Quadrupolar excitons and hybridized interlayer Mott insulator in a trilayer moiré superlattice

Transition metal dichalcogenide (TMDC) moiré superlattices, owing to the moiré flatbands and strong correlation, can host periodic electron crystals and fascinating correlated physics. The TMDC heterojunctions in the type-II alignment also enable long-lived interlayer excitons that are promising for correlated bosonic states, while the interaction is dictated by the asymmetry of the heterojunction. Here we demonstrate a new excitonic state, quadrupolar exciton, in a symmetric WSe2-WS2-WSe2 trilayer moiré superlattice. The quadrupolar excitons exhibit a quadratic dependence on the electric field, distinctively different from the linear Stark shift of the dipolar excitons in heterobilayers. This quadrupolar exciton stems from the hybridization of WSe2 valence moiré flatbands. The same mechanism also gives rise to an interlayer Mott insulator state, in which the two WSe2 layers share one hole laterally confined in one moiré unit cell. In contrast, the hole occupation probability in each layer can be continuously tuned via an out-of-plane electric field, reaching 100% in the top or bottom WSe2 under a large electric field, accompanying the transition from quadrupolar excitons to dipolar excitons. Our work demonstrates a trilayer moiré system as a new exciting playground for realizing novel correlated states and engineering quantum phase transitions.


Introduction
Monolayer TMDCs, as atomically thin direct bandgap semiconductors, offer a unique playground to explore novel optoelectronic phenomena 1,2 , especially with the ability to form heterostructures that enable a new range of control knobs. For example, TMDC heterojunctions in a type-II alignment host long-lived interlayer excitons [3][4][5][6] , with electrons and holes residing in different layers 3,4 . These interlayer excitons possess the valley degree of freedom, as well as a large Stark shift due to the permanent dipole moment, rendering them promising candidates as tunable quantum emitters 6 . Recently, anglealigned TMDC moiré superlattices exhibit strong Coulomb interactions in the electronic flatbands, leading to correlated states [7][8][9][10][11][12][13][14][15][16][17][18][19] such as Mott insulator and generalized Wigner crystal 7,8,12,17 . The moiré coupling also gives rise to flat excitonic bands 20-23 that could potentially be utilized to realize correlated bosonic states 24 , such as Bose-Einstein condensation (BEC) and superfluidity [25][26][27] . The interaction between interlayer excitons is dominated by the repulsive force between their permanent dipoles, whose alignment is dictated by the asymmetry of the heterostructure, with electrons and holes separated in two different layers.
In this work, we report a new interlayer quadrupolar exciton in a symmetric TMDC heterostructure: angle-aligned WSe2/WS2/WSe2 trilayer. The interlayer excitons in the top and bottom bilayers have opposite polarities, which restores the symmetry. Their hybridization then forms a superposition state of interlayer excitons, canceling the dipolar moments and giving rise to a quadrupolar exciton, which has been predicted to enable intriguing quantum phase transition 26,28-30 . In the presence of moiré coupling, this hybridization further gives rise to a new type of correlated electronic state, hybridized interlayer Mott insulator, in which the correlated holes are shared between the two WSe2 layers, and the layer population can be continuously tuned by an electric field.

Quadrupolar Exciton in Angle-Aligned WSe 2 /WS 2 /b-WSe 2 Trilayer
The typical device structure is schematically shown in Fig. 1a, which contains three regions of different stackings among the three monolayers: (I) WS2 over the bottom WSe2, which we denote as WS2/b-WSe2; (II) top WSe2 over WS2 (t-WSe2/WS2); and (III) the t-WSe2/WS2/b-WSe2 trilayer. The whole device is gated by the top and bottom gate electrodes made of few-layer graphene (FLG), which provide independent control of the electric field and doping.
In the bilayer regions I and II, the WSe2/WS2 moiré superlattices host both correlated electrons and interlayer excitons due to the type-II band alignment (Fig. 1c). The interlayer excitons, with holes residing in the WSe2 layer and electrons in the WS2 layer (Fig. 1c), interact with the correlated electrons and can be used to read out the transitions at the correlated insulating states 17,[31][32][33][34] . The doping-dependent photoluminescence (PL) spectra in these regions (Figs. 1e and 1f) clearly reveal these features: the interlayer exciton PL peak has a strong intensity at the charge neutrality point (CNP), which decreases quickly upon doping; the PL energy and intensity are also modulated by correlated insulator states such as the Mott insulator states at both n=1 and -1, consistent with the previous studies 12,31 .
In the trilayer region III, we expect quadrupolar excitons as schematically plotted in Fig.  1b. The quadrupolar exciton is the superposition of the two dipolar excitons of opposite polarities through the hybridization of the valence bands in the top and bottom WSe2 layers, which leads to the splitting of valence bands, ∆ ± , as shown in Fig. 1d, similar to the formation of bonding and antibonding states in a double-well system 29 . As a result, the quadrupolar excitons will have two branches: one at lower energy than the dipolar exciton and the other at higher energy, assuming that all have similar binding energies 29 . Fig. 1g plots the PL in this region, which indeed exhibits a major PL resonance at energies below the dipolar excitons in Figs. 1e and 1f. We have not observed any PL resonance corresponding to the higher energy quadrupolar exciton yet, while some devices show high energy exciton PL with different nature that we are going to explore in the future (details in Supplementary Information Section 18). The doping dependence is also drastically different: the intensity of the lower energy PL peak retains upon hole doping and only starts to decay at n=-1 (we will discuss this in more detail later).
The quadrupole nature of the excitons in the trilayer region is confirmed by the electric field dependent PL spectra. In regions I (Fig. 2a) and II (Fig. 2b), the interlayer exciton PL peaks both shift linearly as a function of the out-of-plane electric field but with opposite signs of the slope. The slope is -0.72 • for WS2/b-WSe2 (region I) and 0.66 • for t-WSe2/WS2 (region II), consistent with the previous results 21, [35][36][37][38][39] . In contrast, the PL from the trilayer region III is symmetric about the electric field, and the resonance energy exhibits a quadratic dependence on the electric field, as shown in Figs. 2c and 2d, clearly demonstrating that the trilayer PL is from quadrupolar excitons. The PL resonance energy can be well fitted by a quadrupolar exciton model (orange curves in Fig. 2d, details in Supplementary Information Section 9). It is worth noting that at large electric fields, the quadrupolar exciton approaches the linear Stark shift of dipolar excitons with a slope around 0.7 • (dashed lines), matching what we extracted from the data in the bilayer regions I and II. We further extract the ∆ , the energy difference between dipolar excitons and quadrupolar excitons under net zero electric field, to be about 12 meV from the fitting in Fig. 2d (Supplementary Information Section 9), consistent with the theoretical calculation for a similar trilayer structure (10-30 meV in WSe2/MoSe2/WSe2) 29 . We have also reproduced similar quadrupolar exciton behaviors in other angle-aligned WSe2/WS2/WSe2 devices, which show a ∆ about 30 meV (device D2, Supplementary Information Section 10) and 9 meV (device D1 and D3, Supplementary Information Section 14). We note that the dipolar exciton resonance energies in regions I and II only serve as a guide of the two dipolar excitons involved in forming the quadrupolar excitons due to dielectric environment difference and possible spatial inhomogeneity. The energies of the two dipolar excitons that form quadrupolar excitons in region III can be extracted from the fitting and are similar in values, typically less than 7 meV (detailed discussion in Supplementary Information Section 14). In fact, the electric field dependence of the quadrupolar exciton can be used to extract the energy difference between the two dipolar excitons involved in the hybridization, which dictates the hybridization to occur at a finite electric field that tunes the two dipolar exciton energies into resonance (details in Supplementary Information Section 14). We also want to mention that the higher energy mode of the predicted quadrupolar excitons (asymmetric quadrupolar exciton mode 29 ) is missing in Fig. 2, likely due to the excited state or even dark state nature 40 of the quadrupolar exciton, which leads to the absence of PL.
The quadrupolar excitons show distinctively different power dependence compared with that from dipolar excitons, as shown in Fig. S5. The integrated PL intensity of quadrupolar excitons exhibits more nonlinear dependence than dipolar excitons, likely due to their larger size. In addition, the PL peak blueshifts as a function of the excitation power (Figs. S5b, e) or exciton density (Fig. S6) is smaller for quadrupolar exciton compared with that of dipolar excitons, consistent with our expectation of reduced exciton-exciton repulsion for quadrupolar excitons. The estimation of the exciton density can be found in Supplementary Information Section 16.

Evidence of An Interlayer Mott Insulator
Next, we study the interaction between the quadrupolar exciton and the correlated electrons in the moiré flatlands. We first revisit the doping dependence of the quadrupolar exciton at zero electric field. Here, the filling factor denotes the number of holes per moiré unit cell ("-" sign for holes), the same as those in the moiré bilayer regions I and II. However, since the trilayer consists of two moiré superlattices, both of which can be filled with carriers, we define their individual filling factors as nt and nb, respectively, and the total filling factor n=nt+nb. We focus on the low energy mode of the trilayer quadrupolar exciton and observe two main features in its PL spectra, at n=-1 and n=-2, respectively. At n=-1, the PL peak energy exhibits a kink (Fig. 3b), and the PL intensity drops sharply upon further hole doping (Fig. 3c). At n=-2, the PL energy exhibits a blueshift. These features correspond to the emergence of insulating states, similar to the previous studies 17,[31][32][33][34] .
The behaviors at these two fillings evolve systematically as a function of the external electric field. Since the device structure is symmetric, the observed PL behaviors are also symmetric with respect to the electric field direction. Fig. 3d-3g plot examples of PL spectra at selected negative electric fields (direction definition in Fig. 1a), while detailed data at both electric field directions are available in Supplementary Information Section 8. Note that the labeled values of external electric fields are calculated based on voltages applied on the top and bottom gates (see Methods). The effective electric fields between the top and bottom WSe2 layers will be different due to carrier populations and layer chemical potentials (Supplementary Information Section 11). As an electric field is applied, the PL spectra of the low energy quadrupolar mode remain largely unchanged concerning the two main features described above in the low field regime (Figs. 3d and 3e). However, it changes drastically at high electric fields (Figs. 3f and 3g): the PL intensity drops quickly when doped away from CNP, and the PL energy exhibits a blueshift at n=-1 instead of n=-2. In fact, the PL spectra at high electric fields resemble that of dipolar excitons in a moiré bilayer (Figs. 1e and 1f, as well as our previous study 26 ). Therefore, the observed change in the PL spectra signals the transition from a quadrupolar exciton to a dipolar exciton. Similar results were reproduced in another device with the same structure (device D3), as shown in Supplementary Information Section 12.
With the understanding of the quadrupolar to dipolar exciton transition, we now discuss the nature of the n=-1 and -2 states and their evolution under electric fields. Figs. 4a and 4c plot the PL intensity and peak energy as a function of both doping (filling factor) and external electric field, respectively. At n=-1, the PL energy and intensity both change abruptly above a certain threshold external electric field ,−1 (about 44 mV/nm), while at n=-2, the PL blueshift disappears when the external electric field exceeds ,−2 (about 32mV/nm). For the n=-1 state, in the absence of an external electric field, each hole is hybridized between the top and bottom WSe2 layers with equal probability, i.e., the hole wavefunction is a superposition of the top and bottom WSe2 valence moiré bands. Laterally it is confined such that there is one hole in the two overlapping moiré unit cells combined (Fig. 4d). This state is a new type of correlated state in the trilayer moiré superlattice, an interlayer Mott insulator. The hole is allowed to tunnel between the top and bottom WSe2 layers in the overlapping moiré cells, but tunneling to neighboring moiré cells is prohibited by the strong Coulomb repulsion. As the electric field increases, for example, in the positive direction defined in Fig. 1a, the probability of holes in the bottom WSe2 layer will increase. Above the threshold electric field ,−1 , the hole will be 100% in the bottom WSe2 layer (nb=-1), leaving the top WSe2 layer empty (nt=0). This state now becomes a Mott insulator in the WS2/b-WSe2 interface only, similar to that in a WS2/WSe2 moiré bilayer. The system should remain insulating, as seen in the behavior of the PL peak energy in Fig. 4c. This transition is the result of the competition between the interlayer and intralayer hopping processes, which we characterize as energy ' and , respectively. The interlayer (intralayer) hopping favors carriers populating both (individual) WSe2 layers. Based on the threshold electric field, we estimate the overall potential difference between the two WSe2 layers is about 0 meV at the transition, which suggests that ' is about the same as (See Supplementary Information Section 11: case 2 for a detailed discussion). We note that it is critical to have similar twist angles to observe the reported hybridized Mott insulator state here. The small difference in the twist angles of the reported device might lead to a moiré superlattice of a much larger period, which is not likely to affect our experimental observation due to the corresponding low density of carriers for the half-filling.
At n=-2, the transition is different. Initially, at zero field, there is one hole per moiré unit cell in each of the two WSe2 layers, forming two separate Mott insulator states at both t-WSe2/WS2 and WS2/b-WSe2 interfaces (Fig. 4e). Application of an electric field will create an energy shift between the two Mott insulator Hubbard bands. However, since both upper Hubbard bands (UHB) are fully occupied by holes, tunneling of holes between the two layers is forbidden, and this carrier configuration (nt=nb=-1) will remain stable until the UHB of the top WSe2 layer starts to overlap with the lower Hubbard band (LHB) of the bottom WSe2 layer (Fig. 4g), and holes from this top layer UHB will start to move to the LHB in the bottom WSe2 layer, resulting in partially filled bands in both layers such that the system will no longer be insulating (see the n=-2 evolution in Fig. 4c). The energy difference between the two WSe2 layer at the transition should be equal to the difference between the onsite Coulomb repulsion, U, and ' − . This potential difference is estimated to be ~20 meV from the threshold field. As ' − is about 0, this suggests a value of about 20 meV for U, consistent with the previous studies 12,41 . We note that the threshold electric field at n=-2 has a large uncertainty due to the weak PL signals, and the resulting estimation of U is a lower bound.
Finally, the temperature-dependent PL spectra (Fig. S7) show that the interlayer Mott insulator transition temperature is about 80 K, consistent with our expectation based on previous studies on Mott insulator state in WS2/WSe2 moiré systems 41,42 . The quadrupolar excitons, however, are still obvious at 100 K.
We note that we have also observed quadrupolar excitons and correlated states in WS2/WSe2/WS2 trilayer moiré devices in which the conduction bands in the two WS2 layers are hybridized (Supplementary Information Section 19). We choose to focus on the WSe2/WS2/WSe2 trilayer system in this work as the hybridization and interlayer Mott insulator only involve one valence band in each WSe2 monolayer instead of two conduction bands in each WS2 monolayer, which simplifies the system.
In summary, our study demonstrates a unique trilayer moiré system that hosts both quadrupolar excitons and correlated states at n=-1 (interlayer Mott insulator) and n=-2 (Mott insulator). In particular, the quadrupolar excitons and interlayer Mott insulator both originate from the valence band hybridization and interact with each other. Here, the flat valence band hybridization, combined with the large spin-orbit coupling, is promising for generating nontrivial topological states and engineering quantum states such as quantum anomalous Hall 43 . The quadrupolar excitons in this unique trilayer moiré system are not only promising for realizing the quantum phase transition of bosonic quasiparticles but also strongly interact with correlated electrons, setting up an exciting platform for engineering new correlated physics of fermions, bosons, and a mixture of both 44 . We also envision that further development in aligning the moiré trilayer to allow different stacking of moiré sites (high symmetry points 45 ) such as AAA, ABA, or ABC will usher in unprecedented opportunities in electronic and excitonic band engineering.

Sample fabrication
We used the same dry pick-up method as reported in our earlier work 32 to fabricate TMDC heterostructures. The gold electrodes are pre-patterned on the Si/SiO2 substrate. The monolayer TMDC flakes, BN flakes, and few-layer graphene (FLG) flakes are exfoliated on silicon chips with 285 nm thermal oxide. It is worth noting that typical large TMDC flakes with one dimension exceeding 50 µm were chosen for the device structure shown in Fig. 1. The polycarbonate (PC)/ polydimethylsiloxane (PDMS) stamp was used to pick up TMDC monolayer and other flakes sequentially. The top WSe2 and bottom WSe2 are aligned with a 0-twist angle (R-stacked configuration). This is achieved either through angle-aligned layer stacking and checking the second harmonic generation (SHG) afterward or using the same WSe2 flake and splitting it into two pieces via the tear and twist technique. The alignment of each layer is achieved under a home-built microscope transfer stage with the rotation controlled with an accuracy of 0.02 degrees. The PC is then removed in the chloroform/isopropanol sequence and dried with nitrogen gas. The final constructed devices were annealed in a vacuum (<10 -6 torr) at 250 o C for 8 hours.

Optical characterizations.
During the optical measurements, the sample was kept in a cryogen-free optical cryostat (Montana Instruments). A home-built confocal imaging system was used to focus the laser onto the sample (with a beam spot diameter ~ 2 µm) and collect the optical signal into a spectrometer (Princeton Instruments). During the measurements, the samples were kept in a vacuum and cooled down to 6-10 K. The PL measurements in Figs. 1 and 2 are performed with 50 µW 633 nm CW excitation. All other PL measurements were performed with 633 nm CW excitation with a power of 200 µW unless specified. The reflectance contrast measurements were performed with a super-continuum laser (YSL Photonics). The polarized SHG measurements were performed with a pulsed laser excitation centered at 900 nm (Ti: Sapphire; Coherent Chameleon) with a repetition rate of 80 MHz and a power of 80 mW. The angle between the laser polarization and the crystal axes of the sample was fixed. The SHG signal was analyzed using a half-waveplate and a polarizer. Additional PL measurements were performed with a 730 nm CW diode laser (Supplementary Information Section 15), which showed similar results as the main text.

Calculation of electric field
The external electric field is defined as The electric field in Fig. 2 is defined as the electric field in the TMDC heterostructure, which is given by 2 ( .

Data availability
Source data are available for this paper. The data in Figs.1-4 are provided in the source data files. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.      For device D5, we obtained 1.4° ± 0.5° alignment between b-WSe2 and WS2 and -0.1°± 0.5° alignment between t-WSe2 and WS2.
The SHG intensity from region I and region II (t-WSe2/WS2) are enhanced compared to the SHG intensities from both WSe2 layers, indicating a close to 0° alignment for both heterobilayers ( Figure S2b Figure S3 shows the PL spectra from region I (WS2/b-WSe2), region II (t-WSe2/WS2) and region III (t-WSe2/WS2/b-WSe2) of device D5 at zero gate voltage. All the PL spectra were taken at 6 K using 50 uW 633 nm excitation and the same integration time (2s).

Supplementary section 7: Reflectance contrast measurement and determination of filling factors for device D1
Top-gate-dependent, back-gate-dependent and dual-gate-dependent reflectance contrast spectra with Vtg = Vbg measured on region I (WS2/WSe2) are shown in Figure  S8a,b,c. The gate voltages corresponding to n=-1, n=+1 and n=0 can be identified by the abrupt change of resonance positions and intensity in the reflectance contrast spectra, consistent with previous report 1,2 . The gate voltages corresponding to different filling factors are the same in the top-gate-only and bottom-gate-only configuration, indicating an equal thickness of the top BN and the bottom BN. The gate dependence in Figure S8 indicates that the gate voltage between the filling -2, -1, 0 and +1 is ~ 3 V. Using the geometry capacitance model, we estimate the thickness of BN layer to be ~ 31 nm.
The filling factor as a function of gate voltages in Figure S8c is used to calibrate the filling factor of the PL spectra measured from region III, which is done using the same gate configuration. It can be found that the peak position and intensity change in the PL spectra is aligned with the filling factors calibrated with this method.  Figure S9 shows the PL spectra as functions of the filling factor in the trilayer region (region III) at different positive electric fields. Figure S10 shows the PL spectra as functions of the filling factor from the trilayer region (region III) at different negative electric fields. The PL from the lower energy branch can be fitted with a single Lorentzian peak. The fitted PL peak positions and integrated PL intensities for different electric fields are displayed in Figure S11 and Figure S12, respectively. Figure S9. PL spectra as a function of the filling factor from the trilayer region at different positive electric fields. Figure S10. PL spectra as a function of the filling factor from the trilayer region at different negative electric fields.   Figure S9 and Figure S10 through the fitting.

Supplementary section 9: Modeling of quadrupolar excitons
We consider the following two-energy-level Hamiltonian describing the coupling between the two dipolar interlayer excitons: Where 1 (E) and 2 ( ) are the dipolar interlayer exciton energies at local electric field E in the WS2/b-WSe2 and t-WSe2/WS2 heterostructures, respectively, ∆ is the coupling between the two interlayer excitons. Solving the eigenvalues of this system, we obtain the expression of the low-energy branch of the hybridized excitons: The energies of the dipolar excitons can be expressed as: Where 1 and 2 are the two dipolar exciton energies at zero electric field.
Supplementary section 10: Electric field dependent PL spectra in WSe 2 /WS 2 /WSe 2 device D2 Figure S13a shows the electric field dependent PL spectra in another WSe2/WS2/WSe2 device. Figure S13b shows the fitting result of the two-level hybridization model. We obtain Δ= 30 ± 9 meV through the fitting shown in Figure S13b.  When the sample is insulating, there's a large potential drop on the quantum capacitance between the device and the ground 3,4 . Therefore, the heterostructure can be seen as floated. Considering the boundary conditions at the boundaries of dielectrics, we obtain the following equations: Considering the potential drop between the top gate and the bottom gate, we have: Here we neglected the potential drop in the TMDC heterostructure because its thickness is significantly smaller than the thickness of the BN layers.
From the above equations, we obtained the electric field inside the heterostructure: Case 2: n = -1 Then we consider the threshold electric field to make one of the bilayers at n = -1 and the other at n = 0. In addition to the applied field between top gate and bottom gate, we need to consider the electric field generated by correlated holes. Let us consider the case where > and holes only occupy the bottom WSe2 layer. The boundary conditions become: where 0 = 2.1 × 10 12 −2 is hole density corresponding to one hole per unit cell and e is the elementary charge. Therefore, we obtain: where is the external electric field defined in the Method Section of the main text. In our calculation we use dielectric constant values = 3.9 0 and = 7. Hence, we obtain: From Fig. 4c, the critical external electric field to make the peak shift at n=-2 disappear is ~ 32.2 mV/nm. Therefore, we have ℎ = 17.4 / in this case. According to the interlayer distance of about 1.2 nm we extracted from the dipolar excitons fitting (Supplementary Section 9), we obtain the energy difference between the two WSe2 layers induced by the electrostatic potential at the critical external electric field to be about 20 meV.
Supplementary section 12: Electric field dependence and doping dependence of PL spectra measured from device D3 Figure S15. Electric field dependent and doping dependent PL spectra of another device D3. The data are taken with a 10 µW CW excitation with phonon energy centered at 1.959 eV and a temperature of 6 K.
Although the two dipolar excitons are of slightly different energies in device D3, they still form quadrupolar excitons due to the electric field tunability (schematically shown in Fig.  S19). We note that the schematic shown in Fig. S19 also offers a way to experimentally determine whether the two dipolar excitons are of similar energies in the trilayer region: whether the quadrupolar excitons energy maximum, indicated by the arrow in Fig. S19, is at zero electric field or finite electric field. It is thus clear that the two dipolar excitons are of similar energies in device D1 (Fig. S18a) but slightly different in device D3 (Fig.  S18b), consistent with the fitting results.  We have observed the signs of interlayer quadrupolar exciton PL in 19 devices, and 9 of them can be well fitted with the quadratic electric field dependence. Among them, the interlayer Mott insulator behavior presented in the main text is observed in 6 devices. We suspect that the observation of the correlated states is more demanding on the relative angle alignment between the top and bottom WSe2 layer, which is needed to ensure the same moiré supercell for the top and bottom moiré bilayer.

Supplementary section 15: PL spectra of device D1 measured with 1.698 eV laser excitation
Here we show the PL spectra of device D1 with the exciton photon energy of 1.698 eV, which is in resonance with the moiré A exciton of WSe2. The spectra are similar to what we show in the main text and supplementary section 4. Both excitation photon energy of 1.959 eV and 1.698 eV are much larger than the interlayer exciton resonance of ~1.4 eV. Figure S20. Electric field dependence and doping dependence of the PL spectra from region III of device D1. CW laser with the phonon energy centered at 1.698 eV, in resonance with the moiré A exciton of WSe2, is used as the excitation. The excitation power is 200 µW, and the data are taken at a temperature of 6 K.

Supplementary section 16: Estimation of exciton density
Here we derive the exciton density as a function of excitation power following the method described in ref. 5. We consider the reflection and absorption of each layer of material. The effective power density at the heterobilayer is expressed as: Where and denote the absorptance of the top graphite and the top BN, respectively. , and denote the reflectance of the top graphite, top BN and the heterobilayer. is the power density at the surface of the sample, which can be calculated using the excitation power and the beam spot size d = 2 µm. For few-layer graphene top gate, the absorptance is negligible. At each dielectric interface, the reflectance R can be calculated by: Where ( ) and ( ) are the real refractive indexes at the reflectance side and the transmittance side, respectively.
Using the refractive index values adapted from previous works 6-9 , we estimate = 0.64 for 1.96 eV excitation.
We estimate the proportion of the power absorbed by the hetero-trilayer using the values in ref. 6, which is 0% for WS2 and 2% for WSe2 at 1.96 eV. For region III, assuming 4% absorption and unitary conversion efficiency, we obtain the exciton generation rate g = 2.6 × 10 18 −2 • −1 for 1 μW 1.96 eV excitation. Using a measured exciton lifetime around 19 ns, the steady-state exciton density can be estimated by = • which is 5.0 × 10 10 −2 for 1 µW excitation. With 200 µW 1.96 eV excitation, the steady-state exciton density is 1.0 × 10 13 −2 for 200 µW 1.96 eV excitation and 2.5 × 10 12 −2 for 50 µW 1.96 eV. Similarly, for region II, 1 µW excitation power corresponds to an exciton density of 2.7 × 10 10 −2 . Figure S21. Time-resolved photoluminescence (TRPL) measured from region II and region III. (a) and (b) are the TRPL data taken from region II and region III of device D1, respectively. The red lines indicate fitting results using an exponential decay function ( ) = 0 ( − 0 )/ . The TRPL measurements were performed using a pulsed laser centered at 1.771 eV with a repetition rate of 10 MHz. The excitation power is 400 µW.

Supplementary section 17: Discussion of the energy scales related to the interlayer Mott state
The n=-2 state in Fig. 3 is the case where the top and bottom moiré superlattices are halffilled by holes (-1 for the top WSe2/WS2 and -1 for the bottom WS2/WSe2 moiré superlattice). Therefore, the bandgap will be the smaller onsite repulsion energy (U) of the top and bottom Mott insulator minus the energy ∆ ± . In Fig. S22, we assume the same U for both valence bands for simplicity. We believe that this U could be more than 50 meV (manuscript under preparation and a recent work on arXiv 10 ).
The n=-1 state in Fig. 3 is the newly discovered hybridized Mott insulator state. It can be understood schematically as shown in Fig. S22. The hybridization of the flat valance band gives to the bonding state (share the origin of the quadrupolar excitons) and antibonding state, separated by the energy gap ∆ ± . For the n=-1 state, the bonding state will be half filled as the LHB, while the UHB of the bonding state will be at the energy U higher. As we extracted ∆ to be 9 meV (SI Section 14) for device D1, which corresponding to a ∆ ± of 18 meV, the UHB will be higher than the antibonding state, so that the next added electron will occupy the antibonding state, leading to the bandgap of ∆ ± . As ∆ ± is smaller than U-∆ ± , we would expect the shift at n=-1 to be smaller than that at the n=-2. The high-energy PL peaks in Fig. 3 and Fig. S4 occur at high excitation power and are not universal among all the devices we have studied. It could be attributed to other exciton modes at higher energy, including possible quadrupolar exciton modes from hybridization of higher energy dipolar excitons. We do not believe that it is from the high energy branch (antisymmetric mode associated with the symmetric quadrupolar exciton mode discussed in the main text) quadrupolar exciton (because of the large energy separation from the ground state quadrupolar exciton. It is likely that the simplified model did not consider moiré effects or possible complications from other dipolar excitons (such as spin-singlet exciton 11 ). As the PL spectra are most sensitive to the ground state, we focus on the symmetric quadrupolar exciton branch (lower energy branch) in this work and leave the investigation of the high energy mode(s) to future exploration.