (d) where an intensered-detuned laser generates a virtual population of moiré excitations. The interaction between this background and a test excitationgenerated with a probe laser are then measured.B. EVRARD et al. PHYS. REV. X 15, 021002 (2025)021002-2splitting ðℏkMÞ2=ð2mexÞ ≈ 27 meV for this system, whichsuggests that the moiré potential is strong enough tosignificantly alter the exciton spatial wave function withinthe moiré unit cell.To reveal the elementary features of moiré excitons, suchas their spatial extent and overlap, we investigate theirmutual interactions using ac Stark spectroscopy. By meas-uring the light shift for co- and cross-circularly-polarizedpump and probe beams, we probe interactions betweenmoiré excitons in the same (cocircular) or opposite valley(cross-circular).A. Cocircularly polarized lightBeginning with the scenario of cocircular polarization,we consider the evolution of ΔR as a function of the delayτ ¼ tprobe − tpump, between the pump and probe pulse, asshown in Fig. 2(b) (here, the pump pulse is red-detunedfrom the MX1 resonance by δ1 ¼ 20 meV). The dominantcoupling mechanisms between two same-valley excitonsare electron and hole exchange interactions [43,44], whichlead to short-range repulsion. Correspondingly, we observea clear blueshift of the brightest exciton MX1 forjτj ⪅ 0.2 ps, that is, when the two pulses overlap in time(for more detail on the experimental setup, seeAppendix A). To better assess the pump-induced modifi-cations of weaker resonances, we use the differentialreflection spectrum ΔRðtÞ − ΔRref , where ΔRref is areference spectrum obtained when the probe pulse hitsthe sample significantly before the pump (τ ⪅ −2.5 ps):Figure 2(c) shows that a smaller blueshift of MX2 isdiscernible in ΔRðtÞ − ΔRref . In a mean-field picture(detailed in Appendix H 4), the light shift Δi of MXi(i ¼ 1, 2) can be expressed asΔ1 ¼ 2u1;1n1 þ 2u1;2n2 þ 4k1ffiffiffiffiffiffiffiffiffiffin1n2p; ð1aÞΔ2 ¼ 2u2;2n2 þ 2u1;2n1 þ 4k2ffiffiffiffiffiffiffiffiffiffin1n2p; ð1bÞwhere ni is the density of MXi excitons and we introducefour different interaction terms: ui;j correspond to exciton-exciton scattering conserving the moiré miniband pop-ulations, i.e., the processes MXi þMXj⇋MXi þMXj,while ki correspond to processes where an exciton isscattered into a different moiré miniband, i.e., MX1þMX2⇋MXi þMXi. The density scales as ni ∝ 1=δ2i withthe detuning δi of the pump from the MXi resonance.Therefore, by tuning the pump laser frequency, one canchange the density imbalance between the two excitonsand, in principle, deconvolve the contribution of eachscattering processes to the light shift. In practice, weobserve signatures of an incoherent response for a blue-detuned pump, so we focus exclusively on red detunings.We consequently always have δ1 < δ2 and, thus, n1 > n2[see Fig. 2(a)]. This imbalance is further amplified by theoscillator strength difference between MX1 and MX2.Figure 2(d) shows the light shifts Δ1;2 as a function of δ1in a range where n1 ≳ 10n2. As a result, the light shift ofMX1 is dominated by MX1-MX1 interactions. From a fit,we obtain the interaction strength u1;1, which we find to belarger by a factor u1;1=uex ≈ 1.6� 0.2 than the moiré-freeexciton-exciton interaction strength uex, measured on amonolayer MoSe2 region of the same device. We tenta-tively attribute this enhancement to the reduced spatialextent of the MX1 center-of-mass wave function within themoiré unit cell, which, in turn, increases the overlapbetween MX1 excitons for a given average density.(a) (b) (c) (d)FIG. 2. Light shift of moiré excitons for cocircularly polarized pump and probe lasers. (a) Schematic of the moiré exciton energy levelat charge neutrality. (b) Reflection spectrum as a function of the pump-probe delay, showing a clear blueshift of the lowest and brightestresonance MX1 at zero time delay. The differential spectrum (c), obtained by subtracting from (b) a reference spectrum (acquired atτ < −2.5 ps), reveals a blueshift of the upper and darker resonance MX2 as well. (d) The dependence of the blueshift of MX1 (bluecircles) on the pump detuning scales as 1=δ21 ∝ n1 (blue line) and stems from MX1-MX1 interactions. The shift of MX2 (red squares) iswell fitted either by a ∝ 1=δ21 dependence (dashed red line) or byffiffiffiffiffiffiffiffiffiffin1n2p ∝ 1=ðδ1δ2Þ (dotted red line). While we are not able to preciselydeconvolve the role of these two potential contributions, both imply significant MX1-MX2 interaction and, hence, spatial overlapbetween these two bright moiré excitons.AC STARK SPECTROSCOPY OF INTERACTIONS BETWEEN … PHYS. REV. X 15, 021002 (2025)021002-3Remarkably and despite the large imbalance n2 ≪ n1,we observe a substantial light shift of MX2, stronglyincreasing as the pump wavelength approaches the MX1resonance. This shift Δ2 is not well reproduced by a 1=δ22dependence and attests to substantial interspecies inter-actions. A fit to our data does not enable us to preciselydisentangle the respective contributions of the MX2þMX1⇋MX2þMX1 and MX2þMX1⇋2MX1 processes.Nevertheless, we estimate that 0.3≲ u1;2=u1;1 ≲ 0.8, whilewe are not able to reliably estimate the ki parameters (seeAppendix C for more details on the fitting procedure).These observations suggest a significant spatial overlapof MX1 and MX2 excitons and invalidates a simplisticpicture of moiré exciton modes that are tightly confinedaround different high-symmetry points of the moirépotential. Moreover, our experiments are in reasonablequalitative agreement with the findings of moiré excitonwave functions obtained for the same structure using thecontinuum model [45]. In particular, we find that the latterpredicts u1;1=uex ¼ 2.2 and u1;2=u1;1 ¼ 0.6 (see Appendix Hfor details).We point out that Eqs. (1a) and (1b) can be understood asarising from first-order perturbation theory. This approachis valid in the limit of sufficiently small pump power andlarge pump detuning. We find an empirical confirmationthat we are indeed working in that regime by observing alinear dependence of the light shifts Δi with the pumpintensity (Appendix B). Nevertheless, we can envisioninteresting higher-order effects arising from the mixing ofthe excitonic states. In particular, mixing of optically brightand dark excitons could be observed as the emergence ofnew resonances in the reflection spectrum.B. Cross-circularly-polarized lightSince electron and hole exchange interactions are sup-pressed for excitons generated in opposite valleys, one maynaively assume that the cross-circularly-polarized scenarioshould yield a significantly smaller light shift. However, thebare interaction of opposite valley excitons is attractive andsupports a bound biexciton state, which has significantimplications.In order to illustrate how the biexciton state affects theexciton-exciton interactions, we briefly review the simplerscenario in which there is no moiré potential (such as in thecase of monolayer TMDs). In this case, it is known that thepresence of the biexciton resonance leads to an additionalcontribution to the light shift, scaling linearly with theinverse of the two-photon detuning δ−1b ¼ ð−Eb þ δexÞ−1,where δex is the detuning of the pump from the excitonresonance, Eb is the biexciton binding energy, and theprobe is assumed resonant with the exciton. A hallmark ofthe biexciton is the ac Stark effect, where the light shiftchanges sign for detunings in the vicinity of Eb [36–40].The two-photon resonance condition δb ¼ 0 can beconsidered as a biexciton Feshbach resonance where theeffective interactions between the pump- and probe-gen-erated excitons change from being attractive (δb > 0) torepulsive (δb < 0).We now extend these concepts to study the nature ofbiexcitons in moiré materials by measuring the ac Starkeffect of MX1 and MX2 excitons under cross-circularly-polarized pump-probe lasers as a function of δ1. We firstobserve that the sign of the ac Stark shift changes from ablueshift at small detunings [δ1 ≈ 22 meV; Figs. 3(a)and 3(b)] to a redshift at large detunings [δ1 ≈ 45 meV;Figs. 3(c) and 3(d)] for both moiré excitons. The fulldetuning dependence in Fig. 3(e) reveals that the signchange occurs at different detunings for MX1 and MX2and is associated with two different zero-quasimomentumbiexcitons. By contrast, a monolayer hosts only a singlezero-momentum exciton. From a heuristic fit with thefunction Δi ¼ a tan−1ðδ1 − Eb;i=bÞ þ c (with i ¼ 1, 2 anda, b, c fitting parameters), we extract the binding energiesEb;1 and Eb;2 of the biexcitonic states determining the acStark shifts of MX1 and MX2, respectively. From the MX1light shift, we find Eb;1 ≈ 36 meV, which suggests that theground state moiré biexciton (MX1-MX1) is more stronglybound here compared to a monolayer MoSe2, where Ebwas measured to be approximately 20 meV [36,38] orapproximately 29 meV [34,46] (these variations couldstem from different device architectures and dielectricenvironments). The light shift of MX2 is related to a secondbiexciton state, with a mixed MX1=MX2 character and abinding energy Eb;2 ≈ 28 meVmeasured with respect to anunbound MX1 and MX2 exciton. Importantly, Eb;2 issmaller than the energy splitting between MX1 and MX2(approximately 40 meV), and this second biexciton ishigher in energy than two unbound MX1 excitons. Thisshould be contrasted with the monolayer scenario, wherethe bound state by definition has lower energy than twounbound zero-momentum excitons.We further investigate the biexciton states of moiréexcitons by modeling their scattering interactions in aperiodic potential (see details in Appendix H). Our modelincorporates short-range interactions calibrated to matchthe biexciton binding energy: We use the lower estimateEb ≈ 20 meV [36,38] to account for screening effects fromthe proximal WS2. Using parameters derived from densityfunctional theory (Appendix H), we identify the brightexcitonic resonances observed in experiment, which weshow in Fig. 3(g). Here, we also show the bound biexcitons,which emerge due to spatial overlap of exciton wavefunctions within the moiré lattice.The biexciton spectrum reveals that higher-energy biex-citons form in gaps within the two-exciton continuum,consistent with experimental observations. Specifically, theboundMX1-MX2 biexciton is enabled by significant spatialoverlap between the exciton wave functions, with MX2B. EVRARD et al. PHYS. REV. X 15, 021002 (2025)021002-4being notably more delocalized. The pump laser energiesfor exciting MX1-MX1 and MX1-MX2 biexcitons arehighlighted in Fig. 3(g).To connect theory with experiment, we calculate theexciton energy shifts induced by the pump laser (seeAppendix H2 and H3). Here, we estimate the light-mattercoupling strength such that the induced exciton density ison the order of 1011 cm−2 (see Appendix H3). The shiftsare shown in Fig. 3(f) and agree qualitatively withmeasurements. The model suggests that only two biexci-tons significantly influence the MX1 and MX2 shifts for themeasured experimental detunings; the other biexcitonshave negligible overlap with these two excitonic resonan-ces. These results highlight how the emergence of inter-species biexcitons in moiré show clear and direct signaturesin ac Stark spectroscopy.III. ATTRACTIVE POLARON LIGHT SHIFTWhen the MoSe2=WS2 heterostructure is electrondoped, we observe an AP resonance that is redshiftedfrom the MX1 by approximately 35 meV. The correspond-ing trion binding energy is about 50% larger than thatobserved in monolayer MoSe2, suggesting that the electronWannier orbitals, as well as the optically generated trions,are strongly localized around the minimum of the moirépotential. Furthermore, the AP oscillator strength followsthe number of singly occupied moiré sites, increasinglinearly until ν ¼ 1 and then decreasing linearly untilν ¼ 2 [Fig. 4(a)]. This observation shows that the APformation is hindered at doubly occupied sites due to theexistence of local electron singlets that Pauli block theformation of (localized) trions. However, linear spectros-copy cannot be used to identify features that distinguish(a)(e) (f)(b) (c) (d) (g)FIG. 3. Coupling to biexcitonic states. (a) Reflection spectrum as a function of the pump-probe delay when the pump is red-detuned by22 meV from MX1 and the pump and probe laser are cross-circularly polarized. (b) The differential spectrum, obtained by subtractingfrom (a) a reference spectrum (acquired at t < −2.5 ps), reveals a blueshift of both resonances MX1 and MX2. (c)(,d) A similarmeasurement carried out for a detuning of 45 meV from MX1 shows instead a redshift of both resonances. (e) The dependence of thelight shifts of MX1 (blue dots) and MX2 (red dots) displays a sign change for a pump detuning δ1 ≈ 36 meV and δ1 ≈ 28 meV,respectively. (f) The corresponding theory simulation for the light shifts of MX1 (blue) and MX2 (red) based on DFT moiré parameters.(g) The calculated spectrum which is shown in three different energy sectors with zero (defined as zero energy), one, and two excitons.In the one-exciton sector, there are the optically bright MX1 and MX2 excitons. In the two-exciton sector (focusing on zero center-of-mass quasimomentum), there exists a band of MX1-MX1 excitons (light blue) as well as other bands (gray) and bound biexcitons (greenlines). We find that the sign change in the light shift of MX1 and MX2 in (f) can be attributed to the blue and red arrows (indicatingexciton-biexciton transitions), respectively.AC STARK SPECTROSCOPY OF INTERACTIONS BETWEEN … PHYS. REV. X 15, 021002 (2025)021002-5APs in deep moiré potentials from their counterparts inweak moiré potentials or in monolayer TMDs.Figure 4(b) shows the pump-probe measurement as afunction of τ for δAP ¼ 12 meV, clearly showing a blue-shift of the AP resonance with negligible alteration of theresonance for τ ≥ 0.2 ps. While this measurement isreminiscent of the result for MX1, Fig. 4(c) shows astriking difference in the δAP dependence: The AP lightshift is better fit using a 1=δAP dependence, typical of the acStark shift observed for an ensemble of noninteracting two-level emitters. We tentatively explain this observation byarguing that the AP resonance can be considered asstemming primarily from a collective excitation of trionslocalized at the M-M sites of the moiré lattice. In theabsence of intersite hopping, the excitation of a trion at agiven site cannot depend on the existence of a trion onany other site, ensuring that moiré trions behave asnoninteracting excitations. The only contribution to thelight shift in this limit comes from the ac Stark shift of eachsite independently, whose magnitude scales as 1=δAP.Small but nonzero hybridization of the collective trionexcitation with the bare exciton could give rise to a finiteinteraction strength, a finite intersite hopping due to long-range electron-hole exchange, and a deviation from thepure 1=δAP contribution to the light shift. We note thatrecent experiments on the same moiré structure yieldedmagnetization signatures consistent with unexpectedlyweak intersite hopping [7].We find the first confirmation of this explanation whenwe measure the electron density dependence of the lightshift: Figure 4(d) shows that varying ν from 0 to 2 results innegligible variation in the magnitude of the light shift,despite large variation in the pump-induced virtual APpopulation. This behavior contrasts with the strong electrondensity dependence of the AP light shift previouslyobserved in a monolayer MoSe2 [34].An additional confirmation of our description of the moiréAP as a collective excitation of noninteracting localizedtrions is provided by complementary measurements wherewe study the saturation behavior of the AP resonance underresonant pump-laser excitation. Figure 5(a) depicts a cartoonof the heterostructure where resonant excitation leads tolocalized trion occupation, which, in turn, Pauli blocksfurther excitation. To investigate the associated saturationof the AP resonance, we measure ΔR as a function of theelectron density in cross-linear configuration for variouspump laser intensities. Figure 5(b) shows that, for pump laserpower Pp ¼ 40 μW, the AP resonance blueshifts andweakens for short pump-probe time delays. Figure 5(c), inturn, shows that increasing Pp leads to saturation of ΔR forall electronic densities in the range 0 ≤ ν ≤ 2. This saturationbehavior, characteristic of an ensemble of noninteractingtwo-level emitters, can be explained by arguing that eachoccupied moiré site is driven into a balanced mixture of itsground (single-electron) and excited (trion) states forPp ≥ 50 μW. In stark contrast, saturation of composite-boson excitations, such as itinerant excitons orAPs, leads to agradual loss of oscillator strength, accompanied by a shift ofthe resonance energy. Figure 5(e) shows that the saturationunder cross-linear excitation with Pp ¼ 40 μW is nearcomplete for pump-probe time delay τ ≲ τpump, where theprobe ΔR is reduced by a factor of ≃2, as compared to itsreference value obtained without the pump laser. We remarkthat the relaxation time of approximately 5 ps of theresonantly excited APs observed in Fig. 5(e) is much longerthan the pump pulse duration τpump ≃ 0.1 ps. This long-timedynamics is most likely due to the generation of a (real) trionpopulation through absorption of resonant photons.The aforementioned cross-linearly-polarized measure-ments ensure that both the pump and probe lasers drivetrion formation on all moiré sites, regardless of the electron(a)(b)(c) (d)FIG. 4. Light shift of moiré polarons. (a) The peak reflectioncontrast as a function of gate voltage, showing that the oscillatorstrength of the AP linearly increases until ν ¼ 1 and thendecreases linearly until ν ¼ 2. (b) Reflection spectrumΔRAPðτÞ of the AP at ν ¼ 1 as a function of the pump-probedelay τ. (c) The observed AP blueshift is not driven byinteractions as shown by its dependence on the pump detuning.(d) The measured light shift is remarkably constant as a functionof ν, up to a small dispersion of approximately 3% (shown as ablue stripe) compatible with statistical fluctuations (shown as anerror bar, obtained for each point by repeating the measurementfour times). The combination of linear 1=δAP dependence of thelight shift and its independence of ν demonstrate the lack ofinteractions between APs or trions localized on differentmoiré sites.B. EVRARD et al. PHYS. REV. X 15, 021002 (2025)021002-6spin. We also perform a test experiment using a cross-circular configuration. In that case, the pump and the probeare addressing singly occupied sites with electrons in theopposite valley. Consequently, the probe reflection isunaffected by the pump [Figs. 5(d) and 5(e)]. This resultstands in contrast to prior measurements on monolayerMoSe2, where in cross-circular configuration we observe aredshift of the AP resonance, which we tentatively attributeto phase space filling induced by the virtual AP populationgenerated by the pump pulse [34].We conclude that the pump-induced light shift andbleaching of the AP transition demonstrates that the APresonance can be described as an ensemble of noninteract-ing two-level emitters and, as such, is inconsistent with theexciton-polaron model that successfully describes theoptical spectrum of monolayer TMDs [12–14] as well asAPs in weak moiré potentials [47]. For electron densitiessatisfying moiré filling factor ν > 1, another resonanceemerges in the spectrum which we term MX3 [Fig. 1(c)].The ac Stark shift of MX3 is similar to that of monolayerAP or exciton resonances, which allows us to tentativelyidentify it as excitons or APs generated at doublyoccupied moiré sites. These measurements are detailedin Appendix F (see Fig. 10): Briefly, MX3 displays adensity-dependent ac Stark shift, contrasting with that ofthe AP discussed above, which we attribute to repulsiveinteractions. Interestingly, the light shift is smaller forinteger fillings ν ¼ 2 and ν ¼ 3, where the oscillatorstrength is larger and where the electrons form incom-pressible states. This counterintuitive behavior is quali-tatively understood from the enhancement of theinteraction when the electron system is compressibleand efficiently mediates interaction between excitonpolarons [34,48].IV. DISCUSSIONOur findings shed new light on the nature of moiréexcitons and polarons, which remains a topic of activeresearch [5,22–27,41,49–57]. In particular, we show hownonlinear spectroscopy unveils the itinerant or localizedcharacter of moiré optical excitations. Our experimentsallow us to assess the extent of spatial overlap betweendifferent moiré excitons or attractive polarons. It is some-what remarkable that this information is accessible to far-field optics given that the moiré length scale is about 2orders of magnitude below the optical resolution. Theinsight we obtain is crucial for the interpretation ofexperiments aimed at optical sensing of correlated elec-tronic states [4–8,16,17]. While the heterostructure westudy exhibits a deep triangular moiré potential, favoringtopologically trivial correlated electronic insulators andlocalized AP, we envision that applying ac Stark spectros-copy to twisted homobilayers exhibiting Chern bands couldreveal features not accessible to linear spectroscopy. Forexample, we expect the AP ac Stark shift to changequalitatively as the system is tuned from a fractionalChern insulator quantum fluid to a Mott-Wigner state withstrongly localized charges, using an applied displacementfield [8,9]. We also expect the ac Stark shift of the APresonance to reveal signatures of the quasigap to chargedexcitations of a composite Fermi liquid at half filling of aChern band [58].The experiments detailed in Fig. 5 present a realizationof a nonequilibrium Bose-Fermi mixture consisting ofelectrons in a flat moiré band and optically injectedexcitons [18–21]. The choice of resonant excitation ofthe AP transition forces the mixture into a state that can bedescribed as a high-density moiré trion gas. Using acircularly polarized resonant pump laser and an externalmagnetic field to valley polarize electrons, it may bepossible to create a trion at each moiré site, therebyrealizing a solid-state analog of the Dicke model [59].(a)(c)(d)(e)(b)FIG. 5. Resonant excitation of moiré polarons. When the pumpand probe lasers are cross-linearly polarized, they drive the APtransition at all moiré sites [see the sketch in (a)]. We observe ablueshift and a reduction of the AP reflection amplitude (b). Thelatter is plotted in (c) as a function of the electronic filling factor νof the moiré lattice. The blue dots are obtained without pump andthe red squares for various pump intensities. For a sufficientlylarge intensity, all sites end up in a statistical mixture of a singleelectron and a moiré trion, and, consequently, the amplitude ofthe AP resonance is divided by a factor of 2 (dashed black line).After the pump pulse is gone, the AP resonance recovers on atimescale of approximately 5 ps [(e), blue line]. In contrast, usingcross-circularly-polarized lasers, the pump and the probe pulseare driving the AP transition on different moiré sites, and,consequently, we do not observe any effect of the pump [(d),(e), red line].AC STARK SPECTROSCOPY OF INTERACTIONS BETWEEN … PHYS. REV. X 15, 021002 (2025)021002-7The dressing of a quantum material with virtual opticalexcitations could be a promising route to engineer newphases of matter. The idea of using an intense laser pulse tomodify material properties has already been demonstrated[60], but it often suffers from incoherent pumping andheating effects when an electronic polarization mode of thesystem is resonantly driven. Similar, albeit much lesssevere, problems in driven atomic systems can be alleviatedusing Rydberg dressing, where an off-resonant laser effectscoherent hybridization of a ground (or long-lived low-energy) state and a Rydberg state: The atoms remain mostlyin the ground state and spontaneous emission is stronglysuppressed, but virtual excitations to the Rydberg state stillensure long-range interactions. Similarly, it was proposedthat virtual excitons in bulk semiconductors could mediateferromagnetic interactions between free electrons in thebulk [61] or trapped in quantum dots [62]. Our experimentsdemonstrate that an intense pump laser can realize anefficient excitonic dressing of a moiré system while keep-ing light absorption negligible thanks to a large detuningfrom the resonances. This approach could provide a newpath toward light-induced magnetism in van der Waalsheterostructures [63].ACKNOWLEDGMENTSWe thank A. Christianen, M. Glazov, M. Hafezi, A.Müller, A. G. Salvador, R. Schmidt, and A. Srivastava forinspiring discussions. This work was supported by theSwiss National Science Foundation (SNSF) under GrantNo. 200020_207520. B. E. acknowledges funding froman ETH postdoc fellowship. H. S. A. acknowledges sup-port from the Swiss Government Excellence Scholarship.T. U. acknowledges support from the Funai OverseasScholarship.DATA AVAILABILITYThe data are available at the ETH ResearchCollection [64].APPENDIX A: EXPERIMENTAL SETUP ANDSAMPLE FABRICATIONThe sample is measured in a dry cryostat (Attodry800,Attocube) at cryogenic temperatures of approximately 5 Kwith free-space optical access and equipped with nano-positioners allowing displacement along the three axes. Forthe pump and probe, we use a mode-locked Ti:sapphirelaser (Tsunami, Spectra-Physics), with a repetition rate of76 MHz and pulse duration of approximately 100 fs. Thepulse is split along two paths for the pump and the probe.The bandwidth of the pump is reduced using a pulse shaper,and its power is controlled using a motorized opticalattenuator. We achieve a larger spectral width for the probeusing a nonlinear fiber (femtowhite 800, NKT Photonics)which produces a quasicontinuum around the investigatedresonances. The length of the probe optical path can bevaried using a retroreflector on a motorized translationstage, enabling a fine-tuning of the time delay between thetwo pulses. Both beams are focused on a diffraction-limitedspot on the sample using an apochromatic microscopeobjective with NA ¼ 0.8 (LT-APO/VISIR/0.82, Attocube).The typical probe power is on the order of a few micro-watts. The reflected light spectra are recorded using aPeltier-cooled CCD camera.For the sample fabrication, few-layer graphite, approx-imately 35 nm h-BN, monolayer MoSe2 and WS2 aremechanically exfoliated. The layers are assembled usingthe dry-transfer technique with a poly(bisphenol A car-bonate) film on a polydimethylsiloxane (PDMS) stamp anddeposited on a 285 nm Si=SiO2 substrate [65]. The crystalalignment of the TMDs is determined prior to the stackingmeasuring the generation of second-harmonic light as afunction of the polarization of an incoming infrared laserpulse. They are then stacked with a negligible twist. Thegraphene top and bottom gates and TMDs are contactedusing gold electrodes deposited using optical lithographyand electron beam deposition.APPENDIX B: DATA ANALYSISIn order to extract the light shift amplitude, we first fit thereflection spectrum for various time delays and extract theresonance position(s). The result is shown in Fig. 6(a) in thecase of the MX1 resonance. We then perform a Gaussian fitof the measured line shift as a function of the pump-probedelay in order to extract the light shift amplitude at zerotime delay. To determine the dependence of the light shifton the pump detuning, we perform for each pump wave-length a measurement at various pump powers as shown inFig. 6(b). In this way, we can use relatively low power for anear-resonant excitation and larger power at larger detun-ings, always making sure that we are in a regime of linearscaling with power. In the figures in the main text, we plotthe slope Δ=Ipump computed from a linear fit at each pumpdetuning δ.FIG. 6. Data analysis. (a) Light shift as a function of the pump-probe delay for MX1 for a pump power of 250 μW and detuningof 11 meV, showing a clear blueshift at zero time delay. The fitcorresponds to a Gaussian envelope. (b) The dependence of thelight shift of MX1 on pump power at different pump detunings δ1exhibits a linear dependence.B. EVRARD et al. PHYS. REV. X 15, 021002 (2025)021002-8APPENDIX C: FIT OF THE WAVELENGTHDEPENDENCEWe discuss here our analysis of the wavelength depend-ance of the light shift. Taking into account a singleresonance, the light shift Δ can be expanded in a seriesof 1=δ using perturbation theory [30]. The first-order 1=δterm comes from light-matter interaction, while exciton-exciton interactions contribute to higher-order terms, in1=δ2 or 1=ðδiδjÞ when several modes contribute. Inprinciple, a fine analysis of the detuning dependence ofthe light shift would enable one to deconvolve the variouscontributions. In practice, this analysis can be challengingdue to the finite range of detuning in which we can takereliable data. Indeed, for all resonances, we had to restrictthe data to a window of approximately [20, 80] meV. Goingcloser to resonance, we face two issues: First, incoherenteffects become more prominent as pump photons carryenough energy to generate a real population of excitonsand, hence, are more likely to be absorbed. Second, theperturbative expansion of the light shift becomes incon-sistent when Δ ∼ δ. Conversely, going further away fromresonance the signal reduces (given the available laserpower) and the extraction of the light shift becomesunreliable. Furthermore, in that regime, the light-matterterms ∝ 1=δ become stronger compared to the interactionterm ∝ 1=δ2 that we are interested in.1. Charge neutralityWith the relatively small detuning we use (compared tothe exciton Rydberg energy), the exciton-exciton interac-tion is expected to be the dominant contribution to the lightshift [34]. This is indeed confirmed by our data, which arenot compatible with a 1=δ law. Let us, thus, focus on theinteraction-induced light shift, which we generalize to ourmoiré system with two bright exciton modes. As shown inAppendix H 4, the light shifts of MX1 and MX2 read,respectively,Δ1 ¼ 2u1;1n1 þ 2u1;2n2 þ 4k1ffiffiffiffiffiffiffiffiffiffin1n2p; ðC1ÞΔ2 ¼ 2u2;2n2 þ 2u1;2n1 þ 4k2ffiffiffiffiffiffiffiffiffiffin1n2p; ðC2Þwith the densities scaling as ni ∝ jΩiϕið0Þj2=δ2i . The ratior ¼ jΩ1ϕ1ð0Þj2=jΩ2ϕ2ð0Þj2 ¼ γrad;1=γrad;2 ≈ 0.48 can beobtained from a fit of the reflection contrast (seeAppendix D). Using this independent estimate and per-forming a joint fit of Δ1;2, we reduce the number of fittingparameters to five: A, u02;2, u01;2, k01, and k02, whereu0i;j ¼ uij=u1;1, k01 ¼ ki=u1;1, andΔ1 ¼ A�1δ21þ r2u01;2δ22þ 2rk01δ1δ2�; ðC3ÞΔ2 ¼ A�r2u02;2δ22þ u01;2δ21þ 2rk02;22δ1δ2�: ðC4ÞFurthermore, within a Born approximation uklij ∝Rdrϕiϕjϕ�kϕ�l (see Appendix H 4) and using theCauchy-Schwarz inequality, we obtain the followingbounds: 0 < u01;2 < ½u02;2�1=2, jk01j < ½u1;2�1=2, and jk20j <½u02;2u01;2�1=2, which we enforce to improve the convergenceof the fit.For MX1 [see Fig. 7(a)], the full fit reveals the dominantcontribution to be the MX1-MX1 interaction. We obtain thefitting parameter A, which we can compare to the valueobtain from fitting the light shift of a monolayer exciton (on(a) (b) (c)FIG. 7. Pump wavelength dependence. From a fit, we infer the origin of the light shift of MX1 (a), MX2 exciton (b), and the attractivepolaron at ν ¼ 1 (c). At charge neutrality [(a),(b)], we are not able to deconvolve the contribution of the u01;2 and k02 fitting parameters[see Eq. (C4)]. We, thus, perform two fits, setting k02 to its upper and lower bounds, shown as a solid blue and red line, respectively. Forthe MX1 exciton (a), the light shift is dominated by the 2u1;1n1 term (shown as a dashed line) and barely sensitive to the MX2-dependentterms, so both fits are nearly identical and we obtain a good estimate of u1;1. For MX2 (b), both fits are in fair agreement with the data,but the contribution of the 2u1;2n1 term (dotted line) is significantly different, and we are, thus, able to provide only a rather broadconfidence interval for u1;2. For the AP, a fit AAP=δ1 þ BAP=δ2AP [solid black line, (c)] suggests that the usual ac Stark shift scaling asAAP=δ1 (dash-dotted line) is occurring instead of an interaction-driven shift scaling as BAP=δ2AP (dotted line).AC STARK SPECTROSCOPY OF INTERACTIONS BETWEEN … PHYS. REV. X 15, 021002 (2025)021002-9a monolayer MoSe2 region of the same device). Weestimate u1;1=uex ≈ 1.6� 0.2, where uex is the monolayerexciton-exciton interaction (see Appendix E for an absolutecalibration).For MX2 [see Fig. 7(b)], the fit is unable to reliablydisentangle the contribution of the second and third termsof Eq. (C4), corresponding to the scattering of an MX1exciton, without (first term) or with (second term) a changeof moiré band. Indeed, performing a fit with each of thesetwo terms independently, we obtain in both cases areasonable agreement with the data over the full detuningrange. We thus perform two fits, where the coupling k02 ¼�½u02;2u01;2�1=2 saturates the Cauchy-Schwarz inequality. Inthis way, we obtain the bounds 0.3 ⪅ u2;2 ⪅ 0.8.2. Moiré attractive polaronFor the AP [Fig. 7(c)], the light shift is typically smallerand more noisy due to the weakness of the transition. As aresult, it is more difficult to discriminate a potential 1=δ and1=δ2 dependence. Nevertheless, the fit does suggest that theformer is here the leading contribution, consistent with thepicture of an ensemble of distinguishable and noninteract-ing two-level systems, as argued in the main text.APPENDIX D: TRANSFER MATRIXSIMULATIONIn order to infer the interaction strength of the variousmoiré excitons from their light shift, we need to estimatethe exciton density that we generate and, hence, the excitonoscillator strength. The latter can be obtained from a fit ofthe reflection spectrum. Such a fit needs to include thereflection of the electromagnetic field on the interfacesbetween the various dielectrics that make our van der Waalsheterostructure. We do this using the transfer matrixmethod [66]. Two fitting parameters for the backgroundare the h-BN thickness of approximately 31 nm andapproximately 37 nm for the top and bottom layer,respectively, and the h-BN refractive index nh−BN ≈ 2.15[67,68]. Then, for each resonance, we have three additionalfitting parameters, namely, its energy and radiative andnonradiative decay rates. The results of this fit are shown inFig. 8. We show here the bare reflection spectrum, obtainedusing a light source that is to a very good approximationspectrally flat in the energy range shown in the figure. Weare unable to obtain a perfect fit of the background, usingthe h-BN thicknesses and refractive index as free param-eters. The discrepancy that we observe, especially on theedge of the spectrum, could be due to chromatic aberrations(although we are using a microscope apochromatic objec-tive to limit those). Nevertheless, in the center of thespectrum, we are able to reproduce our spectra very well atall fillings.APPENDIX E: INTERACTION STRENGTHThe radiative (γr) and nonradiative (γnr) decay rates canbe used to extract the density of excitons nex induced by thepump. Using the optical Bloch equations within theadiabatic approximation, we have [66]nex ¼2Iiγrðγr þ γnrÞ2 þ δ2ex; ðE1Þwhere Ii is the photon flux. For reference, we first look at theexciton light shiftΔex measured on amonolayer region of thesample, as a function of the exciton density; see Fig. 9.We observe a linear dependence Δex ¼ uexnex, from whichwe extract the exciton-exciton interaction strength uex ≈0.06ð3Þ μeV μm2. The latter is compatible with previousmeasurements [34,48,66,69], although significantly smallerthan theoretical estimates, ∼3Eexa2ex ∼ 1 μeV μm2, whereEex is the exciton binding energy and aex is the exciton Bohrradius [43,44].From the measurement of the light shift of the moiréexcitons and polaron shown in Fig. 7, we obtain u1;1 ≈1.6uex and u1;2 ≈ 0.7uex and, for the MX3-MX3 interaction,u3;3 ≈ 5.3uex. The enhancement of the interaction strengthFIG. 8. Fit of the reflection spectrum using transfer matrixsimulation. The blue dots are the data, and the black solid line is afit (gray area span by varying the fitting parameters within theconfidence interval).B. EVRARD et al. PHYS. REV. X 15, 021002 (2025)021002-10of MX1 (charge neutrality) could stem from the partialconfinement induced by the moiré potential. The largerenhancement for MX3 reflects the polaronic nature of thisresonance, as previously observed in a monolayer sam-ple [34].APPENDIX F: LIGHT SHIFT OF MX3 IN THEREGIME OF LARGE DOPINGEven though MX3 is the dominant excitonic resonancefor electron filling factors ν ≥ 1.5, its identification hasremained unclear. To gain insight, we investigate thenonlinear response of the MX3 resonance. Figures 10(a)and 10(b) show the reflection amplitude and the energy ofMX3 for 1.5 ≤ ν ≤ 3.4 in the absence of a pump laser:Consistent with earlier observations, we find that theresonance energy as well as the reflection strength, or,equivalently, the oscillator strength, of MX3 exhibit localmaxima at integer fillings ν ¼ 2 and ν ¼ 3. These featurescould be explained by partial suppression of dynamicaldressing of excitons by electrons, when the two-dimen-sional electron system (2DES) is in an incompressible state[16,70]. In contrast, when the electrons form a Fermi liquid(ν ≠ 2, 3), the dynamical dressing of MX3 is more effectiveand results in a redshift together with a reduction of theoscillator strength.Figure 10(c) shows the light shift as a function of pump-probe delay τ for four representative filling factorsobtained for δ3 ¼ 80 meV. We observe that the light shiftfor τ ≃ 0 indicates repulsive (attractive) interactionsbetween same (opposite) -valley MX3 excitons generatedby co- (cross-) circularly polarized pump-probe fields.While attractive interactions between opposite-valleyexcitons have been reported before, it is surprising thatthe magnitude of the light shift is comparable in the twocases. The attractive interactions for the cross-polarizedconfiguration may be explained through a near-resonanttwo-photon (pumpþ probe) excitation of the biexcitonresonance at ωXX. Verification of this hypothesis could beachieved by changing the pump detuning δ3 so as to probeboth δ3 ≤ ωXX and δ3 ≥ ωXX, since, for the latter case, thebiexciton-mediated interactions would become repulsive.As we had to choose δ3 < ωXX to avoid strong back-ground absorption, we could not verify the role ofbiexciton in the measured light shift.In contrast to the light shift measurements in the charge-neutral regime, we find that the pump pulse results in a MX3line shift that increases linearly with τ for 0.2 ≤ t ≤ 3.0 ps.Moreover, the linear shift at a given ν is identical for co- andcross-circularly-polarized pump-probe configurations buthas a different sign for compressible (ν ≠ 2; 3) andFIG. 9. Light shift as a function of the exciton density. The bluedots are the data, obtained for various powers and detunings. Theerror bars come from the uncertainty of the exciton decay ratesand on the power on the sample. The black solid line is a fit withthe gray area span obtained by varying the fitting parameterswithin the confidence interval.(a)(b)(c)(d)FIG. 10. Light shift of MX3. The resonance MX3 shows distinctive kinks in its amplitude (a) and energy (b) at integer fillings of themoiré lattice with electrons, following the changes in the 2DES compressibility. The light shift of the moiré exciton MX3 at variousfillings (c) is well fitted by a line on top of a Gaussian function (black line). The latter captures the coherent response of the system,which corresponds to an interaction-induced blueshift for cocircularly polarized pump and probe and possibly to a redshift stemmingfrom the coupling to the biexciton states in a cross-circular configuration. The amplitudes of these two shifts are sensitive to the polarondressing of the exciton and, consequently, are extremal at integer filling of the moiré lattice (d).AC STARK SPECTROSCOPY OF INTERACTIONS BETWEEN … PHYS. REV. X 15, 021002 (2025)021002-11incompressible (ν ¼ 2; 3) electron states. We tentativelyexplain this feature by generation of free carriers by non-resonant absorption of pump photons that change theelectron density for timescales well exceeding the pumpduration.Since the MX3 resonance energy has maxima (minima)for ν ¼ 2; 3 (ν ¼ 3=2; 5=2), any pump-induced change inelectron density will result in a redshift (blueshift) of theresonance energy. While we do not understand why theredshift (blueshift) depends linearly on τ for τ > τpump, wespeculate that pump-induced charges are initially generatedin high-energy bands and that they influence the nonlinearresponse only as they relax to the lowest-energy availablemoiré band.We also observe in Fig. 10(c) that the magnitude of thelight shift for the cocircularly polarized pump-probe con-figuration is smaller for incompressible states. PlottingΔ3 asa function of ν [Fig. 10(d)] shows that the blueshift is indeedminimal for ν ¼ 2, 3. This is at first glance surprising, giventhat the oscillator strength, and, consequently, the generatedMX3 population, is maximal for these incompressible states.However, it was recently shown that interactions betweenattractive exciton polarons [34,48,71] mediated by theirdressing cloud are dramatically enhanced compared to thoseof bare excitons. Such an enhancement of interaction strengthmayovercome the reductionof theoscillator strengthofMX3for ν ≠ 2, 3. This tentative explanation suggests that theMX3mode may be identified as a second attractive polaron modewhere the exciton is dressed by electrons in the upper moiréband. Last but not least, we find that the redshift of MX3 inthe cross-circularly-polarized configuration is maximalwhen the electronic state is incompressible; we currentlydo not have an explanation for this observation.APPENDIX G: LIGHT SHIFT OF MX02 AT ν= 1At a unity filling ν ¼ 1 of the moiré potential, weobserve two bright resonances, the AP which we discussin detail in the main text, and MX01, emerging from MX2and which we now investigate. Depending on the pump andprobe polarizations, we observe different behaviors. Incocircular polarization, we obtain the usual blueshift whichwe attribute to MX02-MX02 interactions. Contrary to otherresonances, we cannot confirm this claim by an analysis ofthe detuning dependence. Indeed, we observe strongincoherent behavior for a pump blue detuned from theAP, and we, therefore, explore only the red-detunedsituation. Specifically, we explore the range δMX20 ≈½80; 110� meV in which we observe no significant evolu-tion of the light shift, as expected from a scaling as 1=δ2MX02;see Fig. 11(a). By contrast, in cross-circular polarization,we observe a distinct redshift which diverges close to theAP resonance, in excellent agreement with a 1=δ2AP scalingand suggesting an attractive interaction between AP andMX02 in opposite valleys. We point out that a similarbehavior was observed in a monolayer system and tenta-tively attributed to the reduction of the phase-space fillingupon the generation of an AP, for an opposite-valleyexciton [34]. The pump power dependence [Fig. 11(b)]of the light shift shows an interesting behavior for a near-resonant excitation of the AP δAP ≈ 8 meV. In that case, athigh intensity, we expect a saturation of the AP density asthe moiré potential is filled up, as described in the main text(although here, keeping a finite δAP, we are unable to fullysaturate the transition). Indeed, we observe a sublinearincrease of the cross-polarized light shift attributed to AP-MX02 interactions. On the contrary, the copolarized lightshift which we attribute to MX02-MX02 interactions is linearin the pump power, as the density of MX02 remains far fromsaturation of the moiré lattice (δMX02≈ 80 meV).FIG. 11. Light shift of MX02 at ν ¼ 1. (a) Detuning dependenceof the light shift of the resonance MX02 at a filling ν ¼ 1. Red dotscorrespond to cross-polarized pump and probe and show aredshift of the MX02 resonance which can be well fitted (redline) by a dependence ∝ 1=δ2AP stemming from MX02-AP inter-actions. Blue dots correspond to copolarized pump and probe andshow a blueshift of the MX02 resonance which can be well fitted(blue line) assuming MX02-MX02 interactions and a scaling∝ 1=δ2MX02. (b) Light shift of MX02 as a function of pump powerin different polarizations. Cross-circular polarization shows adistinct redshift; the deviation from a linear dependence is due tosaturation of the AP population at large powers. Collinearpolarization shows a smaller blueshift which is well in the linearregime. (c) MX02 shift vs pump-probe time delay in cocircular(blue) for a pump power of 500 μW and cross-circular (red)polarizations for 50 μW.B. EVRARD et al. PHYS. REV. X 15, 021002 (2025)021002-12APPENDIX H: THEORETICAL MODELIn this section, we primarily provide a detailed descrip-tion of the theoretical model used to analyze the charge-neutral, cross-circular polarization data. The foremost goalof this model is to qualitatively capture the energy shiftsobserved in the MX1 and MX2 exciton resonances whensubjected to the influence of a pump laser. To this end, wecalculate the exciton-exciton T matrix in the presence of amoiré potential, which is inspired by a recent treatment oftwo interacting atoms in a 2D square optical lattice [72].This approach allows us to capture the essential physics ofthe interaction under the influence of the pump.We conclude the section with a brief discussion on amodel for the cocircular polarization data. For ease ofnotation, we set ℏ ¼ 1.1. Model for interactions of two distinguishable excitonsWe begin by modeling the interactions between twodistinguishable rigid (1s) excitons in the presence of moiré.The Hamiltonian we consider consists of four terms:H ¼ HX↑ þHX↓ þHd þ V: ðH1ÞThe first two terms represent the exciton Hamiltonians foreach valley (σ ¼ ↑;↓):HXσ ¼XKϵK;XX̂†KσX̂Kσ þXKQṼXðQÞX̂†KþQσX̂Kσ: ðH2ÞHere, X̂Kσ annihilates an exciton of type σ with energyϵK;X ¼ ϵX þ jKj2=2mX, where mX is the exciton mass andϵX is the 1s exciton energy. The term ṼX is the Fouriertransform of the exciton moiré potential, which is approxi-mated in real space by [23]VðxÞ ¼X6j¼1VjeiGj·x; ðH3Þwhere Gj are the first six reciprocal lattice vectors. Thethreefold symmetry and the realness of the periodicpotential imply that V1 ¼ V3 ¼ V5, V2 ¼ V4 ¼ V6, andV1 ¼ V�4, parametrized by V1 ¼ Veiψ, with V determiningthe potential depth and ψ its shape.Table I summarizes the relevant parameters for our systemas determined by large-scale density functional theory (DFT)calculations [7]. Here, we also include the exciton moiréparameters which are determined by assuming a tightlybound electron-hole pair such that VXeiψX ≡ VeeiψeþVheiψh , where Vi and ψ i are the moiré parameters for theexciton (i ¼ X), electron (i ¼ e), and hole (i ¼ h).The third term in the Hamiltonian describes the closed-channel molecule, which mediates interactions betweenexcitons:Hd ¼XKðϵK;dþδccÞd̂†Kd̂KþXKQ2ṼXðQÞd̂†KþQd̂K; ðH4Þwhere d̂K annihilates a closed-channel molecule withenergy ϵK;d ¼ 2ϵX þ jKj2=2M (M ¼ 2mX) and detuningδcc. The closed-channel experiences both exciton poten-tials, hence the factor of 2 in the moiré potential.The interactions are mediated according toV ¼ gffiffiffiffiApXKQχðKÞðd̂†QX̂Q−K;↑X̂K;↓ þ H:c:Þ; ðH5Þwhere χðKÞ regularizes the ultraviolet (UV) divergence andA is the system area. Throughout this section, we useχðKÞ ¼ ΘðΛ − jKjÞ, where Λ is the UV cutoff. The barecoupling g and detuning δcc are renormalized as [73]δccg2¼ 1AXKχðKÞ 1EBX þ 2ϵK;X; ðH6Þwith EBX being the energy of the biexciton without moiré.We point out that the two-channel model is equivalent tocontact interactions with coupling constantU ¼ −g2=δcc inthe single-channel model limit (δcc; g → ∞). Throughoutthis work, we exclusively work in the single-channel modellimit, since, for our purposes, the two-channel model isused only as a tool to simplify the calculation of theexciton-exciton T matrix.2. T matrixTo study the interactions we calculate the T matrix,which provides an exact solution to the full two-bodyproblem. To begin, we introduce the free exciton andclosed-channel Green’s functions:Ĝð0ÞðEÞ ¼ 1E − ĤX↑ − ĤX;↓; ðH7ÞD̂ð0ÞðEÞ ¼ 1E − Ĥd: ðH8ÞThe two-body T matrix is given by the infinite seriesTABLE I. Parameters for electron, hole, and exciton fromdensity functional theory calculations [7]. The mass of theparticles is given in units of the bare electron mass me. Theassumed moiré length is aM ¼ 8.2 nm.Particle V (meV) ψ m (me)Electron −6.3 0° 0.45Hole 1.9 59° 0.55Exciton 5.6 163° 1AC STARK SPECTROSCOPY OF INTERACTIONS BETWEEN … PHYS. REV. X 15, 021002 (2025)021002-13T ¼ V̂D̂ð0ÞV̂ þ V̂D̂ð0ÞV̂Ĝð0ÞV̂D̂ð0ÞV̂ þ � � �¼ V̂ðD̂ð0Þ þ D̂ð0ÞV̂Ĝð0ÞV̂D̂ð0Þ þ � � �ÞV̂¼ V̂ D̂ V̂; ðH9Þwhere we suppress the energy dependencies for brevity andD̂ðEÞ is the closed-channel Green’s function. Thus, byfinding the closed-channelGreen’s function,we can immedi-ately calculate the T matrix. This approach simplifies thecalculation and provides easier access to the biexcitonenergies, which are both the poles of the T matrix and D̂ðEÞ.The closed-channel Green’s function is given byD̂ðEÞ ¼ 1½D̂ð0ÞðEÞ�−1 − g2A Π̂ðEÞ; ðH10Þwhere we introduce the polarization bubble, which hasmatrix elementsΠqλλ0 ¼Xk;λ↑;λ↓Vq;λkλ↑λ↓1E − Ek;q;λ↑;λ↓Vq;λ0�kλ↑λ↓; ðH11Þwhere Ek;q;λ↑;λ↓ ≡ E↑q−k;λ↑ þ E↓k;λ↓andVqλdkλ↑λ↓≡ hq; λd; djVjq − k; λ↑;↑;k; λ↓;↓iffiffiffiffiAp=g:Here, we introduce a Bloch basis, which satisfiesHX;σjk; λ; σi ¼ Eσk;λjk; λ; σi; ðH12ÞHdjk; λ; di ¼ ðEdk;λ þ δccÞjk; λ; di; ðH13Þwhere k is the quasimomentum and λ is the band index.3. Modeling light shiftsWithin our model for cross-circular-polarization data, weconsider the interaction-induced shift of the probe exciton(denoted by ↑) by virtual excitons (↓) created by the pump.To incorporate the effects of the pump laser, we considerthe following modification to the Hamiltonian ĤX↓ [30]:HX↓ ¼Xk;λE↓k;λX̂†k;λ;↓X̂k;λ;↓ þΩ�e−iωLtXλϕð0;λ;↓Þ�0 X̂†0;λ;↓þΩeiωLtXλϕð0;λ;↓Þ0 X̂0;λ;↓; ðH14Þwhere ϕð0;λ;↓Þ0 ≡ h0jX̂0;↓j0; λ;↓j and X̂k;λ;↓ annihilates a ↓exciton with quasimomentum k and band index λ. Here, Ωrepresents the light-matter interaction strength; we assumethat the light couples only to the zero-momentum excitonand use the rotating wave approximation (jϵX − ωLj ≪ϵX þ ωL). By moving into the rotating frame, it can be seenthat the excitons form a coherent state, jΦi ¼ D̂ðβÞj0i(ignoring normalization), where we introduce the multi-mode shift operatorD̂ðβÞ ¼ exp�XλðβλX̂†0;λ;↓ − β�λX̂0;λ;↓Þ�ðH15Þand βλ ¼ −ϕð0;λ;↓Þ�0 Ω�=ðE↓0;λ − ωLÞ.We can then approximate this energy shift of the λthresonance of the ↑ probe exciton using the T matrix:ΔλðωLÞ ≃Xλ↓λ0↓βλ↓Tλλ↓;0λλ0↓;0ðE0;λ þ ωL þ 2iγÞβ�λ0↓ ; ðH16Þwith γ ≈ 1 meV the inverse lifetime of the exciton. Here,we have the matrix elements of the T matrix defined byTλ↑λ↓;0λ0↑λ0↓;0ðEÞ≡ h0jX̂0λ↓↓X̂0λ↑↑T̂ðEÞX̂†0λ0↑↑X̂†0λ0↓↓j0i: ðH17ÞIn Eq. (H16), we explicitly include the laser frequency ωLas a parameter in the self-energy to emphasize its signifi-cance: We observe that the laser light shifts the scatteringoff shell, similar to the effect discussed in the context ofpolariton-electron scattering [74]. It is important to notethat Eq. (H16) captures the ac Stark effect, whereby theshift can change sign when the laser frequency ωL is tunedinto resonance with the biexciton bound state, i.e.,E0;λ ¼ ωL − EBX;α, where EBX;α denotes the energy of azero-quasimomentum biexciton state. However, unlike thesimpler case of a monolayer TMD, this sign change doesnot occur at all biexciton energies, as not all biexciton statescouple to the λth exciton mode. We can, therefore, expectthat in the experiments, which observe shifts only in theoptically active MX1 and MX2 modes, it will not bepossible to detect all biexciton states.The magnitude of the shift in Eq. (H16) depends on ourchoice of Ω=ffiffiffiffiAp. Since our primary interest is in the zerocrossings of the shift (and not its absolute value), we use asimple estimate for this quantity. In particular, we takeΩ=ffiffiffiffiAp ¼ ffiffiffi2pmeV=nm, which is chosen to correspond toa reasonable induced exciton density of 5 × 1011 cm−2 of1s excitons at a detuning of 20 meV (in the absenceof moiré).4. Same-valley interactionsTo conclude our work, we briefly consider interactionsfrom excitons in the same valley, which do not support abound state. Owing to this, we simply take the Bornapproximation of the T matrix assuming repulsive contactinteractions. Focusing on zero quasimomentum, the inter-actions between the excitons are given byB. EVRARD et al. PHYS. REV. X 15, 021002 (2025)021002-14V̂ ¼ uex2XUλ1λ2λ3λ4X̂†0;λ1;↓X̂†0;λ2;↓X̂0;λ3;↓X̂0;λ4;↓: ðH18ÞHere, uex is the strength of the repulsion of two K ¼ 0excitons (i.e., in the absence of moiré), and we introducethe overlap integralUλ1λ2λ3λ4¼ AZd2rφ�λ1;↓ðrÞφ�λ2;↓ðrÞφλ3;↓ðrÞφλ4;↓ðrÞ; ðH19Þwhere φλ;↓ is the Bloch wave function in real space at zeroquasimomentum with band index λ and valley index ↓. 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INTRODUCTION II. INTERACTIONS BETWEEN MOIRÉ EXCITONS A. Cocircularly polarized light B. Cross-circularly-polarized light III. ATTRACTIVE POLARON LIGHT SHIFT IV. DISCUSSION ACKNOWLEDGMENTS DATA AVAILABILITY APPENDIX A: EXPERIMENTAL SETUP AND SAMPLE FABRICATION APPENDIX B: DATA ANALYSIS APPENDIX C: FIT OF THE WAVELENGTH DEPENDENCE 1. Charge neutrality 2. Moiré attractive polaron APPENDIX D: TRANSFER MATRIX SIMULATION APPENDIX E: INTERACTION STRENGTH APPENDIX F: LIGHT SHIFT OF MX3 IN THE REGIME OF LARGE DOPING APPENDIX G: LIGHT SHIFT OF MX2&prime; AT &nu;=1 APPENDIX H: THEORETICAL MODEL 1. Model for interactions of two distinguishable excitons 2. T matrix 3. Modeling light shifts 4. Same-valley interactions References