Luminescence anomaly of dipolar valley excitons in homobilayer semiconductor moiré superlattices

In twisted homobilayer transition metal dichalcogenides, intraand inter-layer valley excitons hybridize with the layer configurations spatially varying in the moiré. The ground state valley excitons are trapped at two highsymmetry points with opposite electric dipoles in a moiré supercell, forming a honeycomb superlattice of nearest-neighbor dipolar attraction. We find that the spatial texture of layer configuration results in a luminescence anomaly of the moiré trapped excitons, where a tiny displacement by interactions dramatically increases the brightness and changes polarization from circular to linear. At full filling, radiative recombination predominantly occurs at edges and vacancies of the exciton superlattice. The anomaly also manifests in the cascaded emission of small clusters, producing chains of polarization entangled photons. An interlayer bias can switch the superlattice into a single-orbital triangular lattice with repulsive interactions only, where the luminescence anomaly can be exploited to distinguish ordered states and domain boundaries at fractional filling. Long-wavelength moiré patterns by van der Waals stacking of two-dimensional crystals has made possible the exploration into a new realm of physics. The local-tolocal variation of the interlayer registry leads to the spatial modulation of the local band structure, creating an artificial superlattice with a period in the range from a few to a few tens nanometers. In twisted graphene moiré, interlayer coupling at magic angles turns the massless Dirac cones into flat mini-bands, where a plethora of correlation phenomena are observed [1-9]. Another exciting moiré platform is the semiconducting transition metal dichalcogenides (TMDs) which host massive Dirac fermions at a timereversal pair of valleys located at the K and −K corners of the hexagonal Brillouin zone [10]. Moiré pattern in TMDs heterostructures introduces a triangular superlattice landscape for valley electrons [11], in which a variety of correlated insulating states have been observed at various filling factors [12-17]. Excitons formed by the Coulomb binding of valley electron and hole make possible a distinct many-body system in the TMD moiré superlattices with optical addressability and the bosonic statistics. The intralayer exciton (Xintra) has a large optical dipole with a fixed valley polarization selection rule, which underlies the versatile optical controls of valley dynamics [18-21]. In type-II heterobilayers, excitons also form in an interlayer configuration (Xinter), where the separation of constituent electron and hole into adjacent layers leads to a permanent electric dipole that turns on strong dipolar interactions and coupling to electric field [22-27]. The layer separation of the electron and hole leads to long recombination lifetime and valley lifetime of Xinter, favorable for valleytronic applications [28], as well as for the exploration of quantum many-body phenomena such as exciton Bose-Einstein condensation [29] and self-assembled crystal phase of the two-dimensional dipolar excitons [30]. In the moiré pattern, Xinter experiences a strong superlattice potential with spatial variation in the order of ~100 meV, and features an optical dipole that spatially varies on the scale of the moiré period, determined by the local stacking registry [31-41]. The lowest energy Xinter are then trapped at the potential minima, which can form an ordered array of quantum emitters. The radiative recombination of these Xinter ground states results in sharp emission lines with spectral width of ~ 0.1 meV, as observed in MoSe2/WSe2 heterostructures under low excitation power and temperature [34-36]. Photon antibunching experiment indicates that these trapped Xinter can serve as highly tunable single-photon emitters [36]. The repulsive dipolar interaction between the trapped Xinter manifests as blue shifts of their resonances observed in the photoluminescence spectrum [25-27]. At the potential minima, the 2 /3-rotational symmetry ( ) of the local atomic registries dictates circularly polarized optical selection rules of Xinter, with the emission polarization jointly determined by the valley, spin and stacking registry [31,42]. The polarization properties of the exciton emission (coor cross-polarized with respect to the excitation at Xintra resonance) can be used to identify the trapping location within a moiré supercell, which is switchable by an outof-plane electric field [31]. While these properties of Xinter in the heterostructure moiré are highly intriguing for novel optoelectronic applications, the very weak optical dipole has however placed an intrinsic limitation on their exploitation [39]. Here we present an exotic exciton system in twisted TMDs homobilayers. We show that because of the registry dependent interlayer coupling, the energies of intraand inter-layer valley excitons cross each other as functions of position in the moiré, giving rise to a texture of spatially varying exciton hybridization. In a moiré supercell, the ground state valley exciton is trapped at two degenerate high-symmetry points with opposite out-of-plane electric dipole moments, forming a honeycomb superlattice with repulsive on-site and attractive nearest-neighbor interactions. The spatial texture of the Xintra-Xinter hybridization leads to a luminescence anomaly of excitons in the traps. At equilibrium positions, they are as dark as Xinter, while a tiny lateral displacement in the order of Å dramatically increases the brightness, and the emission polarization is changed from circular to linear, with polarization angle reflecting the displacement direction. At full exciton filling of the honeycomb superlattice, radiative recombination predominantly occurs at edges and vacancies where excitons are displaced by the unbalanced dipolar interactions from neighboring sites. An interlayer bias can switch the excitonic superlattice to a single-orbital triangular one with repulsive interactions only. We show that the luminescence anomaly can serve as signatures for different correlated states and domain boundaries at fractional exciton filling. The anomaly also manifests in the cascaded emission of small clusters, producing chains of polarization entangled photons. Consider the near 0◦ twisted homobilayers with moiré period b much larger than the monolayer lattice constant, such that local regions can be approximated by commensurate bilayers with various R-stacking registries (Fig. 1a). For the conduction and valence band edge electrons at K valley, interlayer coupling has a sensitive dependence on the stacking registry and therefore becomes a periodic function of position in the moiré [43,44]: ( ) = ( ) h ( ) h∗ ( ) ( ) , ( ) = ( ) h ( ) h∗ ( ) ( ) . Here / ( ) is the energy shift of the conduction/valence band edge in the upper layer due to the interlayer coupling, while / ( ) is the one in the lower layer. h / ( ) is the interlayer hopping of the electron/hole. Their -dependences underlie the spatially-varying layer hybridizations of the valley electrons and holes in the moiré of TMDs homobilayers [43,44]. The phenomena to be explored are more pronounced for smaller b (≲ 10 nm), where the lattice reconstruction effect is insignificant [45], not considered here. We focus on the valley excitons formed by a pair of electron and hole at these band edges. The strong electron-hole Coulomb interaction leads to a binding energy of a few hundred meV, which is one order larger than the interlayer hopping (|h| ~ 20 meV). This allows one to start from the exciton basis obtained in the limit of vanishing interlayer hopping. The basis states in a given valley include the two Xinter, denoted as | ⟩ and | ⟩ which have opposite electric dipoles (Fig. 1b), and the two Xintra confined in the two layers respectively, denoted as | ⟩ and | ⟩ (Fig. 1b). In this basis, the exciton Hamiltonian reads, ( ) = − ħ ∇ 2 + ( ), (1)

Long-wavelength moiré patterns by van der Waals stacking of two-dimensional crystals has made possible the exploration into a new realm of physics. The local-tolocal variation of the interlayer registry leads to the spatial modulation of the local band structure, creating an artificial superlattice with a period in the range from a few to a few tens nanometers. In twisted graphene moiré, interlayer coupling at magic angles turns the massless Dirac cones into flat mini-bands, where a plethora of correlation phenomena are observed [1][2][3][4][5][6][7][8][9]. Another exciting moiré platform is the semiconducting transition metal dichalcogenides (TMDs) which host massive Dirac fermions at a timereversal pair of valleys located at the K and −K corners of the hexagonal Brillouin zone [10]. Moiré pattern in TMDs heterostructures introduces a triangular superlattice landscape for valley electrons [11], in which a variety of correlated insulating states have been observed at various filling factors [12][13][14][15][16][17].
Excitons formed by the Coulomb binding of valley electron and hole make possible a distinct many-body system in the TMD moiré superlattices with optical addressability and the bosonic statistics. The intralayer exciton (Xintra) has a large optical dipole with a fixed valley polarization selection rule, which underlies the versatile optical controls of valley dynamics [18][19][20][21]. In type-II heterobilayers, excitons also form in an interlayer configuration (Xinter), where the separation of constituent electron and hole into adjacent layers leads to a permanent electric dipole that turns on strong dipolar interactions and coupling to electric field [22][23][24][25][26][27]. The layer separation of the electron and hole leads to long recombination lifetime and valley lifetime of Xinter, favorable for valleytronic applications [28], as well as for the exploration of quantum many-body phenomena such as exciton Bose-Einstein condensation [29] and self-assembled crystal phase of the two-dimensional dipolar excitons [30].
In the moiré pattern, Xinter experiences a strong superlattice potential with spatial variation in the order of ~100 meV, and features an optical dipole that spatially varies on the scale of the moiré period, determined by the local stacking registry [31][32][33][34][35][36][37][38][39][40][41]. The lowest energy Xinter are then trapped at the potential minima, which can form an ordered array of quantum emitters. The radiative recombination of these Xinter ground states results in sharp emission lines with spectral width of ~ 0.1 meV, as observed in MoSe2/WSe2 heterostructures under low excitation power and temperature [34][35][36]. Photon antibunching experiment indicates that these trapped Xinter can serve as highly tunable single-photon emitters [36]. The repulsive dipolar interaction between the trapped Xinter manifests as blue shifts of their resonances observed in the photoluminescence spectrum [25][26][27]. At the potential minima, the 2 /3-rotational symmetry ( ) of the local atomic registries dictates circularly polarized optical selection rules of Xinter, with the emission polarization jointly determined by the valley, spin and stacking registry [31,42]. The polarization properties of the exciton emission (co-or cross-polarized with respect to the excitation at Xintra resonance) can be used to identify the trapping location within a moiré supercell, which is switchable by an outof-plane electric field [31]. While these properties of Xinter in the heterostructure moiré are highly intriguing for novel optoelectronic applications, the very weak optical dipole has however placed an intrinsic limitation on their exploitation [39].
Here we present an exotic exciton system in twisted TMDs homobilayers. We show that because of the registry dependent interlayer coupling, the energies of intra-and inter-layer valley excitons cross each other as functions of position in the moiré, giving rise to a texture of spatially varying exciton hybridization. In a moiré supercell, the ground state valley exciton is trapped at two degenerate high-symmetry points with opposite out-of-plane electric dipole moments, forming a honeycomb superlattice with repulsive on-site and attractive nearest-neighbor interactions. The spatial texture of the Xintra-Xinter hybridization leads to a luminescence anomaly of excitons in the traps. At equilibrium positions, they are as dark as Xinter, while a tiny lateral displacement in the order of Å dramatically increases the brightness, and the emission polarization is changed from circular to linear, with polarization angle reflecting the displacement direction. At full exciton filling of the honeycomb superlattice, radiative recombination predominantly occurs at edges and vacancies where excitons are displaced by the unbalanced dipolar interactions from neighboring sites. An interlayer bias can switch the excitonic superlattice to a single-orbital triangular one with repulsive interactions only. We show that the luminescence anomaly can serve as signatures for different correlated states and domain boundaries at fractional exciton filling. The anomaly also manifests in the cascaded emission of small clusters, producing chains of polarization entangled photons.
Consider the near 0 • twisted homobilayers with moiré period b much larger than the monolayer lattice constant, such that local regions can be approximated by commensurate bilayers with various R-stacking registries (Fig. 1a). For the conduction and valence band edge electrons at K valley, interlayer coupling has a sensitive dependence on the stacking registry and therefore becomes a periodic function of position in the moiré [43,44]: Here / ( ) is the energy shift of the conduction/valence band edge in the upper layer due to the interlayer coupling, while / ( ) is the one in the lower layer. ℎ / ( ) is the interlayer hopping of the electron/hole. Their -dependences underlie the spatially-varying layer hybridizations of the valley electrons and holes in the moiré of TMDs homobilayers [43,44]. The phenomena to be explored are more pronounced for smaller b (≲ 10 nm), where the lattice reconstruction effect is insignificant [45], not considered here.
We focus on the valley excitons formed by a pair of electron and hole at these band edges. The strong electron-hole Coulomb interaction leads to a binding energy of a few hundred meV, which is one order larger than the interlayer hopping (|ℎ| ~ 20 meV). This allows one to start from the exciton basis obtained in the limit of vanishing interlayer hopping. The basis states in a given valley include the two Xinter, denoted as | ⟩ and | ⟩ which have opposite electric dipoles (Fig. 1b), and the two Xintra confined in the two layers respectively, denoted as | ⟩ and | ⟩ (Fig. 1b). In this basis, the exciton Hamiltonian reads, Here is the exciton energy in a pristine monolayer, and is the exciton effective mass. Δ is the binding energy difference between Xintra and Xinter. The electron/hole interlayer hopping causes the off-diagonal couplings between Xintra and Xinter. ( ) ≡ ( ) − ( ) ( = / , = / ) accounts for the energy shift of Xintra and Xinter by the position dependent interlayer coupling in the moiré, which can be determined from fitting first-principles calculations [46], are the lowest harmonics that satisfy the rotational and translational symmetry, which turn out to be sufficient in accounting for the energy modulation in the moiré [46]. = (0,1), and are the three corners of the mini Brillouin zone related by 2 3 ⁄ -rotation ( ). For the example of bilayer MoTe2 which has a direct bandgap, DFT calculations give ∆ ≈ 31 meV, ∆ ≈ 42 meV, ≈2 meV, ≈ 0.5 meV [44].
The energies of Xintra and Xinter (i.e., diagonal terms of ( ) in Eq. (1)) as functions of R obtained from Eq. (2) are shown in Fig. 1c, where the three high-symmetry locations in a supercell have the coordinates =0, = √ ,0 and = √ ,0 , respectively. The energy minima are located at B and C, where Xinter has the energy: − ( − ) − (Δ +Δ )+Δ , a n d X intra has the energy: , then the energies of Xintra and Xinter cross each other in the moiré, and the lowest energy branch is of the mixed nature, being Xinter near B and C, and majorly Xintra elsewhere. Numerical calculations of Δ fall in the range of 20-90 meV [40,[47][48][49], while experiments in various hetero-and homobilayer TMDs report Δ values in the range of 10-60 meV [48][49][50][51][52]. The plot in Fig. 1 has used Δ = 40 meV as an example of the Δ < 2Δ regime which we will discuss first. In the opposite limit of 2Δ <Δ , the lowest energy branch is Xintra at zero interlayer bias, while a finite interlayer bias can also introduce the energy crossing of Xintra and Xinter in the moiré, which will be discussed next (c.f. Fig. 4a).
The interlayer hopping of electron and hole then lead to hybridization of Xintra and Xinter, and anti-crossing of the exciton branches [38,[53][54][55]. The position-dependence of the interlayer hopping can be similarly obtained [43,44] / are real numbers in the order of a few meV (see Supplementary Section I).
( ) can then be solved by first diagonalizing its ( ) part, which leads to four decoupled branches (| = 1,2,3,4⟩) of layer-hybridized excitons with both Xintra and Xinter components. The eigenvalues, ( ), ( ), ( ), ( ), correspond to the moiré superlattice potentials for these decoupled branches, which, together with the kinetic energy − ℏ ∇ , lead to exciton minibands. The moiré potentials for the four branches are shown as solid thick curves in Fig. 1d, where the colormap indicates strength of the optical dipole, determined by the Xintra weighting (c.f. Fig. 1e). The electric dipole determined by the Xinter weighting is shown in Fig. 1f. At A, the local stacking has the out-of-plane mirror symmetry, where the exciton branches are either layer symmetric with large optical dipole, or antisymmetric which is dark. At B and C both ℎ and ℎ vanish, so the four branches at these two locations reduce to either Xintra or Xinter, without layer hybridization.
Below we focus on the lowest energy branch |1⟩, where the moiré potential ( ) has two degenerate minima at B and C with opposite electric dipoles, corresponding to the interlayer configurations | ⟩ and | ⟩ , respectively [56]. Away from these potential minima, branch |1⟩ starts to pick up Xintra component, because of the finite interlayer hopping which takes the chiral forms ℎ / ( + ) = / + ( ), ℎ / ( + ) =− / + ( ), with = (cos , sin ) the small displacement vector. ~25 meV and ~110 meV are determined by the parameters and in Eq. (3). The perturbative expansion of |1⟩ in the neighborhood of B/C reads It is important to note that Xintra and Xinter components in the above expressions feature distinct valley polarization selection rules. Consider excitons in K valley, Xintra (| ⟩ and | ⟩ ) always emit + circularly polarized photons [10], while the Xinter's emission polarization is location and layer-configuration dependent [31,42]. The emission polarizations are − and ( and −) for | ⟩ and | ⟩ (| ⟩ and | ⟩ ), respectively, as indicated in Fig. 1d. The Xinter components involved in Eq. (4a) and (4b) both emit − photons.
Near B/C, the moiré potential ( ) has a parabolic f orm wi th a harmonic oscillator frequency ∝ (ℏ ~10 meV when =8 nm, in agreement with the results in Ref. [57]). The ground state valley exciton is then a gaussian hopping between the trapping sites becomes negligible. We examine the ground state , and analyze how its optical properties vary upon displacement of the potential minima by an in-plane force. The optical dipole of such a wavepacket is determined by its central location (c.f. Supplementary Section II), Here ± * ≡ ± √ * is the unit vector for the ± polarization, ( ) the dipole strength of an Xintra (Xinter) wavepacket of half-width . It is interesting to note that | ⟩ centered right at the B/C point already has a non-negligible Xintra fraction (see Fig. 1e), which, however, does not contribute to the dipole because of the destructive interference due to the factor (also see Appendix A for a symmetry analysis). The contribution of the Xintra fraction to the optical dipole is linearly proportional to the displacement of the wavepacket center from B/C, wi th th e f actor ≡ ∆ + ∆ ~ 3.5.
Because of the opposite polarization of the Xintra and Xinter contributions, the photon emission of the displaced wavepacket is elliptically polarized. Fig. 2a shows the calculated degree of circular-polarization as a function of central location in a moiré supercell, for a wavepacket in branch |1⟩.
is rather close to +1 in most of the supercell except the two small regions around B and C, where it quickly decreases to −1 at B/C. Such a behavior is due to the large ratio of | / |. The plot used | / | = 20 from first-principles result [31,42], consistent with experiments [58,59]. Thus a tiny displacement of the wavepacket center can result in a substantial change to the optical dipole in Eq. (5). Fig. 2b is an enlarged view of the polarization patterns around B/C. As can be seen, in contrast to the − polarization at B/C, wavepacket centers located on a ring centered at B/C exhibit 100% linearly polarized emissions. The ring radius can be as small as 0.1 nm for a typical moiré period of = 8 nm. The degree of linear polarization and the angle of its major axis are obtained from Eq. (5): =− + ± .
Here +/− is for B/C. The polarization angle has a one-to-one correspondence with the displacement direction ( Fig. 2b).
as a function of ⁄ for three different values of Δ is shown in Fig. 2c. Besides the linear polarization, the magnitude of the optical dipole, given by , = | | + , also increases rapidly with as indicated in Fig. 2d. The oscillator strength of the exciton, proportional to its radiative recombination rate, is proportional to the modular square of the optical dipole. We expect our perturbative results of Eq. (4-6) to hold up to ~ 0.1, which corresponds to a ~50-fold increase for / . Thus upon a tiny displacement of the exciton wavepacket from B/C, there can be orders of magnitude increase in its radiative recombination rate.
The electric dipoles of the trapped excitons lead to strong repulsive onsite, and attractive nearest-neighbor interactions. Consider a nearest-neighbor BC pair of trapped excitons, their attractive dipolar interaction can be approximated by − = − √ ⁄ in a suspended bilayer (~− 10 meV for = 8 nm), with the vacuum permittivity and ≈ 0.7 nm the interlayer distance. Taking into account the small but finite Bohr radius , will get larger by a small percentage ∝ as compared to the form above (see Appendix B for a more detailed discussion about the exciton-exciton interaction). The total energy of the pair of excitons is minimized through a symmetric displacement towards each other (c.f. Fig. 3a), with a magnitude = .
as a function of b for three different Δ values are shown in Fig. 3b. The displacement is small in the order of 0.1 nm, but can already lead to a substantial change in the emission polarization which can become 100% linear, and a dramatic increase in the emission rate by up to two orders of magnitude (c.f. Fig. 3b inset).
We note that this rapid change of optical properties is correlated with the change in the exciton trapping potential (due to the occupation of its near neighbors), in which the exciton energy is discretized. And the sharp contrast of polarization and oscillator strength is between the excitonic ground states in the displaced and undisplaced moiré potential traps. Upon the change of the potential trap (e.g., due to the annihilation of a neighboring exciton), the original exciton ground state wavepacket is no longer an eigenstate of the new potential trap. The emission properties are then determined by the competition between the radiative decay and the relaxation to the new ground state. As the latter relaxation time scale is much faster than the radiative decay, the radiative emission will predominately come from the ground state of the new potential trap (see Appendix C).
Below we first give a summary of the selection rule and emission polarization of the trapped excitons in the twisted homobilayer moiré. For an exciton wavepacket centered right at B/C, the combination of and time reversal symmetries requires K and −K valleys to emit photons with − and + circular polarizations, respectively. Thus the photon emission from valley unpolarized excitons exhibits neither circular nor linear polarization. Time reversal symmetry breaking can come from circularly polarized pumping or applying an out-of-plane magnetic field, which introduces valley polarized exciton population that emits with circular polarization.
A tiny displacement from B/C breaks the -symmetry of the wavepacket, and the corresponding photon emission becomes elliptically polarized for either valley. The K and −K valleys are still related by a time reversal, which dictates that their emission polarizations exhibit opposite helicities but the same linear polarization, with the linear polarization angle determined by the displacement direction (c.f. Fig. 2b). A 360°anticlockwise rotation of the displaced wavepacket around B/C results in a 360°clockwise rotation of the linear polarization angle. It is important to note that, upon the same displacement vector, exciton wavepackets near B and C exhibit orthogonal linear polarization directions, which comes from the opposite signs for ℎ / ( + ) and ℎ / ( + ), as a consequence of spatial symmetry of the homobilayer moiré supercell.
This leads to the following emission patterns for a pair of trapped excitons taking into account their dipolar interaction: Next we consider how the dramatic change of optical properties by the tiny displacement of these dipolar excitons can manifest in the light emission of their manybody states. In a many-body configuration, each trapped exciton experiences the overall dipolar force from others, and can be displaced depending on the occupation of its neighboring sites. The leading order displacement are determined by the nearestneighbors. When the three nearest-neighbors are all occupied or unoccupied, the force is balanced and the exciton is centered at B/C with zero displacement, which is essentially dark. The other six configurations shown in Fig. 3c all correspond to unbalanced forces. The center exciton at equilibrium is then displaced by along six possible directions with a substantially increased brightness, and the angle of the linear polarization distinguishes the six cases. Taking = 5 nm as an example, we estimate the trapped exciton has a radiative lifetime of 800 ns at a balanced site, and 10 ns at an unbalanced site [31]. On the other hand, the phonon-assisted equilibration timescale for Xinter is estimated to be in the order of picosecond [47] (also see Appendix C for a detailed analysis on the relaxation process of trapped excitons), orders faster than the radiative lifetime.
The recombination of a trapped exciton breaks the dipolar force balance of its nearest-neighbors, which go to the new equilibrium in a time scale significantly shorter than the radiative lifetime (see Appendix C for details). The radiative emission of an exciton thus changes dramatically the emission timescale and polarizations of its nearest-neighbors, a new form of optical nonlinearity which leads to unique luminescence phenomena. At full filling of the lattice, unbalanced sites appear only at the edges or vacancy defects (c.f. Fig. 3d). The radiative annihilation of the excitons will predominantly occur at the edges and vacancies. Under partial filling, the attractive nearest-neighbor interaction leads to clusterization. Different recombination pathways of a small exciton cluster are characterized by the energies and linear polarizations of the cascaded photon emission, resulting in a chain of entangled photons. Fig. 3e illustrates two sets of recombination pathways for a lowest energy 4-site cluster, each can generate a polarization-entangled photon pair. Other possible recombination pathways (not shown) also exist, which can generate entangled photons with frequencies different from those in Fig. 3e. The atomically thin geometry of the bilayer and the dipolar nature of the exciton further point to various possibilities to isolate a small cluster of excitons for entangled photon generation, e.g. through using a biased metal tip or local gate [60], or creating strained local regions [26,61], or engineering the surrounding dielectric environment [62] (see Appendix D).
The degeneracy of B and C sites can be lifted by a finite interlayer bias, which lowers (increases) the energy of the exciton trapped at B (C) by . Below we focus on a large enough , such that only the B sites will be occupied which then form a triangular lattice with repulsive dipolar interactions only. In Fig. 4a, we show such a case with Δ = 80 meV > 2Δ and = 50 meV. In the lowest energy branch, exciton is dark at B, and the brightness increases fast with the displacement from this energy minimum. The emission polarization also changes dramatically under a tiny displacement from B (see Fig. 4b). Fig. 4c shows the equilibrium emission properties for a nearest-neighbor pair of trapped excitons, which are displaced away from each other by the dipolar repulsion. The two excitons emit with orthogonal linear polarizations, with enhanced emission rate. The equilibrium displacement magnitude is shown in Fig. 4d  For an unbalanced site in the triangular lattice, the different occupations of its six nearest-neighbor sites result in three different displacement magnitudes ( , √3 and 2 ). The degree of the linear polarization and the optical dipole strength under the displacement magnitude (2 ) are shown as solid (dashed) curve in Fig. 4d inset. Fi g. 4e shows the di splacements and the correl ated linear pol arizati ons of unbalanced sites at the edges and vacancies of a full filling exciton lattice, where exciton radiative recombination is much faster than the balanced site inside the bulk.
Such a triangular exciton superlattice with the repulsive interactions can host the various ordered states under fractional fillings, as those observed for electrons and holes in WS2/WSe2 heterobilayer moiré superlattices [13,[15][16][17]. Remarkably, the luminescence anomaly of the unbalanced sites can be exploited as a fingerprint for certain configurations. In Fig. 4f  Finally, we note that the luminescence anomaly is the property of the K valley excitons. The above quantitative analysis uses homobilayer MoTe2 as the example, which has a direct bandgap with conduction and valence band edges both located at K (as evidenced by experiments in [63][64][65]). This could be a highly promising system to exploit the luminescence anomaly. WSe2 bilayer moiré induced by heterostrain can be another candidate system. Pristine WSe2 bilayer has an indirect band gap (K to Q), but the energy is not far from the direct K to K gap. Experiments have shown that, under 1 % uniaxial tensile strain, the band gap of bilayer WSe2 becomes the direct K to K one [66,67]. Having the bilayer subject to heterostrain (different strain magnitude in the two layers, e.g. when strain is applied through substrate) can be an alternative way to create homobilayer moiré superlattices other than the twisting [68]. In such case, we note that the lowest energy excitons trapped at the B/C sites in R-type WSe2 homobilayer moiré will be the spin-triplet at K valleys, which, at the unbalanced sites, are subject to similar displacements due to the dipolar interactions as described in Fig. 3a-c. Their emission can be the phonon assisted process via the spin-singlet exciton discussed here [69,70], or through hybridization with the spin-singlet in an in-plane magnetic field [71,72], so the sharply contrasted emission dipoles of the singlet at balanced and unbalanced sites will also manifest in the luminescence.
The relative orders of the energy scales for exciton in the moiré trap are underlying the perturbative treatment of the layer hybridization (Eq. (4)) and exciton relaxation (see Appendix C), and allow us to focus only on the ground state in the trap for the radiative emission. The radiative recombination rate (≤ 0.1 ns −1 ) is several orders of magnitude slower than the rate for the exciton to relax to the ground state in the trap (p s −1 , see Supplementary Section III). The latter rate is also much smaller than the energy detuning and off-diagonal coupling between Xinter and Xintra, and the quantization energy in the trapping potential ℏ (~10 meV for =8 nm). It is worth noting that the detuning of Xinter and Xintra at the B/C trap is controllable by an interlayer bias and can also be affected by the dielectric environment (through their distinct binding energies). When Xinter and Xintra are brought closer in energy such that their detuning is no longer significantly larger than the phonon and photon induced broadening, corrections beyond the perturbative treatment becomes non-negligible. In such case, the luminescence phenomenon can be a more complex manifestation of the interplay between the interlayer hybridization, relaxation and dephasing by phonon, and radiative emission processes.

Appendix A. Intralayer component of the trapped exciton wavepacket and its contribution to optical dipole: a symmetry analysis
Both Xintra and Xinter have Wannier type wavefunctions with the electron and hole localized around ± in momentum-space, so here we use the envelope approximation to write the involved conduction (valence) band Bloch function as , ( ) ≈ ⋅ , ( ) ( , ( ) ≈ ⋅ , ( )). The Xinter momentum eigenstate (also eigenstate of the electron-hole Coulomb interaction) is [39] ( Here ≡ + is the center-of-mass (COM) coordinate of the exciton, and ≡ − corresponds to the electron-hole relative coordinate.
is the wavefunction of the electron-hole relative motion. Since the exciton has a large binding energy ~200 meV, the slowly modulating moiré potential barely affects Φ ( ). So only the lowest energy electron-hole relative wavefunction (1s state) need to be considered.
An Xinter wavepacket is then the linear superposition of the momentum eigenstates: Here is the COM envelope in real space, with denoting the ground state (s-type), first excited state (p-type), etc., and denoting its center. is the half-width of the COM envelope.
Besides the slowly varying ( − ) and Φ ( ) parts, the Xinter wavepacket also contains the Bloch part , ( ) is at a high-symmetry point, i.e. A, B or C in the moiré (Fig. 1a), and the eigenvalues are +1, −1 and 0 respectively for being at A, B and C [31].
An Xintra wavepacket, on the other hand, has a simpler structure: With the electron and hole Bloch functions in the same layer, the entire wavefunction is always an eigenfunction under -rotation about the wavepacket center (regardless of whether is displaced from B/C or not), and eigenvalues is determined by the index n only (i.e. s-type or p-type COM envelope). Xintra with an s-type envelop has = +1 and emits with + polarization instead, therefore is not allowed in the wavepacket by the -symmetry. (c) The atomic registry underlying an exciton wavepacket slightly displaced from B, with the red arrow denoting the displacement vector. The three circled areas are no longer related byrotation, thus the wavepacket is not -symmetric. (d) The exciton wavepacket with a small displacement from B. As the restriction imposed by the -symmetry is lifted, both the p-and s-type envelope Xintra components are allowed in the wavepacket wavefunction. In (b) and (d), grey color denotes the envelope function of the primary Xinter component, and yellow color denotes that of Xintra. The latter can only have a small contribution in the ground state wavefunction as they are far detuned in energy.
Therefore for an Xinter wavepacket with its center being at a high-symmetry point (B/C as concerned in this work), albeit the spread out in real space as determined by ( − ) and Φ ( ), the wavefunction does have the perfect -symmetry about its center , as can be seen in Fig. 5a. This -symmetry does not allow an s-type envelope Xinter to be hybridized with an s-type envelope Xintra that has a difference eigenvalue. It can hybridize only with a p-type envelope Xintra having the same eigenvalue (see Fig. 5b). The p-type envelope Xintra however, has a negligibly small optical dipole. As a result, the overall optical dipole of the exciton wavepacket remains circularly polarized, with a magnitude essentially being just that of the Xinter optical dipole.
When slightly displaced from B/C, however, because of the incommensurability of the two layers, the Bloch part , ( ) , * ( ) is no longer an eigenfunction ofrotation about the displaced wavepacket center , so the overall Xinter wavefunction no longer has the -symmetry (c.f. Fig. 5c). The broken -symmetry lifts the restriction on the coupling between Xinter and Xintra, and the s-type envelope Xinter wavepacket can also hybridize in a small portion of the s-type envelope Xintra wavepacket (see Fig. 5d, and Supplementary Section II). It is this displacementdependent coupling together with the large optical dipole of s-type envelope Xintra that dramatically change the wavepacket optical properties, even if the displacement is much smaller than the spread of the wavefunction in real space.

Appendix B. Interaction between two exciton wavepackets
We consider two s-type envelope Xinter wavepackets intralayer Coulomb interactions, respectively, see Fig. 6a.
( ) < ( ) due to the vertical separation between the two layers.

i. The dipole-dipole interaction part
The strength of the dipole-dipole interaction is Here ≡ + and ≡ + . Since both the wavepacket half-width and the Bohr radius are much larger than the monolayer lattice constant, in the last step above we have approximated the fast oscillating terms , ( ) and Note that the two nearest-neighbor wavepacket centers are separated by √3 ⁄ ≫ , . For an order of magnitude estimation we can consider the limit of →0 and →0, and write | ( )| ≈ ( ) , |Φ ( )| ≈ ( ) . Then the dipole-dipole interaction is of a simple form Here is the average dielectric constant of the surrounding substrate.
The corrections from the small but finite and are analyzed by a series expansion of Note that both | ( )| and |Φ ( )| are even functions, thus all the odd terms of − , − , and vanish after integration. The dipole-dipole interaction up to the 2nd order of and can then be derived as On the other hand, the trapped exciton near B/C contains a small component of Xintra, which reduces its electric dipole. Using the perturbative wave function Eq. (4), we find the electric dipole is reduced by a fraction After taking into account this contribution, the dipole-dipole interaction up to the 2nd order of and is The wavepacket width scales as ∝ √ ( ~ 1.4 nm when =8 nm from our estimation), whereas ~1-2 nm is nearly independent of . So increases faster than when decreasing . Furthermore, the coefficient of the term is much smaller than that of the t e r m ( ∆ + ∆ ~7 from our estimated parameters).
Therefore, for a relatively small moiré period, we expect the correction from dominates over that from , resulting in an overall enhancement to the interaction strength.

ii. The exchange interaction part
The strength of the exchange interaction is Since |Φ ( )| ≪1 and ≪1, is thus exponentially small compared to . Furthermore, Xinter trapped at B and C have opposite electric dipole orientations, thus ( 1, ℎ2) and ( 2, ℎ1) correspond to Xintra (see Fig. 6c), and the energy detuning further suppresses this off-diagonal process by exchange. Thus the exchange interaction between two exciton wavepackets located at B and C can be safely ignored.
On the other hand, for two excitons in the same trapping site, both the dipole-dipole and exchange parts can be important and dependent on and of the trapped exciton wavepacket [23]. However, such configuration is energetically unfavorable due to the strong onsite repulsion. So when the moiré superlattice is loaded with excitons at filling factor ≤1, one can just focus on the many-body configurations with exciton per trapping site less or equal to one, where the exciton-exciton interaction is dominated by the dipole-dipole interaction . Concerning the radiative emission from the excitonic ground state in the moiré trap, we note that both the energy detuning and off-diagonal coupling between Xintra and Xinter are much larger than the broadening induced by their interaction with either the phonon or photon bath. The low energy excitonic eigenstates (e.g. | ⟩ and | ⟩) are spectrally well separated from each other by ℏ ~10 meV, each having a definite form of superposition between the primary Xinter component and a small Xintra component. Experiments have found that the moiré trapped excitons in heterobilayers have spectral widths ~ 0.1 meV [34][35][36]. In this regime, the approach of diagonalizing the Hermitian Hamiltonian (Eq. (1)) and adding decay perturbatively is justified. Energy relaxation occurs between these spectrally well-separated excitonic eigenstates (c.f. Fig. 7c-d), and the radiative emission of the excitonic ground state is described by a single decay rate determined by its wavefunction, in which the form of the superposition between the primary Xinter and the small Xintra components dictates the optical dipole.

Appendix D. Possible experimental schemes for isolating a small cluster of excitons
The atomically thin geometry of the bilayer and the dipolar nature of the exciton point to several possibilities to isolate a small cluster of excitons trapped in a few moiré supercells, as schematically illustrated in Fig. 8: (1) Using a biased metal tip (c.f. Fig. 8a), or local gate [60], to create a low energy region of a few moiré periods for isolating a cluster of dipolar excitons. (2) Creating a strained region, by placing the bilayer on engineered substrate (c.f. Fig. 8b). Tensile strain lowers the energies of both Xintra and Xinter [61], so the strained region can act as a large scale potential trap that contains a few moiré supercells. In Ref. [26], such locally strained regions are engineered by placing heterobilayer on nanopillars (c.f. Fig. 8b), which can trap Xinter with a tunable number from 1 to 5. (3) Engineering the surrounding dielectric environment (c.f. Fig. 8c). For example, a~ (10) nm scale graphene coverage can be realized by electron beam lithography to pattern a mask, and followed by oxygen plasma etching to etch the unwanted graphene. Ref. [62] shows that covering TMDs by a bilayer graphene (with a large dielectric constant ) can significantly reduce the band gap as well as the exciton energy. The few moiré supercells underneath the graphene coverage are then energetically favored by the dipolar excitons, which can trap an exciton cluster.