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Charalambos Louca, Armando Genco, Salvatore Chiavazzo, Thomas P. Lyons, Sam Randerson, Chiara Trovatello, Peter Claronino, Rahul Jayaprakash, Xuerong Hu, James Howarth, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Stefano Dal Conte, Roman Gorbachev, David G. Lidzey, Giulio Cerullo, Oleksandr Kyriienko, Alexander I. Tartakovskii

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[Interspecies exciton interactions lead to enhanced nonlinearity of dipolar excitons and polaritons in MoS2 homobilayers](https://mdr.nims.go.jp/datasets/beee4a30-f1c1-4017-af3d-1c90c22ce70b)

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Interspecies exciton interactions lead to enhanced nonlinearity of dipolar excitons and polaritons in MoS2 homobilayersArticle https://doi.org/10.1038/s41467-023-39358-9Interspecies exciton interactions lead toenhanced nonlinearity of dipolar excitonsand polaritons in MoS2 homobilayersCharalambos Louca 1,11 , Armando Genco 2,11 , Salvatore Chiavazzo3,Thomas P. Lyons 1,4, Sam Randerson 1, Chiara Trovatello 2,5,Peter Claronino1,6, Rahul Jayaprakash 1, Xuerong Hu1, James Howarth 7,8,Kenji Watanabe 9, Takashi Taniguchi 10, Stefano Dal Conte 2,Roman Gorbachev 7,8, David G. Lidzey 1, Giulio Cerullo 2,Oleksandr Kyriienko 3 & Alexander I. Tartakovskii 1Nonlinear interactions between excitons strongly coupled to light are key foraccessing quantum many-body phenomena in polariton systems. Atomically-thin two-dimensional semiconductors provide an attractive platform forstrong light-matter coupling owing to many controllable excitonic degrees offreedom. Among these, the recently emerged exciton hybridization opensaccess to unexplored excitonic species, with a promise of enhanced interac-tions. Here, we employ hybridized interlayer excitons (hIX) in bilayer MoS2 toachieve highly nonlinear excitonic and polaritonic effects. Such interlayerexcitons possess an out-of-plane electric dipole as well as an unusually largeoscillator strength allowing observation of dipolar polaritons (dipolaritons) inbilayers in opticalmicrocavities. Compared to excitons and polaritons inMoS2monolayers, both hIX and dipolaritons exhibit ≈ 8 times higher nonlinearity,which is further strongly enhanced when hIX and intralayer excitons, sharingthe same valence band, are excited simultaneously. This provides access to anunusual nonlinear regime which we describe theoretically as a mixed effect ofPauli exclusion and exciton-exciton interactions enabled through chargetunnelling. The presented insight into many-body interactions provides newtools for accessing few-polariton quantum correlations.Excitons in two-dimensional transition metal dichalcogenides (TMDs)have large oscillator strengths and binding energies1, making themattractive as a platform for studies of strong light-matter coupling inoptical microcavities2–5. A variety of polaritonic states have been rea-lised using monolayers of MX2 (M=Mo, W; X=S, Se) embedded intunable3,5–7 and monolithic microcavities8–12.One of the central research themes in polaritonics is the study ofnonlinear interactions leading to extremely rich phenomena suchas Bose-Einstein condensation13,14, polariton lasing15,16 or optical para-metric amplification17. Polaritons formed from tightly bound neutralintralayer excitons in TMDs are not expected to show strong non-linearity. However, pronounced nonlinear behaviour was observed fortrion polaritons7,18 and Rydberg polaritons19. Enhanced nonlinearitycan be achieved by employing excitonic states with a physicallyseparated electron and hole, e.g. in adjacent atomic layers20 or quan-tum wells21–25. Such interlayer excitons have a large out-of-planeelectric dipole moment, and thus can strongly mutually interact26.Typically, however, interlayer or ‘spatially indirect’ excitons possessReceived: 11 May 2023Accepted: 9 June 2023Check for updatesA full list of affiliations appears at the end of the paper. e-mail: charalambos.louca@polimi.it; armando.genco@polimi.it; a.tartakovskii@sheffield.ac.ukNature Communications |         (2023) 14:3818 11234567890():,;1234567890():,;http://orcid.org/0000-0002-1122-3133http://orcid.org/0000-0002-1122-3133http://orcid.org/0000-0002-1122-3133http://orcid.org/0000-0002-1122-3133http://orcid.org/0000-0002-1122-3133http://orcid.org/0000-0002-1292-2614http://orcid.org/0000-0002-1292-2614http://orcid.org/0000-0002-1292-2614http://orcid.org/0000-0002-1292-2614http://orcid.org/0000-0002-1292-2614http://orcid.org/0000-0001-5569-7851http://orcid.org/0000-0001-5569-7851http://orcid.org/0000-0001-5569-7851http://orcid.org/0000-0001-5569-7851http://orcid.org/0000-0001-5569-7851http://orcid.org/0000-0001-7993-8218http://orcid.org/0000-0001-7993-8218http://orcid.org/0000-0001-7993-8218http://orcid.org/0000-0001-7993-8218http://orcid.org/0000-0001-7993-8218http://orcid.org/0000-0002-8150-9743http://orcid.org/0000-0002-8150-9743http://orcid.org/0000-0002-8150-9743http://orcid.org/0000-0002-8150-9743http://orcid.org/0000-0002-8150-9743http://orcid.org/0000-0002-2021-1601http://orcid.org/0000-0002-2021-1601http://orcid.org/0000-0002-2021-1601http://orcid.org/0000-0002-2021-1601http://orcid.org/0000-0002-2021-1601http://orcid.org/0000-0001-7434-9940http://orcid.org/0000-0001-7434-9940http://orcid.org/0000-0001-7434-9940http://orcid.org/0000-0001-7434-9940http://orcid.org/0000-0001-7434-9940http://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/0000-0002-1467-3105http://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/0000-0001-8582-3185http://orcid.org/0000-0001-8582-3185http://orcid.org/0000-0001-8582-3185http://orcid.org/0000-0001-8582-3185http://orcid.org/0000-0001-8582-3185http://orcid.org/0000-0003-3604-5617http://orcid.org/0000-0003-3604-5617http://orcid.org/0000-0003-3604-5617http://orcid.org/0000-0003-3604-5617http://orcid.org/0000-0003-3604-5617http://orcid.org/0000-0002-8558-1160http://orcid.org/0000-0002-8558-1160http://orcid.org/0000-0002-8558-1160http://orcid.org/0000-0002-8558-1160http://orcid.org/0000-0002-8558-1160http://orcid.org/0000-0002-9534-2702http://orcid.org/0000-0002-9534-2702http://orcid.org/0000-0002-9534-2702http://orcid.org/0000-0002-9534-2702http://orcid.org/0000-0002-9534-2702http://orcid.org/0000-0002-6259-6570http://orcid.org/0000-0002-6259-6570http://orcid.org/0000-0002-6259-6570http://orcid.org/0000-0002-6259-6570http://orcid.org/0000-0002-6259-6570http://orcid.org/0000-0002-4169-5510http://orcid.org/0000-0002-4169-5510http://orcid.org/0000-0002-4169-5510http://orcid.org/0000-0002-4169-5510http://orcid.org/0000-0002-4169-5510http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-39358-9&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-39358-9&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-39358-9&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-39358-9&domain=pdfmailto:charalambos.louca@polimi.itmailto:armando.genco@polimi.itmailto:a.tartakovskii@sheffield.ac.uklow oscillator strength20,27. Thus, in order to strongly couple to cavityphotons, hybridization with high-oscillator-strength intralayer exci-tons is required11,22,24,25,28.An attractive approach for realization of dipolar excitons andpolaritons is to employ the recently discovered exciton hybridizationin MoS2 bilayers29,30. This approach allows realization of uniformsamples suitable for the observation of macroscopic many-bodyphenomena31. Interlayer excitons unique to bilayer MoS2 possess alarge oscillator strength, comparable to that of the intralayer exciton,arising from interlayer hybridization of valence band states, aided by afavourable orbital overlap and a relatively small spin-orbit splittingamong semiconducting TMDs29. Such hybridized interlayer excitons(hIX) are highly tunable using out-of-plane electric field32,33 and theirvalley degree of freedom persists up to room temperature34.Here we use hIXs in bilayer MoS2 to realize highly nonlinearexcitonic and dipolaritonic effects. We unravel a previously unex-plored interaction regime involving intra- and interlayer excitonsstemming from the fermionic nature of the charge carriers in a valenceband shared between different excitonic species. This regime, acces-sible using broadband excitation resonant with both hIX and intralayerexciton transitions, provides strong (up to 10 times) enhancement ofthe exciton nonlinearity, already enhanced by up to 8 times in MoS2bilayers compared with monolayers. We support our experimentalfindings with a theoretical discussion of nonlinear contributions toenergy shifts and exciton broadening, which become enabled in thepresence of charge tunnelling.ResultsInter- and intra-layer exciton hybridization in bilayer MoS2Our heterostructure samples consists of a MoS2 bilayer (BL) sand-wiched between hBN and placed on a distributed Bragg reflector(DBR). Figure 1a shows a bright field microscope image of the encap-sulated BLMoS2. A sketch of the side view of the device is displayed inFig. 1b. The reflectance contrast (RC) spectrum of the studied MoS2bilayer, displayed in Fig. 1c, shows three peaks: the intralayer neutralexcitonsXA at at 1.937 eV (see Fig. 1d), hybridized interlayer excitonhIXat 2.004 eV and hybridized B-exciton at 2.113 eV. Due to the quantumtunnelling of holes, B-excitons hybridize with an interlayer exciton (IX)(Fig. 1d), which is a direct transition in the bilayer momentum space29.The ratio of the integrated intensities of XA and hIX is 4.5. Based onthese data, we estimate the electron-hole separation d =0.55 nm (seedetails in Supplementary Note S1) in agreement with previousstudies34.We further confirm the natureof the hIX states byplacing theBLMoS2 inmagneticfieldwhere the valley degeneracy is lifted (Fig. 1e).In agreement with recent studies33,35, we measure a Zeeman splittingwith an opposite sign and larger magnitude in hIX compared with XA(−3.5 versus 1.5 meV at 8 Tesla).Hybridised interlayer excitons in the strong-coupling regimeWe study the strong coupling regime in a tunable planar microcavity(Fig. 2a) formed by a silver mirror and a planar DBR3. RC scans as afunction of the cavitymode detuningΔ = Ecav − Eexc, where Ecav and Eexcare the cavity mode and the corresponding exciton energy, respec-tively, are shown in Fig. 2b, c. Characteristic anticrossings of the cavitymode with XA and hIX are observed, resulting in lower, middle andupper polariton branches (LPB, MPB, and UPB, respectively). Theextracted Rabi splittings areΩXA= 38 meV for XA andΩhIX = 19meV forhIX (Supplementary Note S2). Figure 2d shows the RC spectra in thevicinity of the anticrossing with hIX, providing a more detailed view ofthe formation of theMPB andUPB. The intensity of the polaritonpeaksis relatively low for the states with a high exciton fraction at positive(negative) cavity detunings for the MPB (UPB). As the Rabi splittingscales as a square root of the oscillator strength, the ratioΩXA=ΩhIX = 2is in a good agreement with the RC data for integrated intensities of XAand hIX. From the Rabi splitting ratio we can estimate the tunnellingconstant J leading to the B-exciton hybridization. The correspondingcoefficient is J = 48 meV (see Supplementary Note S1 for details),matching the density functional theory predictions29. In polarization-1.90 1.95 2.00Energy (eV)0.00.20.40.6Refl. Contr. (arb. units) σ −σ +1.9 2.0 2.1Energy (eV)0.00.10.20.30.40.50.6Reflectance Contrast (arb. units)10 μma cedLayer 1IXXA XBLayer 2hBNMoS2hBNBLMoS2BottomhBNTop hBNT: 4KXAhIXhXBB: 8TXAhIXRC =Rsub - RBLRsubX*AbFig. 1 | Homobilayer MoS2 and its optical response. a Bright field microscopeimage of an encapsulated BL MoS2 transferred on top of a DBR. b Schematic side-view of the fabricated heterostructure comprising a BLMoS2 sandwiched betweenfew-layer hBN. cReflectance contrast (RC) spectrumof the samplemeasured at lowtemperature (4 K) showing three distinct absorption features at 1.937 eV, 2.004 eVand 2.113 eV for XA, hIX and hXB, respectively. Themeasured linewidths for XA, hIX,and hXB are 20, 23 and 64 meV, respectively. RC is calculated using the formula inthe top-right corner of the graph. d Sketch of the conduction and valence bands intwo adjacent layers of MoS2, displaying the allowed optical transitions of A and Bdirect intralayer excitons (XA and XB) and interlayer excitons (IX) for spin-up states(black lines) at the K point in the bilayer momentum space. IX hybridizes with XBthrough the hole tunnelling between the two layers (red dashed arrow). At the K'point of the bilayer Brillouin zone, the same configuration applies for the stateswith the opposite spins. e RC spectra of excitons in BL MoS2 detected in twocircular polarizations in an out-of-plane magnetic field of 8 T at T=4 K. Zeemanshifts of opposite signs are observed for XA and hIX. The absorption peak of thecharged intralayer exciton (X*A) shows near unity circular polarization.Article https://doi.org/10.1038/s41467-023-39358-9Nature Communications |         (2023) 14:3818 2resolved cavity scans in an out-of-plane magnetic field (Fig. 2e, f),similarly to hIX behaviour, we observe opposite and larger Zeemansplitting for dipolaritons relative to the intralayer polaritons (seeSupplementary Fig. S4). Chiral dipolariton states are observed dis-tinguished by their opposite circular polarization (Fig. 2f).Nonlinear properties of hybridized interlayer excitonsWe investigate the nonlinear response of XA and hIX in the bare BLflake performing fluence-dependent reflectivity experiments, illumi-nating the sample with ultrashort (≈150 fs) pump pulses (in a singlebeam experiment) in both narrow band (NB, full-width at half max-imum, FWHM = 28nm) and broad-band (BB, FWHM=50nm) config-urations (see “Methods”). Our separate resonant pump-probeexperiments have confirmed that the lifetimes of the hIX and XAstates are considerably longer than the pulse duration of ≈ 150 fs(Supplementary Note S3). Measured RC spectra are shown in Fig. 3a, bfor theNB and in Fig. 3c for BB excitation. In theNB case, the excitationwas tuned to excite either XA or hIX independently, while in the BBcase, both resonances were excited simultaneously.As seen in Fig. 3a, b both XA and hIX spectra behave similarly uponincreasing the power of the NB excitation: a blueshift of several meV isobserved, accompaniedby thepeakbroadening andbleaching. For theBB excitation, however, a different nonlinear behaviour is observed asshown in Fig. 3c: the broadening and complete suppression of the hIXpeak is observed at much lower powers, accompanied by a redshift.This is in contrast to XA, whose behaviour is similar under the twoexcitation regimes.The resulting energy shifts and intensities are shown in Fig. 3d as afunction of the exciton density (see details in Supplementary Note S4and S5). Figure 3d quantifies the trends observed in Fig. 3a–c showingfor the BB excitation an abrupt bleaching of the hIX peak above the hIXdensity 5 × 103 μm−2 accompaniedby a redshift of ≈ 4meVanda 12meVbroadening (see Supplementary Note S6). For the NB case, a similardecrease in peak intensity is observed only around 4 × 104 μm−2,accompanied with a peak blueshift of ≈ 7 meV and a broadeningexceeding 15 meV (see Supplementary Figure S10). However, it isapparent that the observed behaviour under the two excitationregimes is similar for XA. A similar blueshift, broadening and saturationa bce fdMPBLPBXAΩXA=38 meV1.9751.9501.9251.900-100 -75 -50Cavity Detuning (meV)RC (arb. units)UPBMPBhIXT: 4KΩ IX=19 meVh2.042.022.001.98Energy(eV)0 10 20 30-10-20Reflectance Contr . (arb. units)101.98 2.00 2.021.982.002.028T UPBMPBhIXCavCavity Mode Energy(eV)Energy(eV)σ +σ -1.98 2.00 2.020.000.050.100.15 8Tσ +σ -Energy(eV)Refl. Contr. (arb. units)UPBMPBhIXIncreasingcavitydetuning1.963 2.000 2.035Energy (eV)0.20.40.60.81.01.21.4Reflectance Contrast (arb. units)01Energy(eV)Silver mirrorhBN/BL MoS2/hBNΔL ΩR 12x SiO2/Nb2O5Piezo CrystalPiezo movementsCavity Detuning (meV)Fig. 2 | Strong exciton-photon coupling in MoS2 bilayers. a Schematics of thetunable open microcavity composed of a bottom DBR and a top semi-transparentsilver mirror. b, c Low temperature (4K) RC spectra measured as a function of thecavity-exciton detuning (Δ = Ecav − Eexc) for cavity scans across b XA and c hIXenergies. White dotted lines show the fitting obtained using the coupled oscillatormodel providing the Rabi splittings ΩhIX = 19 meV and ΩXA= 38 meV. d RC spectrameasured for the cavity-exciton detunings in the vicinity of the anticrossingbetween hIX and the cavity mode. e Dipolariton dispersion measured with circu-larly polarized detection for 8 T magnetic field. The orange and black solid curvesare the coupled oscillator model fits for σ+ and σ− detection, respectively. Thepositions of the Zeeman-split hIX peaks are shownbydashed lines. f σ+ (orange) andσ− (black)RC spectrameasured at 8 T at the hIX-cavity anticrossing. Fittingwith twoLorentzians (solid lines) is shown.Article https://doi.org/10.1038/s41467-023-39358-9Nature Communications |         (2023) 14:3818 3areobserved at slightly higher densities compared to the hIXunder theNB excitation (Supplementary Note S7). We also find that due to theincreased excitonic Bohr radius, the onset of the nonlinear behaviourfor XA in bilayers occurs at a lower exciton density than for XA inmonolayers (Supplementary Note S8).We develop a microscopic model to describe the contrastingphenomena under the NB and BB excitation. Under the NB excitation,either XA or hIX excitons are created as sketched in Fig. 3e. In this case,nonlinearity arises fromCoulomb exciton-exciton interactions causingthe blueshift and dephasing36. For simplicity, in the main text we willuse a Coulomb potential VCoul combining the exchange and directterms further detailed in Supplementary Note S9. We confirm that forthe intralayer exciton-exciton interaction (XA-XA) the dominant non-linear contribution comes from the Coulomb exchange processes, asin the monolayer case36,37, while for the hIX-hIX scattering the domi-nant contribution is from the direct Coulomb (dipole-dipole) interac-tion terms22. We find that for the modest electron-hole separationd =0.55 nm in the bilayer, VCoul is overall 2.3 times stronger for hIXcompared with XA. For XA and hIX of the same dipole orientation theCoulomb interaction is repulsive, corresponding to the experimentallyobserved blueshifts. At the same time, opposite dipole orientationleads to the negative energy shifts, and the total contribution ofinterlayer scattering depends on possible asymmetry in the system(Supplementary Note 11).Analysing the shapes of the reflectance spectra in the NB case, wenote that they depend on the rates of radiative (ΓR) and non-radiative(ΓNR) processes. The area under RC curves is described by the ratio ΓR/(ΓR + ΓNR). This ratio changes under the increased excitation if the ratesdepend on the exciton densities. Specifically, we account for thescattering-induced non-radiative processes thatmicroscopically scale asΓNR∝ ∣VCoul∣2n, i.e. depend on the absolute value of the combinedmatrixelements for the Coulomb interactions and the exciton density n36. Thisprocess allows reproducing the RC behaviour and bleaching at increas-ing pump intensity. Moreover, it explains stronger nonlinearity for XA inbilayers compared tomonolayers. Namely, the scattering scales with theexciton Bohr radius, VCoul∝α, which is larger in the bilayers due to theenhanced screening (Supplementary Note S9). We note that the fullanalysis of the observed behaviour should take into account the inho-mogeneous broadening, the effect of which nonetheless can beneglected for the qualitative analysis presented in the paper.1.95 2.00Energy(eV)Reflectance Contrast (arb. units)bBBa dfeLayer 1 Layer 2XANB excitation BB excitationLayer 1 Layer 2ElectronsHolesTunnelling-enabled phase space fillinghIXhIXchIXXAhIX32.5μJ/cm20.1 μJ cm232.5μJ/cm20.1 μJ cm247.8 μJ/cm20.3 μJ cm2XABBNBBBNBExciton Density (μm-2) 529.1 1.950Energy(eV)Reflectance Contrast (arb. units)2.000 2.025Energy(eV)10 104−50510201 01 3 1040.00.51.0ΔE (meV)RC/RC0(arb. units)Exciton Density (μm-2)hIXAXhIX XABBNBBBNBNB NB3Fig. 3 | Exciton nonlinearity in MoS2 bilayers. RC spectra measured with the NB(FWHM=28 nm) excitation for the XA (a) and hIX (b), and with the BB (FWHM=50nm) excitation (c) at different fluences. The dashed curves are guide for the eye.d The energy shift ΔE (top) and normalized integrated RC intensity (bottom) as afunctionof the excitondensity for the hIX (left) andXA (right). Solid (open) symbolsshow the results for the BB (NB) excitation. For the normalized RC we divide thespectrally integrated RC at each laser fluence by its maximum value measured forthe whole power dependence. Schematic diagram showing exciton generationunder the NB (e) and BB (f) excitation. In (e) only generation of hIX is shown. In (f),the holes of the two excitonic species share the same valence band.Article https://doi.org/10.1038/s41467-023-39358-9Nature Communications |         (2023) 14:3818 4In the BB case, both XA and hIX excitons are generated simulta-neously, and together with intraspecies scattering (XA-XA and hIX-hIX),interspecies scattering (XA-hIX) occurs, similarly to the direct-indirectexciton Coulomb scattering in double quantum wells38. Since XA andhIX are formed by the holes from the same valence band (Fig. 3f), anadditional contribution arises from the phase space filling, i.e. thecommutation relations for the excitons (composite bosons) start todeviate from the ideal weak-density limit once more particles arecreated39. For particles of the same flavour, the phase space fillingenables nonlinear saturation effects in the strong coupling regime,similar to polariton saturation observed in Ref. 18. However, in thepresence of several exciton species, we reveal a distinct mixture ofthe phase space filling and inter-exciton interactions which we term thetunnelling-enabled nonlinearity. Specifically, we note that the commu-tator of theXA annihilationoperator (X̂) andhIX creationoperator (̂Iy) isnon-zero, ½X̂ ðpÞ,̂IyðqÞ�= � B̂p,q, meaning that modes are not indepen-dent. Here p,q are exciton momenta and B̂p,q is an operator denotingthe deviation from the ideal commuting case (B̂p,q =0) of distinctbosons where holes do not compete for the valence band space.This statistical property of modes that share a hole, andexperience phase space filling, has consequences for the nonlinearresponse. Namely, the total energy is evaluated as an expectationvalue over a many-body state with both XA and hIX excitons,∣NX,NhIX�: = ðQNXp X̂yÞðQNhIXq ÎyÞ∣Ω0�, where NX and NhIX particles arecreated from the ground state ∣Ω0�. If the excitonic modes areindependent, the contributions from XA and hIX simply add up.However, the hole coexistence in the valence band induces theexcitonic interspecies scattering. The phase space filling then affectsboth Coulomb-related processes, radiative lifetime of excitonicmodes, and scattering amplitudes, which can lead to redshifts forhIX (see Supplementary Notes S9–S11). The interplay of these effectsis unique to bilayers as tunnelling is absent in monolayers. More-over, in contrast to standard epitaxially-grown double quantumwells where the layers are separated up to 10 nm, much smallerinterlayer separation in TMD bilayers (0.5 nm) allows for strongertunnelling-enabled nonlinearity.We note that the magnitude of shifts depends on the modepopulations as well as interaction constants. For instance, the redshiftfor XA scales as ΔEXA= gD�InhIX. The population of hIX is much smallerthan XA for a given power and the imbalance grows further as thefluence is increased (see Supplementary Note S12). This can explainthat the redshift induced by tunnelling-enabled nonlinear processesfor XA is less apparent than for hIX under BB illumination. At the sametime, the shifts for hIX are proportional to the density of intralayerexcitons, nXA, and thus the effect of the BB excitation shall be mostpronounced for the interlayer mode. In addition, hIX-hIX scatteringplays a significant role for increasing nonradiative rates. In this case,interactions of both repulsive and attractive sign lead to enhancedscattering channels (irrespective of dipole orientation), and thusresults into RC spectra bleaching at lower hIX exciton densities (Sup-plementary Note S9). Using the estimated tunnelling-enabled non-linearity coefficients, we model the RC in the BB regime andqualitatively reproduce the strong bleaching and redshift for hIX at theincreased density (see Supplementary Fig. S13).We observe the same BB-enhanced tunnelling-enabled non-linearity in samples with lower inhomogeneous broadening (Supple-mentary Note S13), confirming that our findings are not masked bydisorder. In our analysis on the excitonic nonlinearities, we neglect theeffect of trions and biexcitons40,41, since the natural doping is relativelysmall in our sample (see Fig. 1c, e) and the excitation densities requiredto generate a substantial population of biexcitons are considerablyhigher than those used in our experiments42–44. Moreover, we observea redshift only for interlayer excitons and specifically under broadbandexcitation, while an excess of charges would have produced the sameeffect on both intra and interlayer excitons. In fact, the interspeciesinteractions in our study are mediated through charge tunnelling andcannot be explained as a consequence of the presence of many-bodycomplexes.Nonlinear behaviour of dipolar polaritonsWe investigate nonlinear properties of dipolar polaritons in a mono-lithic (fixed-length) cavity created by a silver mirror on top of a PMMAspacer (245 nm thick) covering the hBN-encapsulated MoS2 homo-bilayer placed on the DBR. The cavity mode energy can be tuned byvarying the angle of observation (0 degrees corresponds to normalincidence). We use a microscopy setup optimized for Fourier-planeimaging, thus allowing simultaneousdetection of reflectivity spectra ina range of angles as shown in (Fig. 4a) displaying the measuredpolariton dispersion. In this experiment, the cavity mode is tunedaroundhIX andonly twopolariton branches LPB andUPBare observedat low fluence of 0.6 μJ cm−2 with a characteristic Rabi splitting of 17.5meV. In Fig. 4b, at an increased fluence of 58.5μJ cm−2, only a weaklycoupled cavity mode is visible.Figure 4c shows RC spectra taken at ~6.5∘ around the anticrossingat different laser fluences. The collapse of the two polariton peaks intoone peak signifying the transition to the weak coupling regime isobserved above 25μJ cm−2. The LPB and UPB energies extracted usingthe coupled oscillator model (Supplementary Fig. S5) are shown in(Fig. 4d). As the polariton density is increased, the LPB and UPBapproach each other almost symmetrically, converging to the excitonenergy. The corresponding normalized Rabi splitting (Ω/Ωmax, whereΩmax is measured at low fluence) is shown in Fig. 4d, e as a function ofthe total polariton density.In this experiment, the cavity mode is considerably above the XAenergy, which therefore is not coupled to the cavity. Hence, theextracted Rabi splittings are fitted with a theoretically predicted trendof Ω for the NB excitation regime (Supplementary Note S9). A non-linear polariton coefficient β =0.86 μeVμm2 is extracted by differ-entiating the fitted function with respect to the polariton density.Comparing our results to XA intralayer-exciton-polaritons in mono-layers in similar cavities11, we observe that the nonlinearity coefficientfor dipolar interlayer polaritons is about an order of magnitude larger.This is in a good agreement with the theoretically predicted intrinsicnonlinearity of hybridized interlayer polaritons (SupplementaryNote S9), and with our experimental data for XA taken on amonolayerembedded in a microcavity (Supplementary Note S14).DiscussionIn summary, we report the nonlinear exciton and exciton-polaritonbehaviour in MoS2 homobilayers, a unique system where hybridizedinterlayer exciton states can be realized having a large oscillatorstrength. We find that nonlinearity in MoS2 bilayers can be enhancedwhen both the intralayer and interlayer states are excited simulta-neously, the regime that qualitatively changes the exciton-excitoninteraction through the tunnelling-enabled nonlinearity. This approachenriches the exciton nonlinear behaviour in atomically thin semi-conductors, leading to unique effects compared to any other semi-conducting system. In this broad-band excitation regime, the bleachingof the hIX absorption occurs at 8 times lower hIX densities compared tothe narrow-band excitation case when the interlayer excitons are gen-eratedon their own. In addition to this, wefind that the dipolar nature ofhIX states in MoS2 homobilayers already results in 10 times strongernonlinearity compared with the intralayer excitons inMoS2monolayers.Thus, we report on an overall enhancement of the exciton nonlinearityby nearly two orders of magnitude.Thanks to the large oscillator strength, hIX can enter the strongcoupling regime inMoS2 bilayers placed inmicrocavities, as realized inour work. Similarly to hIX states themselves, dipolar polaritons alsoshow 10 times stronger nonlinearity comparedwith exciton-polaritonsin MoS2 monolayers. We conclude that dipolaritons in MoS2 bilayersArticle https://doi.org/10.1038/s41467-023-39358-9Nature Communications |         (2023) 14:3818 5uniquely combine strong nonlinearity, large Rabi splitting and fabri-cation reproducibility among other TMD systems studied so far9,11,18(see Supplementary Note 14). We note that in the monolayer samples,the nonlinearity arising from Pauli blocking is only pronounced in thestrong light-matter coupling regime18,45, and is not significant in theweak coupling. On the contrary, for bilayers we discover other non-linear processes that crucially arise due to the presence of interlayertunnelling. This is specific to the broad band (BB) illumination regime,where the simultaneous occupation of intra- and interlayer excitonstates effectively activates this channel of nonlinearity. We expect thatinmicrocavitieswhere the cavitymode is coupled tobothhIXandXA inMoS2 bilayers, and the excitation similar to the broad-band regime canthus be realized, the nonlinear polariton coefficient will be dramati-cally enhanced owing to the tunnelling-enabled nonlinearity effect,allowing highly nonlinear polariton system to be realized. We thuspredict that MoS2 bilayers will be an attractive platform for realizationof quantum-correlated polaritons with applications in polariton logicnetworks46 and polariton blockade47,48.After the submission of this manuscript another article reportingstrongcoupling andnonlineareffects inMoS2bilayerswaspublished49.MethodsSample fabricationThe hBN/MoS2/hBN heterostructures were assembled using a PDMSpolymer stamp method. The PMMA spacer for the monolithic cavitywas deposited using a spin-coating technique, while a silver mirror of45 nm was thermally evaporated on top of it.Optical studiesBroad-band excitation was used to measure the reflectance contrast(RC) spectra of the devices at cryogenic temperatures, defined asRC = (Rsub −RBL)/Rsub, where Rsub and RBL are the substrate and MoS2bilayer reflectivity, respectively. For the magnetic field studies thesame RC measurements were performed using unpolarized light inexcitation and with λ/4 and λ/2 waveplates and a linear polarizer incollection to resolve σ+ and σ− polarized light. The low temperaturemeasurements using the tunable cavity were carried out in a liquidhelium bath cryostat (T = 4.2K) equipped with a superconductingmagnet and free beam optical access. We used a white light LED as asource. RC spectra were measured at each cavity length and wereintegrated over the angles within 5 degrees from normal incidence.TheRC spectrameasured in the cavity arefitted using Lorentzians. Thepeak positions are then used to fit to a coupled oscillator model,producing the Rabi splitting and the exciton and cavitymode energies.The measurements on the monolithic cavity were performed in ahelium flow cryostat (T = 6K). For the power-dependent RC experi-ments, we used supercontinuum radiation produced by 100 fsTi:Sapphire laser pulses at 2 kHz repetition rate at 1.55 eV propagatingthrough a thin sapphire crystal. The supercontinuun radiation wasthen filtered to produce the desired narrow-band excitation.Density calculationsAll the exciton and polariton densities were calculated following theprocedure introduced by L. Zhang et al.11, taking into account thespectral overlap of the spectrum of the excitation laser and theinvestigated exciton peak (see further details in SupplementaryNote S4).Data availabilityThe data that support the findings of this study are available in theMARVEL public repository (MARVEL Materials Cloud Archive: https://archive.materialscloud.org) with the same title as this paper.a1.98 1.99 2.00 2.01 2.02 2.03Energy (eV)−0.3−0.2−0.10.00.10.20.30.40.5Reflectance contrast(arb.units)cbeLPBCavhIX58.5 μJ/cm2d0.6 μJ     cm239 μJ/cm20.6 μJ/cm2UPBhIX1.992.002.012.022.03Energy (eV)1.992.002.012.022.03Energy (eV)0 5 10Angle(°)1.9952.0002.0052.010 LPB UPBEnergy (eV)0.5 51 10 (μJ/cm2) Fluence103 1040.20.40.60.81.0Ω/ΩmaxPolariton density (μm-2)CavRefl. contr. (arb.units.)Refl. contr. (arb.units.)0.550.0--0.550.0--0.550.0--Fig. 4 | Nonlinear behaviour of dipolaritons. a, b Reflectance contrast spectrameasured at different laser fluences for the MoS2 bilayer placed in a monolithiccavity. a The low fluence case (0.6μJ cm−2). A clear anticrossing at 6.5∘ is observed.Dashed red lines show the results of the fitting using a coupled oscillator model,with two polariton branches LPB andUPB formed.White and orange lines show theenergies of the uncoupled cavity mode and hIX state, respectively. The vertical linemarks the anticrossing angle. b The high fluence case (58.5μJ cm−2). A completecollapse of the strong coupling regime is observed, with the disappearance of theanticrossing and transition into the weak coupling regime. c RC spectra measuredat the anticrossing at6.5∘ as a functionof the laser fluence.dMeasuredUPB and LPBpeak energies at 6.5∘ as a function of the laser fluence (see top axis) and the cor-responding polariton density (bottom axis). e Symbols show the Rabi splittingsnormalized by the Rabi splitting measured at the lowest power (Ω/Ωmax) asdeduced from (d). The line shows the fitting using our theoretical model (Sup-plementary Note S9).Article https://doi.org/10.1038/s41467-023-39358-9Nature Communications |         (2023) 14:3818 6https://archive.materialscloud.orghttps://archive.materialscloud.orgReferences1. Wang, G. et al. Colloquium: Excitons in atomically thin transitionmetal dichalcogenides. Rev. Mod. Phys. 90, 21001 (2018).2. Liu, X. et al. Strong light-matter coupling in two-dimensional atomiccrystals. Nat. Photonics 9, 30 (2014).3. 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T.P.L. acknowledgesfinancial support from the EPSRC Doctoral Prize Fellowship scheme.Article https://doi.org/10.1038/s41467-023-39358-9Nature Communications |         (2023) 14:3818 7C.T., S.D.C. and G.C. acknowledge support by the the European Gra-phene Flagship Project under grant agreement 881603. A.G. and G.C.acknowledge support by the European Union Marie Sklodowska-CurieActionsproject ENOSISH2020-MSCA-IF-2020-101029644. P.C., R.J. andD.G.L. thank EPSRC Programme Grant ‘Hybrid Polaritonics’ (EP/M025330/1). K.W. and T.T. acknowledge support from the JSPSKAKENHI (Grant Numbers 20H00354, 21H05233 and 23H02052) andWorld Premier International Research Centre Initiative (WPI), MEXT,Japan. R. G. acknowledges support from Royal Society, ERC Con-solidator grant QTWIST (101001515), EPSRC grant numbers EP/V007033/1, EP/S030719/1 and EP/V026496/1, the European GrapheneFlagship Project (881603) and the European Quantum TechnologyFlagship Project 2DSIPC (820378). O.K. and S.C. acknowledge EPSRCgrants EP/V00171X/1 and EP/X017222/1, and NATO SPS projectMYP.G5860. C.L., A.G., G.C. and A.I.T also acknowledge EuropeanUnion’s Horizon 2020 research and innovation programme under grantagreement no. 654148 Laserlab-Europe.Author contributionsC.L., S.R., J.H., X.H. and R.G. fabricated hBN-encapsulated MoS2samples. K.W. and T.T. synthesized high quality hBN. C.L. and A.G.designed the microcavity samples. P.C., R.J., D.G.L. fabricated themicrocavity samples. C.L., A.G., C.T., T.L., S.R. and S.D.C. carriedout optical spectroscopy experiments. S.C. and O.K. developedtheory. A.G. calculated polariton densities. C.L. and A.G. analyzedthe data with contribution from A.I.T., T.L., S.C., O.K., C.T., S.D.C.and G.C. C.L., A.G., S.C., O.K. and A.I.T. wrote the manuscript withcontribution from all other co-authors. A.I.T., O.K., D.G.L., G.C.managed various aspects of the project. A.I.T. supervised theproject.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-023-39358-9.Correspondence and requests for materials should be addressed toCharalambos Louca, Armando Genco or Alexander I. Tartakovskii.Peer review information Nature Communications thanks the anon-ymous reviewer(s) for their contribution to the peer review of this work.A peer 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 license, and indicate ifchanges were made. The images or other third party material in thisarticle are included in the article’s Creative Commons license, unlessindicated otherwise in a credit line to the material. 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To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 20231Department of Physics and Astronomy, The University of Sheffield, Sheffield S3 7RH, UK. 2Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo daVinci, 32, Milano 20133, Italy. 3Department of Physics, University of Exeter, Stocker Road, Exeter EX4 4PY, UK. 4RIKEN Center for Emergent Matter Science,Wako, Saitama 351-0198, Japan. 5Department of Mechanical Engineering, Columbia University, NY 10027 New York, USA. 6Department of Physics andMathematics, University of Hull, Rober Blackburn Hull HU6 7RX, UK. 7National Graphene Institute, University of Manchester, Manchester, UK. 8Department ofPhysics and Astronomy, University of Manchester, Manchester, UK. 9Research Center for Electronic and Optical Materials, National Institute for MaterialsScience, 1-1 Namiki, Tsukuba 305-0044, Japan. 10Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki,Tsukuba 305-0044, Japan. 11These authors contributed equally: Charalambos Louca, Armando Genco.e-mail: charalambos.louca@polimi.it; armando.genco@polimi.it; a.tartakovskii@sheffield.ac.ukArticle https://doi.org/10.1038/s41467-023-39358-9Nature Communications |         (2023) 14:3818 8https://doi.org/10.1038/s41467-023-39358-9http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/mailto:charalambos.louca@polimi.itmailto:armando.genco@polimi.itmailto:a.tartakovskii@sheffield.ac.uk Interspecies exciton interactions lead to enhanced nonlinearity of dipolar excitons and polaritons in MoS2 homobilayers Results Inter- and intra-layer exciton hybridization in bilayer MoS2 Hybridised interlayer excitons in the strong-coupling regime Nonlinear properties of hybridized interlayer excitons Nonlinear behaviour of dipolar polaritons Discussion Methods Sample fabrication Optical studies Density calculations Data availability References Acknowledgements Author contributions Competing interests Additional information