Cavity-QED-controlled two-dimensional Moiré excitons without twisting

Domain: physics
Published: 2025-12-24
Editorial outcome: minor_revision

Abstract
We propose an all-optical Moiré-like exciton confinement by means of spatially periodic optical cavities. Such periodic photonic structures can control the material properties by coupling the matter excitations to the confined photons and their quantum fluctuations. We develop a low energy non-perturbative quantum electro-dynamical description of strongly coupled excitons and photons at finite momentum transfer. We find that in the classical limit of a laser driven cavity the induced optical confinement directly emulates Moiré physics. In a dark cavity instead, the sole presence of quantum fluctuations of light generates a sizable renormalization of the excitonic bands and effective mass. We attribute these effects to long-range cavity-mediated exciton-exciton interactions which can only be captured in a non-perturbative treatment. With these findings we propose spatially structured cavities as a promising avenue for cavity material engineering.

Introduction
Two-dimensional van der Waals materials have emerged as a versatile platform for opto-electronic devices thanks to their highly tunable properties, such as optical and electronic band gaps and dielectric response1–5. Among this class of materials, transition metal dichalcogenides (TMDs) have attracted particular interest due to their strongly bound excitons and related optical activity in the visible range6–8. Excitons do indeed dominate the dynamics of several phenomena of TMDs, such as valley polarization and non-linear optical response9,10. Hence, controlling the behavior of excitons offers a direct route to program optical functionalities in this class of materials. Twist engineering11–13, a technique in which atomically thin TMD layers are stacked with a predefined twist angle, allows for spatial control of excitons and consequent modification of their optical activity14. This is possible because twisting two-dimensional (2D) crystals with respect to each other leads to the formation of Moiré patterns, which generate spatially periodic electrostatic potentials strong enough to influence the motion and confinement of the otherwise free excitons, c.f. Fig. 1a.

Aside from excitonic control, twist engineering, also dubbed twistronics, allows for generating novel quantum phases5,15, including correlated insulator states16, unconventional superconductivity12, and fractional Chern insulators17. In this work, we propose an alternative strategy for spatial confinement of excitons that utilizes optical cavities instead of twist engineering. Optical cavities confine the electromagnetic field in a small volume, strengthening its intensity and making it possible to strongly couple to excitons of embedded TMDs. Such interaction can in turn lead to the formation of exciton-polaritons18–22. Previous works show that due to the strong light-matter interaction, planar optical cavities allow the control of the creation and mixing of polaritons made of composite excitons18, as well as excitons, phonon and photon quasi-particles, the so-called phonoriton23. Cavity confinement not only alters the excitonic spectrum24,25 but can also give rise to new equilibrium quantum phases due to the coupling of the host material with the vacuum fluctuations of light26–33. Here, we propose using spatially structured optical cavities, such as those sketched in Fig. 1, to produce an all-optical Moiré-like exciton confinement without twisting. This is a different paradigm compared to previous works that deal with exciton-polaritons in grated systems as they aim to reshape the properties of emitted light20,34. With our approach, instead, we demonstrate that we can control the matter properties by specifically structuring the excitonic quasi-particle, using both quantum fluctuations and classical fields.

Results
Theory
To study the behavior of Moiré excitons in an optical cavity, we first model the two uncoupled systems (excitons and photons) and subsequently describe their interaction. Then, we formulate and discuss the QED Hamiltonian for Moiré excitons, which constitutes one of the main results of the paper. All throughout this work, we use atomic units.

The uncoupled matter Hamiltonian H^M of a twisted bilayer heterostructure can be formulated in an excitonic representation following ref. 14 as

H^M = Eν ll0,ij,Q X^ν ll y 0,ij,Q X^ν ll0,ij,Q + Mν ll0,ij,q X^ν ll y 0,ij,Q+q X^ν ll0,ij,Q

where i,j are band indexes which span over conduction (C) or valence (V) band states, and l,l0 are layer indexes. The operators X^y, X^ create (annihilate) an exciton between any pair of bands of either the same layer (when l=l0, in which case one has intralayer excitons) or different layers (interlayer excitons), Q is the momentum associated with the center of mass of the exciton, and q is the momentum transferred by the Moiré potential. The index ν refer to the bound state (i.e., 1s, 2s…). Eν ll0,ij,Q = Q2/2mll0,ij + Eg,ll0 - Eb,ll0 encodes the dispersion relation of a free exciton, which we assume parabolic, where m is the excitonic effective mass, Eg is the energy gap and Eb is the binding energy, which we obtain by solving the Mott–Wannier equation (c.f. Supplementary Information Section V.A). M is the matrix element of the Moiré potential in the excitonic basis, describing the Moiré scattering of excitons with different momenta. Note that M carries the information on the Mott–Wannier states (c.f. Supplementary Information Eq. A25). Refer to the Supplementary Information Section I for a detailed derivation.

We describe the uncoupled light system with a Hamiltonian consisting of a set of effective harmonic oscillators (the photon modes of the cavity):

H^ph = ω q(cid:2) a^y q(cid:2),λ a^ q(cid:2),λ + 1/2

where ω q(cid:2) represents the energy of the photon mode q(cid:2) and λ is the polarization. q(cid:2) is the momentum of the photon in the xy plane of the 2D material. In a planar setup, electromagnetic waves propagate in the in-plane direction, while they are standing waves in the z-direction (which is orthogonal to the cavity mirrors). We assume that the matter couples with the in-plane component of the electric field and therefore only consider modes for which it is finite. The momentum of the photon in the z-direction is set by the standing wave condition for the fundamental cavity mode. Additionally, when a periodic grating is present, the xy plane momentum is finite and determined by the periodicity of the grating. When studying optical properties of materials in the far field, one usually makes the approximation that the light does not carry any momentum, the long-wavelength approximation (LWA). We recently showed that, also within a QED framework, the LWA is applicable35 and reduces the description of the electromagnetic Hamiltonian to one with a single effective mode at q(cid:2)=0. In practice, the LWA implies that an electron cannot scatter to another k-point following the absorption of a photon (vertical transition in k-space). In contrast, if the cavity is spatially structured, the light-matter interaction can transfer momentum to the excitons, which is the key to our proposal. We define the effective modes as those in which the in-plane momenta are not averaged around q(cid:2)=0, but around the finite momenta q(cid:2) set by the cavity. We stress here that the term “effective”, when referring to the cavity modes, indicates that such modes represent a summation over a continuum of modes centered around a certain momentum q(cid:2). Working with effective modes is necessary to guarantee a finite light-matter coupling strength in the bulk limit of extended cavity-matter systems when working with a finite number of modes for the description of the electromagnetic field.

To describe the light-matter coupling, we start by performing the canonical momentum substitution p^ -> p^ + A^ to the uncoupled matter Hamiltonian, obtaining the second-quantized Pauli-Fierz Hamiltonian35. The Hamiltonian in the excitonic representation reads (c.f. the Supplementary Information Section I for the complete derivation):

H^ = ω q(cid:2) a^y q(cid:2),λ a^ q(cid:2),λ + 1/2 + Eν ll0,ij,Q X^ν ll y 0,ij,Q X^ν ll0,ij,Q + Mν ll0,ij,Qq X^ν ll y 0,ij,Q+q X^ν ll0,ij,Q + A ~ 0,q(cid:2) Bν ll λ 0,ij,Qq(cid:2) X^ν ll y 0,ij,Q+q(cid:2) X^ν ll0,ij,Q a^y q(cid:2),λ + a^ (cid:2)q(cid:2),λ + Iν ll , 0, λ ij,Qq(cid:2) X^ν ll y 0,ij,q(cid:2) a^y q(cid:2),λ + a^ (cid:2)q(cid:2),λ + h:c:

where A ~ 0,q(cid:2) = q(A0,q(cid:2) / Veff,q(cid:2)), A0,q(cid:2) is the coupling strength of the mode q(cid:2) and Veff is the effective mode volume. The first line of the Hamiltonian contains the uncoupled photon and the uncoupled matter, while the second represents the paramagnetic bilinear coupling between photon modes and excitons. Note that we absorbed the diamagnetic term into the uncoupled photon Hamiltonian by performing a Bogoliubov transformation37,38. B and I are matrix elements describing the exciton-photon interactions and are defined in Supplementary Information Section I.C. The former allows an exciton to scatter to another k-point after absorbing or emitting a photon, the implications of which will be discussed in depth in the next sections. Importantly, B=0 when q(cid:2)=0 (see Supplementary Information Section V.C). The matrix elements I deal with the creation or destruction of such particles. In order to access the polaritonic states via full diagonalization, the Hamiltonian in Eq. (3) is projected onto a combined light-matter product state, with excitonic states and the many-body ground state for the matter18, which are written in a Slater determinant representation, and number states for each photonic mode. This basis keeps the N-particles electronic states explicit, so that even if the number of excitons is not conserved, the total number of particles is fixed.

In Eq. (3), we observe that the standard Moiré potential M and the first term of the bilinear coupling B share the same excitonic operators X^y, X^, in case q(cid:2)=q. For the Moiré potential, q refers to the super-lattice Moiré periodicity. The bilinear coupling q(cid:2) refers to the periodicity of the electromagnetic field inside the cavity. Despite having different physical meanings, both indices determine a lattice periodicity, which modifies the symmetry experienced by the excitons. To highlight this equivalence, we let q(cid:2)=q and rewrite the interaction terms of the Hamiltonian in Eq. (3) (which also includes the non-conserving bilinear coupling I) as:

H^int = Mν ll0,ij,Qq X^ν ll y 0,ij,Q+q X^ν ll0,ij,Q + A ~ 0,q Bν λ X^ν ll y 0,ij,Q+q X^ν ll0,ij,Q a^y q,λ + a^ (cid:2)q,λ + A ~ 0,q Iν λ X^ν ll y 0,ij,q X^ν ll0,ij,Q a^y q,λ + a^ (cid:2)q,λ + h:c:

This similarity, and indeed mutual mathematical interchangeability, of the exciton-exciton interaction provided by the Moiré potential and the structured optical modes is the central theoretical result of this work. In the following section, we investigate the effect of H^int on the excitonic system when the radiation field is treated both in a classical and quantum manner.

Classically driven cavity
We start by considering the effect of a driven structured cavity on the excitonic properties of the material in equilibrium, going beyond what is possible to achieve with twisting. We model this excitation for single-polarization (λ=s) modes as a time-dependent coherent state |~λ q(cid:2)(t)i = |~λ q(cid:2)(t)i(cid:3)|~λ (cid:2)q(cid:2)(t)i. Here we define the time-dependent coherent states as |~λ q(cid:2)(t)i = e P (cid:2)iω 2 q(cid:2)t |e(cid:2)iω q(cid:2)t ~λ q(cid:2)i. Since the obtained Hamiltonian is time periodic, we use Floquet theory to solve it39,40. The advantage of this approach is that the Floquet Hamiltonian is time independent, which makes it computationally cheaper while still being able to capture the polaritonic effects from the light-matter coupling. With this semi-classical treatment, light with finite momentum q(cid:2) enters the coupled Hamiltonian in Eq. (4) exactly as the potential created by a Moiré super-lattice, by generating interacting Floquet replicas, as sketched in Fig. 1b. This means that we can use classically driven cavities to induce an all-optical Moiré potential.

Dark cavity
We now analyze the cavity-exciton system for a dark cavity, i.e., in the absence of external fields, where the effect of the photons on the matter is via their quantum fluctuations. We begin by focusing on the quantum effects on the heterostructure embedded in a spatially unstructured cavity (B=0), including in this case the periodic confining potential from the Moiré pattern caused by twisting. In the absence of momentum exchange, light and matter are coupled only through the bilinear coupling associated with I in Eq. (4). Here, we examine how the unstructured planar cavity influences the twist-angle dependence of the excitonic response. In the presence of cavity-matter coupling, when the cavity resonates with the MoSe2 excitonic transition, we expect the Rabi splitting to dominate the behavior of the exciton-polariton response. We observe that compared to the case of uncoupled light and matter, the coupling splits the lowest branch into two, the upper (UP) and lower polariton (LP). The magnitude of this splitting increases linearly with the light-matter coupling A ~ 0. Interestingly, the UP has the same dispersion as in the uncoupled matter system, whereas the LP line is almost constant for all twist angles. This implies that the LP behaves as a low-energy unperturbed and unconfined exciton. Conversely, the dispersion of the UP is steeper and linear at small twist angles and then flattens out for larger angles, where it behaves similarly to the LP, i.e., as nearly free particles. In short, the effect of the twisting angle can be modified by the electromagnetic field of a cavity, which influences the excitonic response by causing Rabi splitting of the resonant transition. This results in a different localization for the upper and lower polariton, and may pave the way for new quantum phases41.

We then consider a spatially structured cavity, where both creation of excitons and Moiré-like exciton-scattering are included, i.e., both B and I are finite in Eq. (3), while we let M=0 to isolate the effect of the periodic grating of the cavity. The striking result is the emergence of a four-excitonic operator term, i.e., a term that describes an interaction between two excitons, which is a fundamental difference with respect to the Moiré potential. This alters the excitonic dynamics as manifested by our calculated spectral functions and the non-trivial linear response features. These excitons-excitons interactions, absent in classical treatments, highlight the potential of quantum cavities to engineer correlated excitonic phases.

Methods
In this work, we distinguish between spatially unstructured cavities, of the Fabry–Perot kind, where the electromagnetic field is treated with a single effective mode description in the long-wavelength approximation35 (q(cid:2) = (qx, qy) = (0,0)), and spatially structured cavities, whose description requires two effective momentum-carrying modes (q(cid:2)1 = (qx, 0), q(cid:2)2 = (-qx, 0)) corresponding to a cavity with a one-dimensional periodicity (like the one in Fig. 1d). Based on symmetry arguments, we assume that ω q(cid:2)1 = ω q(cid:2)2 and A ~ 0,q(cid:2)1 = A ~ 0,q(cid:2)2. Such photonic environments can be realized by means of dielectric metasurfaces44,45, or in polaritonic and plasmonic cavities46–48. In order to illustrate the broad potential of cavity-structuring of excitons, we leave the cavity mode energy, volume, and grating wavelength as adjustable parameters.

In order to obtain the values of the modes' momentum used in the previous sections (q(cid:2) = (±0.009, 0), (±0.021, 0), (±0.033, 0) a.u.), one should design the grating of the cavity accordingly. We estimate that the periodicity of the corresponding grating should be d = 2π/q(cid:2) = 40 nm, 16 nm, 10 nm. We expect these extreme grating sizes to be larger for the optical confinement of interlayer excitons.

In order to obtain the full exciton-polariton states, we start by solving the Mott–Wannier equation in the k-space (c.f. Supplementary Information Section V.A), which gives us access to the binding energy and the Mott–Wannier states. We considered only the 1s intralayer exciton in the MoSe2 layer. The Mott–Wannier states are used to construct the excitonic creation and annihilation operators, as well as all the matrix elements appearing in Eq. (3). Refer to Supplementary Information Section I for the complete derivation. We represent the photonic modes using the Fock number states {|0i, |1i}. Finally, we obtain the full exciton-polariton states via the full diagonalization of Eq. (3). The physics of the coupled light-matter system is then investigated by computing excitonic quantities: the linear optical susceptibility (LS) function and the spectral function (SF). The former represents the optical response of the system, obtained from applying linear response theory to the polaritonic states49:

χ(ω, Ωc, θ) = |MI,0|2 / (ω - (EI(Ωc, θ) - E0(Ωc, θ)) + iη)

where MI,0 is the transition matrix element between the polaritonic ground state Ψ0 and an excited state ΨI. η is a small artificial broadening, Ωc is the energy of the cavity modes, ω is the energy of the probe field, θ is the Moiré twist angle, and EI(Ωc, θ) is the energy of the Ith polaritonic state. Note that when we set M=0 (no standard Moiré potential), χ does not depend on θ. Furthermore, note that we only calculate the matter part of such a response by tracing out the photons. To investigate the excitonic dispersion, i.e., their band structure, we employ the spectral function defined as:

S(ω, Ωc, θ, Q) = |<ΨI(Ωc, θ)|X^yQ|Ψ0(Ωc, θ)>|2 δ(ω - (EI(Ωc, θ) - E0(Ωc, θ)) + iη)

This represents the probability of creating an exciton with an energy EI(Ωc, θ) at a certain k-point. Note that when we set M=0 (no standard Moiré potential), S does not depend on θ. Refer to the Supplementary Information Section II for additional insights.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors propose an all-optical realization of Moiré-like exciton confinement using spatially periodic optical cavities. The\nidea of engineering Moiré physics in a purely photonic way is very appealing, as it avoids complications from atomic\nreconstruction and offers a higher degree of tunability. I also find the analysis of the dark cavity scenario particularly\ninteresting and original.\nOverall, I believe this work will attract broad interest at the interface of the photonic and Moiré physics communities. I\ntherefore recommend publication in Nature Communications after the authors address the following points:\ni) The authors correctly point out that the bilinear coupling term formally shares the same structure as the Moiré Hamiltonian,\nleading to an additional periodicity. However, for Moiré physics of real interest to emerge, the induced potential must be\nstrong enough to produce flat bands, i.e., to substantially modify the band curvature, with miniband separations large\ncompared to the exciton thermal distribution. Otherwise, excitons will experience a spatial modulation without true\nlocalization.\nIn the present case, the folding vector q is small, so the ground state appears flat only because it is very close to the\nparabolic minimum, and already the third band in Fig. 2c shows a more pronounced dispersion. This raises the question: to\nwhat extent does localization actually occur, given that at low temperatures a thermal distribution spans ∼20 meV, whereas\nthe miniband separation in Fig. 2c is less than 10 meV?\nii) Related to the above, if one increases the cavity parameters to enhance the strength of the bilinear term to overcome the\npoint mentioned above, is it still valid to use the unmodified Wannier exciton states to define the exciton basis? In Moiré\nsystems, the effective potential originates from atomic displacements and does not significantly alter the underlying Wannier\nexciton problem in first approximation, making it possible to treat it via an effective static continuum potential. By contrast, in\nthe photonic case the periodic potential arises from light–matter interaction itself, and stronger coupling may renormalize the\nexciton wavefunctions directly. Strictly speaking, such a term should then be included in the Wannier equation to obtain new\nexciton states. I do not expect the authors to perform such a calculation, but I suggest that they comment on the range of\nvalidity of their approximation.\niii) At the end of the first paragraph of the “Dark cavity” section, the authors cite only Ref. 14. I suggest also citing Nano Lett.\n2022, 22, 11, 4468–4474, where exciton–polariton Moiré structures were computed. Although that work addresses a non-\ndark cavity, it provides a useful comparison and would strengthen the discussion.\nThe authors might find the following list of minor corrections useful (some may be due to PDF conversion):\nPage 1, first column, first line: \"Van der Waals\" → \"van der Waals\".\nPage 5, second column, last line: “form” → “from”.\nUse of hyphens: \"hetero-structure\" → \"heterostructure\", \"bi-layer\" → \"bilayer\", \"bi-linear\" → \"bilinear\", \"intra-layer\" →\n\"intralayer\".\nIn the SI, Eqs. (17) and (19): the exciton annihilation operator is missing the index ν.", "reviewer_2": "This work proposes and theoretically demonstrates an original approach for creating moiré systems without twists using\nstructured planar optical cavities. The proposed schemes uses a TMD heterostructure embedded in a planar cavity with\nfaces parallel to the structure. The authors show that, in the case of a structured cavity, the excitonic dispersion relations\nhave characteristics of a moiré. The authors study the case of cavities driven by a classical field or without driving. In the\nsecond case, the interaction between the van der Waals heterostructure and the cavity induces long-range interactions.\nThis is an original approach, likely to open a new avenue in the field of moiré systems, avoiding twisting of solid-state\nstructures. Moiré systems are attracting growing attention and the field is very active at the interface of condensed matter\nphysics, quantum optics, and quantum gases. This new work enriches it and I am convinced that it will be of great interest to\nthe community. The presentation of the theoretical model is clear and sound. Useful technical details of the calculations are\nwell described in the supplementary material. In addition, the paper is clearly written and will be accessible even for non-\nspecialists in the field.\nI recommend the publication of the manuscript after the authors have considered the following points:\nIn section II.A, it would be useful to say something about the method used in numerical calculations.\nIt is written that the results \"show that it gives confinement signatures similar to a twist induced Moir´e potential\" but the\nauthors do not write precisely what makes the signature unambiguous. A discussion of these signatures would be very\nuseful, particularly for the non-specialist reader.\nIt is also written that \"B shares the same excitonic operators of the Moir´e potential M\" but in a paragraph where it is written\nthat M=0 a few lines above. This is very misleading.\nA few lines below eq 2, please specify that z is the direction orthogonal to the cavity planes.\nThe authors write “We assume that the matter couples with the in-plane component of the electric field and therefore only\nconsider modes for which it is finite.” What does justify this assumption? To what extent is this justified in practical\nimplementation?\nPage 6, right-hand-side column, it is written “In the range of small twist angles, however, we can identify localized polaritonic\nstates [14].” What does induce localization? In fact, I am surprised that the localization appears at low angles but apparently\nnot at larger angles, according to the text. I would expect the following phenomenology: For certain particular angles, the\nsystem is periodic and cannot induce localization; For most angles, on the other hand, the system is non-periodic and can\ninduce localization. For very low angles but inducing a periodic structure, the moiré period is very large so that the couplings\nbetween subsequent cells are very weak and very small fluctuations can overcome them and give a false apparent\nlocalization. This was recently analyzed in a slightly different context in PRR 6, L042066 (2024) and PRA 111, 043305\n(2025). Can the results observed in the manuscript be explained in a similar way?"}, "author_response": "Responses to the reviewers comments\nWe sincerely thank the two reviewers for their efforts to carefully evaluate our manuscript. We are\nvery pleased that both reviewers recognize the importance, significance and validity of our work,\nand support the conditional acceptance of the paper. Their constructive and insightful comments are\nhelpfulforustoimprovethequalityofthisworkandclarifysomepointsinthemanuscript. Below,we\nprovideourresponses,inbluetext,toeachoftheindividualcomments,alongwithalistofchangeswe\nmade in the revised manuscript. We also thank the Editor for their consideration of our work and for\ninviting us for a revision. We added the Data availability and Code availability sections as requested\nbytheEditor. Additionally,weaddedtheAuthorscontributionsandtheCompetingInterestssections\tocomplywiththeeditorialpolicies.\nThe authors propose an all-optical realization of Moire´-like exciton confinement using spatially peri-\nodicopticalcavities. TheideaofengineeringMoire´physicsinapurelyphotonicwayisveryappealing,\nas it avoids complications from atomic reconstruction and offers a higher degree of tunability. I also\nfind the analysis of the dark cavity scenario particularly interesting and original. Overall, I believe\nthisworkwillattractbroadinterestattheinterfaceofthephotonicandMoire´ physicscommunities. I\ntherefore recommend publication in Nature Communications after the authors address the following\npoints:\nResponse: We appreciate the reviewer’s recognition of the importance and correctness of our work,\nand recommendation for publication in Nature Communications. We also thank the reviewer for the\nconstructive comments to improve our work. In the following, we respond each individual comment\nindetail,alongwithcorrespondingchangesmadeintherevisedmanuscript.\ni) The authors correctly point out that the bilinear coupling term formally shares the same structure\nas structure\nas the Moire´ Hamiltonian, leading to an additional periodicity. However, for Moire´ physics of real\ninteresttoemerge,theinducedpotentialmustbestrongenoughtoproduceflatbands,i.e.,tosubstan-\ntially modify the band curvature, with miniband separations large compared to the exciton thermal\ndistribution. Otherwise, excitons will experience a spatial modulation without true localization. In\nthe present case, the folding vector q is small, so the ground state appears flat only because it is very\nclose to the parabolic minimum, and already the third band in Fig. 2c shows a more pronounced\ndispersion. Thisraisesthequestion: towhatextentdoeslocalizationactuallyoccur,giventhatatlow\ntemperatures a thermal distribution spans ∼ 20 meV, whereas the miniband separation in Fig. 2c is\nlessthan10meV?\nResponse: As the reviewer points out, at room temperature the thermal energy (≈ 25meV) is larger\nthan the minibands separation. However, at cryogenic temperatures (for instance liquid nitrogen,\n𝑇 = 77K),thethermalenergyis≈ 6.6meV,hencetheseparationinFig. 2canberesolved.\nWestressthatourfocusisontheconfinementoftheexcitons,notontheirlocalization. Thesamegoes\nfor the discussion of Fig. 2, and more generally in Section II.B. By excitonic confinement we mean\nthatexcitonsfeelapotentialwhichcomesfromtheinteractionwiththedrivingfield,andthepotential\nhas the effect of folding the bands and splitting them at the border of the Moire BZ, as opposed to a\nfree particle dispersion characteristic of unconfined (free) excitons. Both signatures can be seen in\nFig. 2(c).\nConversely,byexcitoniclocalizationwemeanthatexcitonsaretrappedinapotentialwell,whosesig-\nnatureisthepresenceofaflatdispersion. Theminibandflatnessisdeterminedbytheratiobetweenthe\nstrength of the interaction and the parabolic free excitonic dispersion 𝐸 𝑅 = Q2/(2𝑀 𝑒𝑥𝑐 ). Practically,\nusing a longer-period grating (smaller q) and/or increasing 𝐴˜ 0,q (e.g., higher field enhancement or\nsmallermodevolume)flattensthebandsandenlargestheminibandgaps. NotethatsincethisaFloquet\ndrivencavity,itisalwayspossibletoincreasethecouplingbyincreasingthedriving. However,asthe\nreviewerpointsout,theBZissmallandoversuchanarrowrangeaparabolicbandnearitsminimum\ncanappearflat. Forthisreasonwedonotassociatetheseeminglyflatbranchtoalocalizedphase.\nFinally, let us consider the experimental feasibility of the parameters used for Fig. 2. The strength\nof interaction Hamiltonian responsible for the photon-induced band depends, as discussed in the\nSupplementaryInfoIII,on 𝐴˜ 0,q B 𝑙 𝜈 𝑙 , ′ 𝜆 ,𝑖𝑗,Q,q ,whereB 𝑙 𝜈 𝑙 , ′ 𝜆 ,𝑖𝑗,Q,q dependslinearlyonq (c.f. Supplementary\nInfo V.C) and 𝐴˜ 0,q = 𝐸 0,q /𝜔 q , where 𝐸 0,q is the amplitude of the electric field. As we discuss in\nSection III of the main text, the value of the mode momentum q is determined by the periodicity\nof the grating. In particular, to achieve q = [±0.009,0.0]a.u. one needs a grating with periodicity\n𝑑 ≈ 40nm, which should be accessible given the current state-of-the-art fabrication techniques. As\nfor the coupling strength, typical values for Fabry-Perot cavities are 𝐸 0,q ≈ 10 ÷ 100kV/m and\n𝐸 0,q ≈ 1÷10MV/mforsub-wavelengthcavities,asitcanbecalculatedfrom[Phys. Rev. Research7,\n033163]. Theconversionrateforthecouplingstrengthis 𝐴˜ 0,q [a.u.] = 𝐴˜ 0,q [Vs/m]/1.24e-5. Hence,\nto obtain 𝐴˜ 0,q = 0.04[a.u.] with 𝜔 q = 1.5eV one needs 𝐸 0,q > 10MV/m which is very challenging\nforthecurrentcavitytechnologybothwithFabry-Perotandsub-wavelengthcavitiesunderthevacuum\nfluctuating photon conditions. Conversely, to obtain 𝐴˜ 0,q = 0.02[a.u.] with 𝜔 q = 0.05eV one needs\n𝐸 0,q ≈ 3MV/m, which is feasible using a sub-wavelength cavity. Note that Fig. 2 refers to a driven\ncavity,hencetheaforementionedconditionscanbeachievedbyincreasingtheclassicalpump.\nii) Related to the above, if one increases the cavity parameters to enhance the strength of the bilinear\nterm to overcome the point mentioned above, is it still valid to use the unmodified Wannier exciton\nstates to define the exciton basis? In Moire´ systems, the effective potential originates from atomic\ndisplacementsanddoesnotsignificantlyaltertheunderlyingWannierexcitonprobleminfirstapprox-\nimation, making it possible to treat it via an effective static continuum potential. By contrast, in the\nphotonic case the periodic potential arises from light–matter interaction itself, and stronger coupling\nmay renormalize the exciton wavefunctions directly. Strictly speaking, such a term should then be\nincluded in the Wannier equation to obtain new exciton states. I do not expect the authors to perform\nsuchacalculation,butIsuggestthattheycommentontherangeofvalidityoftheirapproximation.\nResponse: In our formalism the light couples to the excitons via the bilinear coupling terms that, in\nthe exciton representation, act on the center-of-mass degree of freedom (scattering Q → Q±q). As\nthe reviewer points out, issues may arise when the coupling starts to significantly affect the excitonic\nwavefunctionsuchthattheexcitonstateitselfmustberecomputed.\nOnephenomenonthatrequiresrecomputingtheexcitonicwavefunctionisthemixingofthe1𝑠exciton\nwithhigherexcitonicstates(e.g. 2𝑝,3𝑑). Infact,relaxingthetruncationontheexcitonicbasisallows\nfor internal wavefunctions rearrangements via mixing with the higher excited states (c.f. Nano Lett.\n2019, 19, 6, 3473–3479). The theory we present in Section II.A also accounts for this effect, while\nin our calculation we used only the 1𝑠 state due to computational reasons. Here, we justify such\nnumerical approximation. For Fig. 2, the binding energy of the intralayer exciton is 𝐸 𝑏 ≈ 0.368eV,\nwhile the energy difference between the 1𝑠 and 2𝑠 state is Δ𝐸 𝑏,1𝑠−2𝑠 ≈ 0.193eV. The mixing of the\nexcitonic ground state with higher states is controlled by the ratio between the interaction energy\nand the energy difference between such states. If such ratio is significantly less than 1, then the\napproximationofconsideringonlythe1𝑠statehasasmallinfluenceontheresults. Forthevaluesused\nin Fig. 2, 𝐴˜ 0,q B 𝑙 𝜈 𝑙′ ,𝜆 ,𝑖𝑗,Q,q ≈ 0.3. Including the 2𝑠 exciton would therefore yield a minor improvement\nΔ𝐸\n𝑏,1𝑠−2𝑠\nin accuracy that affects only numerical values, not the qualitative conclusions, while substantially\nincreasing computational cost. We thus consider the present approximation justified for the scope of\nthemaintext.\nOne may also consider a spatial argument for using the unmodified Mott-Wannier equation. In the\nexciton relative-coordinate decomposition, the cavity field with in-plane wavevector q varies across\ntheexcitononthescaleoftheexcitonicradius𝑟 𝑏. InthecaseofFig. 2(c)suchradiusis𝑟 𝑏 ≈ 1.64nm,\nin agreement with previous works (Phys. Rev. B 92, 245123). Hence, the periodicity of the grating\nshould be such that the field can be assumed constant over the excitonic radius 𝑟 𝑏. For the values\nof q used throughout this work we have estimated a real-space periodicity of the grating (hence of\nthe field) which ranges from 𝑑 ≈ 10nm to 𝑑 ≈ 40nm, c.f. Section III of the main text. Since these\nvaluesareanorderofmagnitudebiggerthantheexcitonicradius,weexpectthatusingtheunmodified\nMott-Wannier equation is a valid approximation, provided that the q = 0 excitonic problem for the\nequivalent coupling strength shows that the wavefunction is not altered. A grating periodicity of\n𝑑 = 1.64nm would generate a mode momentum of q ≈ [±0.2,0.0]a.u.. It should also be noted that\nrealizing a grating with such small periodicity is experimentally challenging, and unlikely to be seen\ninthenearfuture.\nFinally, one could also argue that even the screened interaction that is used to define the excitonic\nproblem is affected by the coupling with the cavity. However, in order to account for that one would\nneed to build the theory starting from light-dressed orbitals, for example using QEDFT, which is\nbeyondscopeforthepresentwork.\nWeaddedthefollowingparagraphintheSupplementaryInformation,SectionV.A:Theuseofunmodi-\nfiedMott-Wannierexcitonstatesassumesthatthemixingbetweenthe𝜈 = 1𝑠excitonicstateandhigher\nexcitedstatesissmall. Thisapproximationisjustifiedaslongasthetheratiobetweentheinteraction\nenergy and the energy difference between such states is significantly less than 1. Note that including\nthe excitonic 2𝑠 implies a major increase in the computational cost. Moreover, we assume that the\ncavity field does not significantly distort the internal exciton wavefunction. Such an approximation is\njustifiedaslongaslongthelightmatterinteractiondoesnotdistorttheinternaldegreesoffreedom\nof the exciton (i.e. the relative position of the electron and of the hole). Hence, the spatial variation\nof the cavity field, which is determined by the periodicity of the grating, must be small relative to the\nexcitonsize(i.e. totheexcitonicradius),whichisthecasefortheinvestigatedperiodicites.\niii) At the end of the first paragraph of the “Dark cavity” section, the authors cite only Ref. 14. I\nsuggest also citing Nano Lett. 2022, 22, 11, 4468–4474, where exciton–polariton Moire´ structures\nwerecomputed. Althoughthatworkaddressesanon-darkcavity,itprovidesausefulcomparisonand\nwouldstrengthenthediscussion.\nWethankthereviewerforsuggestingtheworkNanoLett. 2022,22,11,4468–4474. Afterreadingit,\nwedecidedtoincludeitasfollowsinthemaintext:\n• In the introduction: ”We find that spatially unstructured cavities, where the electromagnetic\nfield carries no momentum, can alter the twist induced excitonic confinement from the Moire´\npotential when the cavity-mode is resonant with the excitonic transition. This result is in\nagreement with previous works [Nano Lett. 2022, 22, 11, 4468–4474], where they applied a\nHopfield diagonalization to a tight binding model to study how bare photonic and excitonic\nstates contribute to exciton-polariton states in unstructured cavities at different twist angles.\nWe show that the same result can be obtained using a more general theory, starting by a first\nprinciples QED Hamiltonian beyond the RWA approximation, and focusing on controlling the\nlocalizationofexciton-polaritons.”\n• Weaddedthecitationatthepointsuggestedbythereviewer\n• In paragraph 2 of Section II.C: ”Interestingly, the UP in Fig.3(b-c) has the same dispersion\nas in the uncoupled matter system (Fig. 3(a)), whereas the LP line is almost constant for all\ntwist angles, in agreement with previous works [Nano Lett. 2022, 22, 11, 4468–4474]. ···\nRef. [Nano Lett. 2022, 22, 11, 4468–4474] provides with an interesting and complementary\ndiscussion on how bare photonic and excitonic states contribute to the UP and LP;. In short,\n··· ”\n• In the caption of Fig. 3: ”(a) Twisted heterostructure without the cavity (see Ref. [Nano Lett.\n2020, 20, 12, 8534–8540]Fig. 5(a) and Ref. [NanoLett. 2022,22, 11, 4468–4474] Fig. 4(a)).\n(b,c) ··· NotethatforthesetwopanelsΩ𝑐 issignificantlysmallerthantheexcitonicresonance\nduetothediamagneticshift. TheseresultsarequalitativelyinagreementwithRef. [NanoLett.\n2022,22,11,4468–4474]Fig. 4(b).”\nThe authors might find the following list of minor corrections useful (some may be due to PDF\nconversion): Page 1, first column, first line: ”Van der Waals” → ”van der Waals”. Page 5, second\ncolumn, last line: “form” → “from”. Use of hyphens: ”hetero-structure” → ”heterostructure”, ”bi-\nlayer” → ”bilayer”, ”bi-linear” → ”bilinear”, ”intra-layer” → ”intralayer”. In the SI, Eqs. (17) and\n(19): theexcitonannihilationoperatorismissingtheindex 𝜈.\nWethankthereviewerforspottingthesetypos. Wefixedtheminthemanuscript.\nThis work proposes and theoretically demonstrates an original approach for creating moire´ systems\nwithouttwistsusingstructuredplanaropticalcavities. TheproposedschemesusesaTMDheterostruc-\nture embedded in a planar cavity with faces parallel to the structure. The authors show that, in the\ncase of a structured cavity, the excitonic dispersion relations have characteristics of a moire´. The\nauthorsstudythecaseofcavitiesdrivenbyaclassicalfieldorwithoutdriving. Inthesecondcase,the\ninteractionbetweenthevanderWaalsheterostructureandthecavityinduceslong-rangeinteractions.\nThis is an original approach, likely to open a new avenue in the field of moire´ systems, avoiding\ntwisting of solid-state structures. Moire´ systems are attracting growing attention and the field is very\nactiveattheinterfaceofcondensedmatterphysics,quantumoptics,andquantumgases. Thisnewwork\nenriches it and I am convinced that it will be of great interest to the community. The presentation of\nthetheoreticalmodelisclearandsound. Usefultechnicaldetailsofthecalculationsarewelldescribed\ninthesupplementarymaterial. Inaddition,thepaperisclearlywrittenandwillbeaccessibleevenfor\nnon-specialistsinthefield.\nIrecommendthepublicationofthemanuscriptaftertheauthorshaveconsideredthefollowingpoints:\nResponse: We appreciate the reviewer’s recognition of the originality, clearness and accessibility of\nour work. We also thank the reviewer recommending the publication in Nature Communications and\nfor the constructive comments to improve our work. In the following, we respond each individual\ncommentindetail,alongwithcorrespondingchangesmadeintherevisedmanuscript.\nInsectionII.A,itwouldbeusefultosaysomethingaboutthemethodusedinnumericalcalculations.\nResponse: AfterreconsideringSectionII.A,wenoticedthatitwasnotexplicitlystatedthatthematrix\nelements M and E contain information of the Mott-Wannier states. We made it clear by adding\nthe following sentences (in italics): ”𝐸 𝑔,𝑙𝑙′ is the energy gap and 𝐸 𝑏 𝜈 ,𝑙𝑙′ is the binding energy, which\nwe obtain by solving the Mott-Wannier equation (c.f. Supplementary Information Sec. V.A). M is\nthe matrix element of the Moire´ potential in the excitonic basis, describing the Moire´ scattering of\nexcitonswithdifferentmomenta. NotethatM carriestheinformationontheMott-Wannierstates(c.f.\nSupplementaryInformationEq. A25).”.\nWethenaddedmoredetailsregardingtheprocedurewefollowedinSectionIII.Wepreferthislocation\ntoSectionII.Aasitalreadycontainedthedefinitionoftheobservablesusedinthiswork,andbecause\nin Section II.A the reader should focus on the model rather than on the methods used to solve it.\nHere’s what we added: In order to obtain the full exciton-polariton states, we start by solving the\nMott-Wannier equation in the k-space (c.f. Supplementary Information Sec. V.A), which gives us\naccess to the binding energy and the Mott-Wannier states. We considered only the 1𝑠 intralayer\nexciton in the MoSe layer. The Mott-Wannier states are used to construct the excitonic creation and\n2\nannihilationoperators,aswellasallthematrixelementsappearinginEq. 3. RefertoSupplementary\nInformation Sec. I for the complete derivation. We represent the photonic modes using the Fock\nnumberstates {|0⟩,|1⟩}.\nItiswrittenthattheresults”showthatitgivesconfinementsignaturessimilartoatwistinducedMoire´\npotential”buttheauthorsdonotwritepreciselywhatmakesthesignatureunambiguous. Adiscussion\nofthesesignatureswouldbeveryuseful,particularlyforthenon-specialistreader.\nResponse: We thank the reviewer for raising this important point. We clarified in the main text what\nwe mean by ”confinement signatures” by adding the following (in italics): ”... show that it gives\nconfinement signatures similar to a twist induced Moire´ potential, which include the formation of a\nMoire´ BrillouinZone,andtheopeningofminibandgaps.”\nIt is also written that ”B shares the same excitonic operators of the Moire´ potential M” but in a\nparagraphwhereitiswrittenthatM=0afewlinesabove. Thisisverymisleading.\nResponse: We thank the reviewer for pointing out this potential confusion. We decided the remove\n”(i.e. M = 0)”forthatpoint,andweaddeditlaterasfollows:\n”Fig. 2showstheresultsforaclassicallydrivenunstructuredcavityaswellasforaspatiallystructured\ncavity (with a one-dimensional spatial periodicity) both coupled to the 1𝑠 intralayer exciton in the\nMoSe layerwithintheuntwistedheterostructure(i.e. M = 0).”\nResponse: We added the following bracket: ”... are standing waves in the 𝑧 direction (which is\northogonaltothecavityplanes)”.\nThe authors write “We assume that the matter couples with the in-plane component of the electric\nfield and therefore only consider modes for which it is finite.” What does justify this assumption? To\nwhatextentisthisjustifiedinpracticalimplementation?\nResponse: Wecouplelightandmatterthroughminimalcoupling,performingthecanonicalmomentum\nsubstitution p → p+A (in atomic units). Hence, light and matter are coupled through the bilinear\ncouplingp·A,whichyieldmatrixelementswhichareproportionaltop·e,whereeisthepolarization\nvectorofthefield. Ina2Dmaterialsuchtheoneconsideredinthemaintext,aparticlehasasignificantly\nnon-zero value of the momentum only in the in-plane direction (as it cannot move in the orthogonal\none). Since 𝑝 𝑥 , 𝑝 𝑦 ≠ 0, 𝑝 𝑧 ≈ 0,thecouplinghappensonlyforthein-planecomponents.\nThisassumptionisalsowelljustifiedexperimentally(Flattenetal.,Sci. Rep. 6,33134(2016)). Planar\nmicrocavitiesusedwith2DmaterialsareoperatedinTE-likeconfigurationswithstrongin-planefields\nat the layer position, and the observed Rabi splittings in TMD polariton experiments arise from in-\nplane coupling. TM modes can have a finite Ez, but their overlap with the 2D current is small and\ntheircouplingtothebrightexcitonsisnegligibleintypicalgeometries. Includingasmallout-of-plane\ncomponentwouldonlyintroducenegligiblecorrectionsandwouldnotaffectourconclusions.\nPage 6, right-hand-side column, it is written “In the range of small twist angles, however, we can\nidentifylocalizedpolaritonicstates[14].”Whatdoesinducelocalization? Infact,Iamsurprisedthat\nthelocalizationappearsatlowanglesbutapparentlynotatlargerangles,accordingtothetext. Iwould\nexpectthefollowingphenomenology: Forcertainparticularangles,thesystemisperiodicandcannot\ninduce localization; For most angles, on the other hand, the system is non-periodic and can induce\nlocalization. For very low angles but inducing a periodic structure, the moire´ period is very large so\nthat the couplings between subsequent cells are very weak and very small fluctuations can overcome\nthem and give a false apparent localization. This was recently analyzed in a slightly different context\ninPRR6,L042066(2024)andPRA111,043305(2025). Cantheresultsobservedinthemanuscript\nbeexplainedinasimilarway?\nResponse: We thank the reviewer for this insightful comment. The highlighted sentence is indeed\nambiguous. We defined localization according to Ref. 14 (S. Brem, Nano Letters 20, 8534 (2020)),\nwhich states that an excitonic state is localized if its corresponding branch in the linear response is\nbelowtheunperturbedexcitongroundstate(whichisafreeparticle)andifitsshapeisnotparabolic. In\nourFig. 3(a-c),theenergyofsuchstatewouldbeslightlyabove1.48eV. Inotherwords,atlargeangles\n(whentheMoire´ periodissmall),theexcitonicgroundstate(lowestbranchinFig. 3(a))behavesasit\nwasunperturbedbythepotential,henceapproachingtheunperturbedexcitongroundstates. Whileat\nlargeanglesonlythegroundstateisbelowthislevel,atsmallanglesotherstatesappearbelowthefree\nparticlelevel,henceoursentenceIntherangeofsmalltwistangles,however,wecanidentifylocalized\npolaritonic states [14]. As for the other branches, at large angles their shape is parabolic, which is a\nsignatureofdelocalization(S.Brem,NanoLetters20,8534(2020)).\nIn the two interesting works suggested by the reviewer, the authors analyze the localization of Moire´\nstates in terms of tunneling between adjacent Moire´ cells, using a tight-binding approach for quasi-\ncrystallinestructures. Webelievethatsuchmethodsarecomplementarybutoutsideourpresentscope.\nNevertheless,webelievethatreasoningintermsoftunnelingprobabilityfromaMoire´ BZtoanother\nmaybeusefulforaqualitativeunderstandingeveninthecaseofthepresentwork.\nAfterreviewingthediscussion,webelievethatthesentencehighlightedbythereviewerisastand-alone\nsentencewhichdoesnotaddmuchtothediscussiononthetheUPandLPpolaritonicbranchescarried\nout in that paragraph. Therefore, in order to avoid confusion of the reader, we decided to remove\nit. Furthermore, we added a citation to one of the suggested papers at the end of Section II.C: ”In\nshort, the effect of the twisting angle can be modified by the electromagnetic field of a cavity, which\ninfluencestheexcitonicresponsebycausingRabisplittingoftheresonanttransition. Thisresultsina\ndifferentlocalizationfortheupperandlowerpolariton,andmaypavethewayfornewquantumphases\n[PRR6,L042066(2024)]”."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have thoroughly addressed all the comments raised in my previous review. The clarifications and revisions improve the presentation of the manuscript and provide sufficient detail regarding the methodology. I find the current version satisfactory and have no further comments. I therefore recommend the manuscript for publication in its present form.", "reviewer_2": "The authors have carefully answered all my initial concerns and clarified all points. This is a very interesting and useful work. I fully support publication in the present form."}, "author_response": "Responses to the reviewers comments\nWe sincerely thank the two reviewers for their efforts to carefully evaluate our manuscript. We feel\nthat their comments have helped us improve our work and we are very pleased that both reviewers\nrecognizeitsimportance,significanceandvalidity,andsupportitsacceptance.\nTheauthorshavethoroughlyaddressedallthecommentsraisedinmypreviousreview. Theclarifica-\ntions and revisions improve the presentation of the manuscript and provide sufficient detail regarding\nthe methodology. I find the current version satisfactory and have no further comments. I therefore\nrecommendthemanuscriptforpublicationinitspresentform.\nResponse: Wesincerelythankthereviewer1.\nThe authors have carefully answered all my initial concerns and clarified all points. This is a very\ninterestingandusefulwork. Ifullysupportpublicationinthepresentform.\nResponse: Wesincerelythankthereviewer2."}]

Decision letter