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Jakub Jasiński, Joakim Hagel, Samuel Brem, Edith Wietek, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Alexey Chernikov, Nicolas Bruyant, Mateusz Dyksik, Alessandro Surrente, Michał Baranowski, Duncan K. Maude, Ermin Malic, Paulina Plochocka

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[Quadrupolar excitons in MoSe2 bilayers](https://mdr.nims.go.jp/datasets/bb8c4cbb-bd17-4619-90d0-d2b068b8f4dc)

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Quadrupolar excitons in MoSe2 bilayersArticle https://doi.org/10.1038/s41467-025-56586-3Quadrupolar excitons in MoSe2 bilayersJakub Jasiński 1,2, Joakim Hagel 3, Samuel Brem4, Edith Wietek 5,Takashi Taniguchi 6, Kenji Watanabe 7, Alexey Chernikov 5,Nicolas Bruyant 2, Mateusz Dyksik 1, Alessandro Surrente 1,Michał Baranowski 1, Duncan K. Maude2, Ermin Malic 4 &Paulina Plochocka 1,2The quest for platforms to generate and control exotic excitonic states hasgreatly benefited from the advent of transition metal dichalcogenide (TMD)monolayers and their heterostructures. Among the unconventional excitonicstates, quadrupolar excitons—a superposition of two dipolar excitons withanti-aligneddipolemoments—are of great interest for applications in quantumsimulations and for the investigation of many-body physics. Here, we unam-biguously demonstrate the emergence of quadrupolar excitons in naturalMoSe2 homobilayers, whose energy shifts quadratically in electric field. Incontrast to trilayer systems, MoSe2 homobilayers have many advantages,which include a larger coupling between dipolar excitons. Our experimentalobservations are complemented bymany-particle theory calculations offeringmicroscopic insights in the formation of quadrupolar excitons. Our resultssuggest TMD homobilayers as ideal platform for the engineering of excitonicstates and their interaction with light and thus candidate for carrying out on-chip quantum simulations.Two-dimensional (2D) layered semiconductors, such as transitionmetal dichalcogenides (TMDs), have emerged as an ideal playgroundto study exciton physics on the nanoscale, essentially due to theintricate valley physics and the greatly enhanced electron–holeattraction related to the reduced dimensionality and dielectricscreening in the monolayer limit1–7. The subsequent development ofvan der Waals heterostructures significantly enriched this field ofresearch. The absence of the lattice-matching constraints for TMDsandmany other emerging layered 2Dmaterials opens a new paradigmin material engineering, where different materials can be seamlesslystacked into virtually limitless combinations, whose properties can betuned by both the material selection and the relative orientation8–11.For instance, homobilayers and heterostructures support long-lived dipolar interlayer excitons (IXs), where electrons andholes residein different layers12–18, and hence canbe canbe easily tuned by externalelectric field19–22. Transition metal dichalcogenide heterostructureshave emerged as an excellent solid-state platform for exploring many-body physics and quantum phases arising from monopolar23 anddipolar interactions24–33, entering fields traditionally dominated byultracold atoms34–38. Very recently it has been demonstrated that TMDheterostructures can also host more complex quasiparticles, referredto as quadrupolar excitons39–46, stemming from the hybridizationbetween two dipolar excitons with opposite dipole moments. Thehigher-order symmetry causes the quadrupole-quadrupole interac-tions to be substantially different compared to the dipole–dipole ones.In particular, their non-local interactions can be finely tuned by theapplication of electric field. The quadrupolar interactions enable newcollective phenomena beyond monopolar and dipolar interactionssuch as the exotic rotons, new flavors of Bose-Einstein condensate,charge density wave or topological superfluids44,47–50. The solid-stateReceived: 30 October 2024Accepted: 20 January 2025Check for updates1Department of Experimental Physics, Faculty of Fundamental Problems of Technology, Wroclaw University of Science and Technology, Wroclaw, Poland.2Laboratoire National des Champs Magnétiques Intenses, EMFL, CNRS UPR 3228, Université Grenoble Alpes, Université Toulouse, Grenoble andToulouse, France. 3Department of Physics, Chalmers University of Technology, Gothenburg, Sweden. 4Department of Physics, Philipps-Universität Marburg,Marburg, Germany. 5Institute of Applied Physics andWürzburg-Dresden Cluster of Excellence ct.qmat, Technische Universität Dresden, Dresden, Germany.6Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba, Japan. 7Research Center for Electronic and OpticalMaterials, National Institute for Materials Science, Tsukuba, Japan. e-mail: paulina.plochocka@lncmi.cnrs.frNature Communications |         (2025) 16:1382 11234567890():,;1234567890():,;http://orcid.org/0000-0003-0631-9461http://orcid.org/0000-0003-0631-9461http://orcid.org/0000-0003-0631-9461http://orcid.org/0000-0003-0631-9461http://orcid.org/0000-0003-0631-9461http://orcid.org/0000-0002-3858-4174http://orcid.org/0000-0002-3858-4174http://orcid.org/0000-0002-3858-4174http://orcid.org/0000-0002-3858-4174http://orcid.org/0000-0002-3858-4174http://orcid.org/0000-0001-6681-9260http://orcid.org/0000-0001-6681-9260http://orcid.org/0000-0001-6681-9260http://orcid.org/0000-0001-6681-9260http://orcid.org/0000-0001-6681-9260http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-9213-2777http://orcid.org/0000-0002-9213-2777http://orcid.org/0000-0002-9213-2777http://orcid.org/0000-0002-9213-2777http://orcid.org/0000-0002-9213-2777http://orcid.org/0000-0003-4697-4188http://orcid.org/0000-0003-4697-4188http://orcid.org/0000-0003-4697-4188http://orcid.org/0000-0003-4697-4188http://orcid.org/0000-0003-4697-4188http://orcid.org/0000-0003-4945-8795http://orcid.org/0000-0003-4945-8795http://orcid.org/0000-0003-4945-8795http://orcid.org/0000-0003-4945-8795http://orcid.org/0000-0003-4945-8795http://orcid.org/0000-0003-4078-4965http://orcid.org/0000-0003-4078-4965http://orcid.org/0000-0003-4078-4965http://orcid.org/0000-0003-4078-4965http://orcid.org/0000-0003-4078-4965http://orcid.org/0000-0002-5974-0850http://orcid.org/0000-0002-5974-0850http://orcid.org/0000-0002-5974-0850http://orcid.org/0000-0002-5974-0850http://orcid.org/0000-0002-5974-0850http://orcid.org/0000-0003-1434-9003http://orcid.org/0000-0003-1434-9003http://orcid.org/0000-0003-1434-9003http://orcid.org/0000-0003-1434-9003http://orcid.org/0000-0003-1434-9003http://orcid.org/0000-0002-4019-6138http://orcid.org/0000-0002-4019-6138http://orcid.org/0000-0002-4019-6138http://orcid.org/0000-0002-4019-6138http://orcid.org/0000-0002-4019-6138http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-56586-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-56586-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-56586-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-56586-3&domain=pdfmailto:paulina.plochocka@lncmi.cnrs.frwww.nature.com/naturecommunicationsmatrix endows the quadrupolar states with robustness, which insteadeludes theirmolecular counterparts51,52, and is particularly attractive toimplement quantum simulation protocols and to reveal unconven-tional quantum states, many body phases48–50, and phasetransitions44,45. Themultipolar character of excitons in TMD structurescan be continuously tuned between quadrupolar and dipolar statesdue to the nonlinear Stark effect39,40 or mixing with other excitonicspecies (as we show herein), which enables a continuous control overmany–body interactions.So far the formation of quadrupolar states in TMD systems hasbeen explored (both theoretically and experimentally) only for TMDheterotrilayers39–46, which enforce the formation of interlayer excitonswith anti-aligned static dipole moments. Similar conditions can befound in TMD homobilayers15,17,53,54, suggesting that quadrupolarexcitons might also form in these structures. However, they haveremained elusive so far.Here, we demonstrate the existence of quadrupolar states in anatural 2H-stacked homobilayer of MoSe2. In our double gated devicewe identify two types of interlayer transitions with dipolar and quad-rupolar character. Combining many-particle theoretical modelling,and electric field dependent reflectivity measurements, we provide amicroscopic understanding of the complex excitonic landscape in anatural MoSe2 bilayer. We show that the quadrupolar states emergefrom the coupling between the dipolar transitions. The observedexcitonic states, including the quadrupolar excitons, can be effectivelytuned, with the use of electric field, between interlayer, hybrid andintralayer character showing that naturalMoSe2 bilayers are promisingcandidates to studymany-body physics driven by field-tunable electricmultipolar interactions.Results and DiscussionObservation of quadrupolar excitonsWe have investigated a natural 2H-stacked MoSe2 homobilayer, fullyencapsulated in hexagonal boron nitride (hBN). The hBN encapsulatedMoSe2 is grounded by a few layer graphite (FLG) electrodes. Twoadditional FLG electrodes are used as the top and the bottom gates.Schematic andmicroscope images of the device are shown in Fig. 1a, b.During the fabrication of the device by the dry-transfer method, thestackwas annealed after each stamping step. Thegoal of annealingwastominimize the concentration of bubbles and simultaneously improvethe adhesion between the consecutive layers (see Methods section formore details on the fabrication procedure). To reveal the complexexcitonic landscape of the naturalMoSe2 bilayer, we studied its opticalresponse as a function of the out-of-plane electric field (Ez) using thecapacitor-like design of the structurewhich allows for the independentcontrol of the out-of-plane electric field and carrier doping (seeMethods for details).A typical reflectivity spectrum of bilayer MoSe2 (measured attemperature of 5 K), shown as a second derivative is presented inFig. 1c. The spectrum is dominatedby a strong resonance related to theA1s exciton state, accompanied on the high energy side by threeweaker transitions, labelled asQ1, Q2 and IX1s. To understand the originof these transitions, we track their evolution as a function of theelectric field. In Fig. 1d we present the reflectivity spectrum at varying-50 0 50 100-0.03-0.02-0.010.000.010.020.032ndderivativeR/R 0(arb.u)E-E0 (meV)Q1 Q2A1sEz = 0 V/nmc dIX1se-50 0 50 100-202E-E0 (meV)V BG=-1/2V TG(V)-0.00250.0010A1sQ1Q22nd derivativeR/R0 (arb. u.)IX1sIX'1s-50 0 50 100-0.10-0.050.000.050.10E z(V/nm)E-E0 (meV)IntraInterHybridQ2Q1A'1sIX11sIX'11sA1s-0.10-0.050.000.050.10E z(V/nm)1.60 1.65 1.70Energy (eV)1.60 1.65 1.70Energy (eV)BG FLGTG FLGt-hBNb-hBNLayer 2Layer 12H-MoSe2 2LGr. FLGXIX IX’VTGVBGa5 �m2L-MoSe21L-MoSe2Gr. FLG (1)Gr. FLG (2)BG FLGb-hBNb-hBNbFig. 1 | NaturalMoSe2 bilayer device andoptical response under applied out-of-plane electric field. a Schematic of the device. 2H-stacked natural bilayer ofMoSe2encapsulated by insulating bottom and top hBN layers. Few layer graphene (FLG)serve as the bottom (BG) and top (TG) gates, respectively, forming a capacitor-likestructure. Two additional FLG flakes physically touch the MoSe2 bilayer forgrounding purposes (Gr. FLG (1)/(2)). The effect of an applied out of plane electricfield (Ez) indicated by the arrows) is shown on the intralayer exciton (X), and theinterlayer excitons with opposite dipole moments (IX and IX'). bMicrograph of thesample. The white circle indicates the measurement spot. c Second derivative ofnormalised reflectivity (R/R0) at Ez =0 V/nm. d False-colormap of the 2nd derivativeof reflectivity as a function of the gate voltages (left axis) and the corresponding Ez(right axis). The intensity of the strongest neutral A1s exciton transition is inten-tionally saturated to reveal the behaviour of theweaker interlayer transitions. Note,that at negative electric fields, Ez <−0.5 V/nm, the device is unintentionally doped,thus for simplicity, we focus on the analysis of the positive Ez >0 electric field data.eCalculated evolution of the excitonic energy landscape under the influence of theelectric field. The color scale corresponds to the spatial character of the excitonsi.e. intralayer (blue), hybrid (purple) or interlayer (red). The opaque (semi-trans-parent) lines correspond to the spin-singlet (spin-triplet) states. The bottom energyscales (E − E0) in panels c and d are shown with respect to the A1s exciton energyE0 = 1.606eV.Article https://doi.org/10.1038/s41467-025-56586-3Nature Communications |         (2025) 16:1382 2www.nature.com/naturecommunicationselectric field, plotted in the form of a false-color map. Characteristicfeatures can be identified in the false colour map, providing deeperinsight into the exciton landscape and the mutual interaction of theexcitonic states. The states labelled as IX1s,↑↓ and IX01s,#" exhibit a linearStark shift, consistent with their dipolar, interlayer character54–57.Matching theobserved shift to the Stark shift simulatedwith themodeldetailed below,we estimate thedipolemoment tobed≃0.5−0.6 e nm,which is in the range of the dipole length reported for other MoSe2bilayers55,56. In addition, the new states labelled Q1 and Q2 exhibit adistinct, quadratic Stark shift at lowelectricfields. This behaviour is theunequivocal evidence of their quadrupolar nature39,40,43, which stemsfrom the coupling of a pair of anti-aligned dipolar states. The sym-metric arrangement of charges in an electric quadrupole yields a zerodipole moment at Ez = 0. However, increasing electric field displacesthe charges, and the quadrupolar state gradually acquires a dipolemoment, giving rise to the non-linear Stark shift. To corroborate theassignment of the Q1 and Q2 as quadrupolar excitons, we also plottedin Fig. S1 the electric field dependence of the static electric dipolemoment, calculated as dEdEz. The non-linear dependence of the quad-rupolar exciton energy on the electric field translates to a vanishinglysmall electric dipole at low fields. The static dipole moment increaseswith increasing electric field and steadily approaches the dipolemoment obtained for the dipolar spin-triplet interlayer exciton IXA1s"#.At higher fields, around 0.1 V/nm, one can observe the deviation fromthe expected behaviour which stems from the interaction with otherexcitonic states as we discuss in the further part of the manuscript.Additionally, analogous excitonic resonances shifting quadraticallywith the applied electric field, which attests to their quadrupolar nat-ure, were observed on different spots on the main device (Fig. S2) andalso on a 2nd device shown in Fig. S3.Microscopic modelTo provide a detailed microscopic understanding of our observations,we complement our experiments with an effective many-particlemodel that allows for the identification of the key coupling mechan-isms. The symmetric band structure in naturally stacked bilayers hostsa fourfold degeneracy, stemming from the combination of valley andlayer degeneracy. This applies to all intralayer and interlayer excitonspecies, including the A-exciton, interlayer spin-singlet (↑↑, ↓↓) andspin-triplet states (↑↓, ↓↑) (see Supplementary Fig. S7(a,b,c)). Forinstance, the spin-singlet interlayer excitons (SupplementaryFig. S7(b)), IX↑↑ and IX↓↓ have a reversed dipole moment, i.e.,exchangedpositions of the electron andhole, with respect to the othertwo (IX0"" and IX0##), leading to their mixing via the dipole exchangeinteraction (see Section II in SI for further details). The effectiveHamiltonian, which includes all possible interaction channels, can thenbe written asH =H0 +HT +HQC :HereH0 describes the electron and hole Coulomb interaction throughthe generalized Wannier equation,58 together with the excitonresponse to the external electric field.HT is the tunneling contribution,which takes into account both electron and hole tunneling59. The lastterm HQC contains the effective dipole exchange coupling ~J (see Eq.(S5) in SI), giving rise to the formation of quadrupolar excitons.We initially focus on the exchange coupling to explain the non-linear shift of the Q1 and Q2 transitions. We assume that ~J only mixesthe 1s interlayer exciton states of the spin-singlet configuration IXs(↑↑, ↓↓) and opposite dipole moments. This tentative assumption ismotivated by themore pronounced signature ofQ-states compared toIX features, which suggests a higher oscillator strength characteristicfor singlet transition. Moreover, the mixing is assumed to stem fromthe Coulomb interaction, which is a spin-conserving interaction. Suchmixing between the necessary spin-triplet states would not be spin-conserving in a bilayer system. Nevertheless, qualitatively similarquadrupole formation could be expected assuming anequally efficientcoupling between interlayer triplet states. We infer that four spin-singlet IXs mix through two possible interaction paths (~J = J + J0) (SeeEq. (8) in SI) schematically shown in Fig. 2a–c. The first path couplesanti-aligned dipolar IXs corresponding to the same valley but withopposite spin configuration (IX"" + IX0## and IX## + IX0""), schematicallydrawn in Fig. 2a, c and denoted as J. The second path, indicated as J0,mixes anti-aligned dipolar IXs localized in the opposite valleys, butwith the same spin configuration (IX"" + IX0"" and IX## + IX0##), asschematically represented in Fig. 2b, c.The evolution of the energy landscape of IXs under electricfield inthe absence and in the presence of the exchange couplings J=J0 isschematically presented in Fig. 2d–f. When the J=J0 couplings are notaccounted for (Fig. 2d), the application of an electric field gives rise totwo linearly shifting IX states: the higher energy spin-singlet and thelower energy spin-triplet states, offset by the spin-orbit coupling in theconduction band. The inclusion of the first term J in Fig. 2e mixes thespin-singlet IXs, which yields two quadrupolar excitons with oppositecurvature, i.e., the (symmetric – red-shifting) Q and (anti symmetric –blue-shifting) Q’ separated by an energy 2J at zero electric field (and by± J from the IX singlet states in the non-interacting picture). The J0coupling, added in Fig. 2f, leads to a further splitting of the quad-rupolar branches intoQ1 andQ2, andQ01 andQ02, each pair separated by2J0 (see the detailed model description in Section II in the SI). As thelower energy spin-triplet IXs (IX↑↓, IX↓↑, IX0"#, IX0#") are not spin-con-serving, they remain unaffected by the J=J0 coupling and thus theyfollow the standard linear Stark shift.Bymatching the values of J (energy de-tuning from the spin-tripletIX plus spin orbit coupling of ~ 22meV60 at Ez = 0V/nm) and J0 (half ofthe separation between the Q1 and Q2 at Ez =0V/nm) to match theexperimentally observed redshifting quadrupole branch, we obtainJ = 90meV and J0 =8 meV. The results of the simulation are shown inFig. 1e. The good qualitative agreement with the experiment sum-marized in Fig. 1d demonstrates that our effective Hamiltonian suc-cessfully explains the experimentally observed electric field-inducednonlinear energy shift of the Q1 and Q2 states. These states, formedfrom linear combination of spin-singlet IXs (Q1 � IX0"" + IX## andQ2 � IX0## + IX""), correspond to symmetric quadrupole brancheswhich red shift with increasing electric field. According to the pre-sented analysis the IX states (IX"#=IX0#") with resonance around 90meV above the A1s at Ez = 0V/nm are transitions originating fromoptically bright spin-triplet states61. Due to the lack of coupling~J thesepreserve their dipolar character exhibiting a linear Stark shift. Here wenote that the opposite assignment of the dipolar transition origin canalso be found54. Unfortunately, our model does not allow for a defini-tive differentiation between a singlet or triplet origin of quadrupolarstates. Importantly the singlet or triplet nature of the interlayer is notessential to interpret the quadratic shift of Q1 and Q2 transitions.Charge tunnelingTo reveal the importanceof the charge tunnelingmediated interactionbetween different excitonic species we plot the measured reflectivityspectra over an extended energy and electric field ranges as shown inFig. 3a, b. At higher electric fieldweobserve an anti-crossing behaviourof A1s with interlayer exciton species such as the spin-triplet IX1s,↑↓, 2sspin-singlet IX2s,↑↑ and the quadrupolar Q1/2 states as indicated bygreen dashed lines in Fig. 3b. To explain this behaviour we incorporatethe electron tunneling term (te) into our model (schematically shownin Fig. 2a, b by the arrows labeled te). In Fig. 3c we present the results ofthe simulation, taking the electron tunneling into account, whichclearly shows that the anti-crossing behaviour is well captured by themodel (see also Fig. S8 in SI showing progressively the contribution ofthe various coupling mechanisms to the exciton spectrum). Note thatelectron tunneling is usually considered symmetry-forbidden inArticle https://doi.org/10.1038/s41467-025-56586-3Nature Communications |         (2025) 16:1382 3www.nature.com/naturecommunicationsnaturally stacked homobilayers (Hhh stacking)62. Nevertheless, ourresults demonstrate that some electron tunneling occurs, and it iscrucial for the correct description of the excitonic landscape in MoSe2bilayers under the electric field. In the simulation, we assume theelectron tunneling term to be te = 11.9meV, corresponding to the cal-culated value for the HXh stacking59. The much stronger hole tunnelingterm (th = 56.3meV59) predominantly drives the hybridization betweenvarious interlayer and intralayer states53,63. These manifest in theexchangeof the oscillator strengthwith increasing electricfield as theyapproach energetically (see also the extended energy range data inSupplementary Fig. S4b, c, where the observed hole tunneling medi-ated hybridizations are marked).Another significant effect, stemming from the hybridization dueto charge tunneling (both electron and hole), is reflected in the changeof the intra-inter layer character of excitonic states, which is tuned bythe value of electric field. This is shown in the simulated spectra ofFig. 3c as the color coding of the lines, where the blue, purple and redcorrespond to the intralayer, hybrid and interlayer character, respec-tively. For example, the electron tunneling changes the character ofquadrupolar excitons when they approach the A1s transition withincreasing electric field. Around the anti-crossing region, the quadru-poles rapidly change their character from interlayer to intralayer, witha negligible energy shift as a function of the electric field. At the sametime, some of the A excitons acquire a partially interlayer character.This is in contrast to the heterotrilayer case, where the quadrupolarexciton is the lowest state of the system, and its energy shift steadilyapproaches a rate which is characteristic for the dipolar interlayerexciton39,43. Similar electric field induced change of exciton spatialcharacters can be observed for other transitions (see also Fig. S4c inthe Supplementary Information). The charge carrier tunneling alsoexplains the suppression of the blue-shifting anti-symmetric quadru-pole branch in our spectra as a result ofmixing with A2s exciton states.The antisymmetric exciton branch is expected to be at energies veryclose to the A2s intralayer exciton at zero field. Due to this closeproximity, the antisymmetric quadrupolar excitons hybridize withintralayer A2s exciton, and their signature in the optical spectrumvanishes. However, the mixing of these intra- and interlayer statesmanifests as a splitting of A2s exciton at zero electric field. This state isnot expected to exhibit any splitting in the non-interacting picture, butin the presence of quadrupolar exciton complexes, it shows a finitesplitting, following hybridization with interlayer excitons such asIX2s,↑↑ or the antisymmetric branch of the main quadrupolar excitons.Consequently, the A2s state splits into A2s and A02s, even at zero electricfield, as can be seen both in the experimental (Fig. 3b) and theoretical(Fig. 3c) spectra (see also Supplementary Fig. S8, where the influenceof the individual couplings is shown). The conclusions drawn from thereflectivity spectra are supported by PL measurements shown inFig. S5. In the evolution of the PL spectra in electric field, the char-acteristic anti-crossing behaviour when quadrupolar states approachthe A1s exciton can be observed, together with the red shift of the A2semission.DiscussionThe origin of the couplings J=J0 has so far been attributed to lowdensity effects in the Coulomb Hamiltonian. As demonstrated, theexchange coupling does indeed lead to terms which mix differentdipoles and qualitatively fits well within the picture that J≫J0. This issince J includes both long and short range electron-hole exchange,whereas the short-range interaction is symmetry forbidden in J0 due tothe mixing of different valleys (see expression for HQC in Eq. (S8) inSI)64. For intralayer excitons, this coupling has been calculated to bearound 20meV65 and is expected to be smaller for interlayer excitonsdue to the reduced wave function overlap between the layers. Othereffects such as density dependent dipole–dipole attractionmight playa role in enhancing the mixing between different dipoles. Taking intoaccount the device-to-device variability of the properties of TMD-Fig. 2 | Dipolar exchange interaction in natural MoSe2 bilayer. a Schematic ofthe J coupling between one of the oppositely aligned pairs of singlet IXs (samevalley, different spin) and the interplay with electron tunneling to their respective Aexcitons.b Schematic of the J0 couplingbetweenoneof theoppositely alignedpairsof singlet IXs (different valley, same spin) and the interplay with electron tunnelingto their respective A excitons. c Schematic of the real space interaction of the fourdegenerate singlet IXs via J and J0 couplings. d Scheme of IX exciton species energyas a function of the electric field for the singlet (↑↑, ↓↓) and triplet (↑↓, ↓↑) IXspecies without ~J coupling. e including the J coupling term IX singlet states mixforming quadrupole branches Q and Q0. f exciton landscape including both J and J0coupling which splits degeneracy of Q and Q0 branches. The dashed lines in e andf correspond to spin-singlet states in the non-interacting picture d, from which thequadrupolar branches formed.Article https://doi.org/10.1038/s41467-025-56586-3Nature Communications |         (2025) 16:1382 4www.nature.com/naturecommunicationsbased devices, the clear experimental signatures of exciton quadru-pole formation, together with the overall good qualitative agreementwith theory, indicate that the coupling between the different dipolesaremuch stronger than previously thought. Our effective Hamiltoniansuccessfully explains the evolution of the exciton landscape in bilayerMoSe2. The formation of a quadrupolar state is driven by the dipolarexchange interaction between interlayer spin-singlet states. At thesame time, the interlayer triplet states preserve their dipolar character,shifting linearly in the electric field. In addition, the hole and electrontunneling are responsible for the observed avoided crossing beha-viour, and hybridization of the states.In summary, we have investigated the evolution of the excitonenergy landscape under external electric field in natural MoSe2homobilayers. Notably, for the first time, we observe quadrupolarexciton states in a natural MoSe2 bilayer. These excitonic transitions,characterized by nonlinear shift in an electric field, exhibit a muchstronger dipolar exchange interaction than the one observed in het-erotrilayers. Our experimental observations are accurately capturedby the proposedmany-particle effective Hamiltonian.We propose thatdipolar excitons are characterized by spin-triplet configuration, whilequadrupolar states emerge from the exchange coupling of the inter-layer spin-singlet excitons. Moreover, our model highlights theimportance of hole and electron tunneling for understanding theexciton landscape evolution under the electric field.Our research underscores the potential of MoSe2 bilayers toserve as a field-tunable exciton playground, wherein the mutualinteraction of exciton states facilitates the effective tuning of theirspatial and electric multipole characteristics via electric fields.Therefore, we show that natural MoSe2 bilayers display potential tobe considered as a solid-state platform to study many-body physicsdriven by field-tunable electric multipolar interactions. The inherentrobustness of a homobilayer as compared to layer-by-layer stackingof TMD heterobi- or trilayers makes the platform proposed hereeasier to incorporate reliably into devices. This stems from the factthat homobilayers are not prone to imperfect flake alignment, flakerearrangement during deposition and possible post-stacking sur-face reconstruction for lattice commensurate stacks, which una-voidably plague other heterostructures.Fig. 3 | Extended exciton energy landscape under electric field. a Second deri-vative of reflectivity without electric field. b False-color map of 2nd derivative ofreflectivity as a function of applied gate voltages (left axis) and the correspondingEz (right axis). c Simulated exciton energy landscape in electric fieldwhich includesthe hole and electron tunneling as well as the coupling ~J that forms thequadrupoles. The color scale corresponds to the spatial character of the excitonsi.e. intralayer (blue), hybrid (purple) or interlayer (red). The opaque (semi-trans-parent) lines correspond to the spin-singlet (spin-triplet) states. The bottom energyscales (E − E0) in panels a and b are shown with respect to the A1s exciton energyE0 = 1.606eV.Article https://doi.org/10.1038/s41467-025-56586-3Nature Communications |         (2025) 16:1382 5www.nature.com/naturecommunicationsMethodsSample fabricationThe sample was fabricated using mechanically exfoliated flakes andstacked one by one using the dry transfer method. Each step of layerdepositionwas followedby annealing in ambient conditions, by rampingthe temperature from 100° to 150° for the duration of ~15min. At thefinal step, the sample was annealed for ~15min at 200°. The goal of theannealing was to remove or coagulate air bubbles that notoriously formin TMD stacks during dry transfer deposition. The MoSe2 bilayer isencapsulated with hBN and sandwiched in between few layer graphite(FLG) layers acting as the bottom and top gates. Two additional FLGlayers are connected directly to (physically touching) the MoSe2 bilayerserving as the grounding contacts, one as a spare contact. All FLGcontacts overlap the nearby evaporated gold paths through which thevoltage is applied. The gold pads are pre-deposited on the SiO2 sub-strate. Our sample design does not require any additional lithographyafter the stack is transferred on the substrate, which minimizes the riskof introducing defects and lower the optical quality of the sample.MeasurementsThe sample is wire bonded in a chip carrier installed in a custom-madeelectrical adapter for the cold finger inside a helium flow cryostat. Allpresented measurements were performed at cryogenic temperaturesof ~ 5 K.We characterized the influence of the applied gate voltages at thebottom (VBG) and top (VTG) gates. To adjust for theunequal thicknessesof the bottom and top insulating layers of hBN, we found the optimalgate voltage ratio which minimizes the effect of free carrier dopingduring electric field sweep to be VBG = � 12VTG. To make sure that wekeep themost neutral doping level we checked also the ratio of neutralto charged exciton (trion) by applying gate voltages of the samepolarity (VBG =12VTG).We found theVBG =12VTG =0V tobe theoptimalinitial voltages due to the highest ratio of neutral to charged exciton,both in PL and Reflectivity (Supplementary Fig. S6). The reflectivitymeasurements were performed using a Tungsten-Halogen white lightsource, while PL used a 532 nm continuouswave laser at ~1mWpower.Data analysisThe details of the analysis of the reflectivity spectra are described insection III of the Supplementary Information. Fig. S9 shows the schemeof the data processing. The comparison of the reflectivity spectra inthe form of R/R0 and its 1st and 2nd derivatives are shown in Fig. S10.The strength of the applied electric field was calculated bymatching the dipole moments of the measured and simulated spin-singlet interlayer exciton for low electric field/gate voltages, far fromthe crossing region.Theoretical modelExciton energies were modelled using an effective many-particle the-ory based on the density matrix formalism and input from densityfunctional theory57. A two-particle tunnelingHamiltonian is formulatedand the excitonic response to the electric field is included to firstorder53. The exchange interaction giving rise to the quadropole for-mation is included from the low-density Coulomb Hamiltonian andmatching the coupling strength to the experiment.Data availabilityThe experimental and theoretical datasets generated and/or analysedduring this study are available at https://doi.org/10.5281/zenodo.14584540.References1. Chernikov, A. et al. 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Nonanalyticity, valley quantumphases, and lightlike exciton dispersion in monolayer transitionmetal dichalcogenides: Theory and first-principles calculations.Physical review letters 115, 176801 (2015).AcknowledgementsJ.J. acknowledges funding from the National Science Centre Polandwithin the Preludium Bis 1 (2019/35/O/ST3/02162) program. M.B.acknowledges funding from the National Science Centre Poland withinthe Sonata Bis (2020/38/E/ST3/00194) program and OPUS LAP (2021/43/I/ST3/01357). All authors thank Marzia Cuccu and Sophia Terres fortheir support in the laboratory work. P.P acknowledge supportedthrough the EUR grant NanoX no. ANR-17-EURE-0009 in the frameworkof the “Programme des Investissements d’Avenir”. The Marburg group(S.B. and E.M) acknowledges funding from the Deutsche For-schungsgemeinschaft (DFG) via SFB 1083 (project B9). K.W. and T.T.acknowledge support from the JSPS KAKENHI (Grant Numbers21H05233 and 23H02052) and World Premier International ResearchCenter Initiative (WPI), MEXT, Japan. A.C. and E.W. gratefully acknowl-edge funding from the Deutsche Forschungsgemeinschaft via SPP2244grant (Project-ID: 443405595) and the Würzburg-Dresden Cluster ofExcellence on Complexity and Topology in Quantum Matter (ct.qmat)(EXC 2147, Project-ID 390858490). E.M. and A.C. acknowledge DFGfunding via project 542873285.Author contributionsJ.J. has carried out all optical experiments and drafted the text andfigures of the main manuscript and the supplementary information.J.H. and S.B. under the supervision of E.M. developed the theoreticalmodel and performed the simulations. E.W. and A.C. have providedthe necessary training and participated in the fabrication of electricaldevices and electrical measurements. T.T. and K.W. have provided thehigh-quality hexagonal boron nitride for encapsulation of the sample.N.B. has been involved in the optimization of the experimental setupfor electrical measurements and involved in those measurements.M.D.A.S., D.K.M. contributed to data analysis, interpretation of resultsArticle https://doi.org/10.1038/s41467-025-56586-3Nature Communications |         (2025) 16:1382 7www.nature.com/naturecommunicationsand manuscript preparation. M.B. and P.P. have proposed the goal ofthe scientific inquiry, the methodology and refined the manuscripttext and the interpretation of the experimental results.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-025-56586-3.Correspondence and requests for materials should be addressed toPaulina Plochocka.Peer review information Nature Communications thanks the anon-ymous reviewer(s) for their contribution to thepeer reviewof thiswork. Apeer review file is available.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard to jur-isdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in any medium or format, aslong as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons licence, and indicate ifchanges were made. The images or other third party material in thisarticle are included in the article’s Creative Commons licence, unlessindicated otherwise in a credit line to the material. If material is notincluded in the article’s Creative Commons licence and your intendeduse is not permitted by statutory regulation or exceeds the permitteduse, you will need to obtain permission directly from the copyrightholder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2025Article https://doi.org/10.1038/s41467-025-56586-3Nature Communications |         (2025) 16:1382 8https://doi.org/10.1038/s41467-025-56586-3http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/www.nature.com/naturecommunications Quadrupolar excitons in MoSe2 bilayers Results and Discussion Observation of quadrupolar excitons Microscopic model Charge tunneling Discussion Methods Sample fabrication Measurements Data analysis Theoretical model Data availability References Acknowledgements Author contributions Competing interests Additional information