On-chip photonic crystal dressed Rydberg exciton polaritons with enhanced nonlinearity in monolayer WS_2

Domain: physics
Published: 2025-11-24
Editorial outcome: major_revision

Abstract
Rydberg exciton polaritons are promising for quantum optoelectronic devices exploiting their strong optical nonlinearities. Conventional polaritonic devices consist of optical microcavities made by vertically stacking distributed Bragg reflectors (DBR) and/or metallic mirrors and excitonic active regions, which may restrain their applicability in photonic integrated technologies. Here, we demonstrate the first on-chip photonic crystal (PhC) Rydberg polaritons based on monolayer WS2, which survive up to 80 K. These Rydberg polaritons exhibit a nonlinearity 8.0 ± 2.3 times larger than that of the ground polaritonic states due mostly to their extended exciton radii and, additionally, due to a stronger spatially localized electric field distribution compared to the dual-DBR geometry. Furthermore, the coexistence of 2s and 1s polaritonic states enables us to experimentally quantify their cross-interaction constant (14.4 ± 6.8 μeV μm2). The first realization of on-chip Rydberg polaritons represents a solid step towards polaritonic-based integrated circuits.

Introduction
Exciton polaritons, hybrid light−matter bosonic particles induced by the strong coupling of photons and excitons, can exhibit long spatial and temporal coherences, small effective masses derived from their photonic constituent, and strong nonlinearities and long lifetimes inherited from the excitonic constituent1,2. Such hybrid quasi-particles have emerged as ideal candidates for implementing and manipulating many-body physics phenomena such as Bose-Einstein condensation (BEC)1–3, solitons4,5, and quantized vortices6,7, which can be harnessed in ultrafast polaritonic transistors and switches8,9, polaritonic interferometers10, and topological polaritonic architectures11,12. Recently, analog polariton simulators of complex Hamiltonians have been implemented by exploiting the interaction between neighbouring polariton condensates13.

To push these technologies into the quantum regime14, one option consists of enhancing polariton-polariton interactions and attaining the polariton-blockade regime15, in which the interaction strength between two polaritons is large enough to produce a shift of the polariton resonance larger than the polariton linewidth, thereby ensuring the generation of single-polariton states. Polaritonic nonlinear interactions, governed by the exciton oscillator strength, the exciton Bohr radius (aB), and the excitonic fraction of the polaritons employed15–17, are thus crucial for realizing polaritonic quantum technologies. In this context, Rydberg excitons, corresponding to excitonic excited states, constitute a promising approach to enhancing polaritonic nonlinearities. Rydberg exciton polaritons have been investigated in vertical Fabry-Perot cavities formed by distributed Bragg reflectors (DBR) and/or metal mirrors surrounding GaAs18,19, Cu2O20, WSe2 17, ReS2 21, and CsPbBr3 22 active regions, among which the TMD based Rydberg polaritons show a record nonlinearity (Table S1). Furthermore, because of their larger spatial extension, Rydberg excitons can also be regarded as a new means of enhanced optical sensing due to their larger sensitivity to their environment23,24, highlighting potential new applications of Rydberg exciton polaritons.

While vertical microcavities have enabled fundamental polaritonic studies, on-chip photonic structures ease their cascadability and improve the compactness of polaritonic devices25. With this aim, polaritons have been lately generated thanks to the coupling of excitonic resonances and photonic modes confined either in dielectric waveguides26–29 or in one- (1D) and two-dimensional (2D) photonic crystals (PhC) and metasurfaces30–34. Among the wide choice of polaritonic materials35 compatible with on-chip photonic structures, monolayer transition metal dichalcogenides (TMDs) appear as the most natural choice: on the one hand, they display excitons exhibiting very large oscillator strength36, and on the other hand, they can be easily exfoliated and transferred onto any material platform37. Indeed, TMD-based exciton polaritons have contributed in the last years to showcase fundamental polaritonic physics38,39, including BEC40, strong nonlinearities39, and parametric scattering41, as well as to implement optoelectronic devices exploiting the spin properties of TMD-based polaritons42. Different excitonic species, including dipolar exciton polaritons43, trion polaritons44, charged biexciton polaritons45, and Moiré polaritons46, have been widely demonstrated. Exciton-polaritons based on Moiré MoSe2/WS2 excitons46 and hybridized interlayer excitons in MoS2 bilayers43,47, have been reported to display the largest polariton nonlinearities, in the order of tens of μeV μm2.

In this work, we aim to demonstrate the first realization of on-chip Rydberg polaritons, employing WS2 monolayers on a 1D PhC cavity structure. PhCs enable engineering of the photonic constituent of the polaritons and, therefore, the polariton dispersion. By modifying the properties of the polaritons’ excitonic constituents through temperature and magnetic field, we observed 2s exciton polaritons up to 80 K with an increase of their Rabi energy by more than 14% at ± 9 T. Furthermore, the polariton-polariton nonlinearity of 2s Rydberg polaritons in the PhC is 8.0 ± 2.3 times larger than the polariton-polariton nonlinearity of polaritonic ground states. This enhancement arises mostly due to the larger exciton radius of 2s excitons. Additionally, PhC-induced confinement of polaritons in real space enables a 15-20% enhancement of nonlinearity compared with dual-DBR geometry. Furthermore, the simultaneous quenching of the oscillator strength (accounted for by the Fano q parameter) and broadening of the Fano resonances’ linewidth indicates the onset of the nonlinearity. These results provide an alternative strategy for enhanced nonlinearity of polariton species in TMD systems and pave the way for implementing integrated polaritonic quantum technologies.

Results
Photonic crystal dressed 1s and 2s exciton polaritons
The system studied in this work is illustrated in Fig. 1a, where one monolayer WS2 and a thin hBN (~15 nm) flake are dry transferred onto a silicon nitride (SiN) based 1D PhC. The PhC is etched onto a 220 nm thick SiN layer, itself deposited onto a 1.2 μm thick SiO2 layer lying on a Si wafer. The 1D SiN PhC is fabricated via electron-beam lithography and reactive ion etching (Methods). Such 1D PhC exhibits anisotropic photonic dispersions that will, in turn, give rise to anisotropic polaritonic ones. Herein, we define X-polarization as the polarization in which the electric field is parallel to the PhC grooves (x-axis in Fig. 1a), and Y-polarization as the polarization in which the electric field is perpendicular to the grooves (y-axis in Fig. 1a). Figure 1b presents the experimental and simulated angle-resolved reflectance of X-polarized modes of the PhC structure with hBN but without monolayer WS2 on top. They exhibit two highly-dispersive modes (labeled as modes 1 and 2) around the targeted wavevectors and energies. Note that the mode dispersions are modulated by Fabry-Perot modes corresponding to the underlying thick SiO2 layer, as attested by the broad reflectance features from 2.10 to about 2.14 eV. These lossy Fabry-Perot modes display a negligible electric field intensity at the WS2 location and, therefore, barely modify the photoluminescence (PL) of the excitons and polaritons (Fig. S1). The electric field profiles in the y−z plane for the X-polarized mode 1 at the targeted angles and energies (red circles in Fig. 1b), corresponding to 1s and 2s energies at 4 K, are shown in Fig. 1c. The electric field intensity at the position where monolayer WS2 will be located (black dashed lines inside panels) is 0.6 and 0.8 times that of the maximum electric field intensity across the entire photonic structure for 1s and 2s excitons, respectively.

We first present the strong coupling in a hybrid hBN/WS2/PhC structure (device 1) at room temperature (RT), as shown in Fig. 2a. The two X-polarized modes 1 and 2 exhibit an anti-crossing around the ground excitonic state of WS2 (~2.01 eV), while the Y-polarized mode remains in the weak coupling regime due to a reduced overlap between the photonic mode and the exciton wavefunction (Fig. S2). The right panel in Fig. 2a shows the corresponding numerical simulation (Methods). Figure 2b shows the detailed angle-dependent reflection spectra at RT around the 1s exciton energy extracted from Fig. 2a. The spectra are fitted with two Lorentzian functions (solid curves in Fig. 2b) to obtain the upper polariton branch (UP) and the lower polariton branch (LP) peak positions, shown as dots in Fig. 2c (Methods). The obtained peak positions are further fitted with a coupled-oscillator model (solid curves in Fig. 2c) with a Rabi splitting in reflectivity (ΩRef) of 25.0 ± 2.0 meV. The corresponding exciton (X) and photon fractions (C) of UP and LP obtained are shown in Fig. 2d. In comparison, the damping rates for bare excitons and cavity photons are γexc = 22.8 meV and γcav = 19.0 meV, respectively (Fig. S3). Therefore, ΩRef is slightly larger than (γexc + γcav)/2 = 20.9 meV, confirming that the system operates in the strong coupling regime. Note that, taking into account the measured damping rates for excitons and photon modes, the actual Rabi splitting between the polariton resonances, defined by the energy separation between UP and LP, is indeed larger than the splitting observed in reflectivity, further confirming the achievement of strong coupling at RT48. Notably, given that within the same flake the bare exciton energy and γexc of WS2 can fluctuate depending on the actual environment (Fig. S4), the excitonic values used for the corresponding modelling of the polaritonic dispersions are those measured at the same location but at larger angles, typically ~30 degrees, where the exciton is not coupled to any photonic mode. Alternatively, such bare exciton properties could also be obtained from measurements in Y-polarization, for which the system operates in the weak coupling regime (Fig. S5)31.

Upon cooling down the sample to 4 K and performing angle-resolved reflectance in device 1, as shown in Fig. 2e, a second anti-crossing around the energy of 2s exciton state of WS2 appears, confirming the coupling of the 2s excitons to the PhC optical mode. In addition, the X-polarized mode 1 also couples to the 1s exciton state with an energy separation of 35.0 ± 3.0 meV, larger than that at RT due to the narrowing of the excitonic line. The corresponding PL emission further confirms the occurrence of strong coupling for 1s exciton polaritons (Fig. S1). Figure 2f plots the detailed angle-dependent reflection spectra around the 2s Rydberg exciton in Fig. 2e. The spectra are fitted with two Lorentzian functions (solid curves in Fig. 2f) to obtain UP and LP peak positions (Methods), shown as dots in Fig. 2g. The obtained peak positions are further fitted with a coupled-oscillator model (solid curves in Fig. 2g) with a Rabi splitting in reflectivity (ΩRef) of 16.7 ± 1.5 meV. The corresponding X and C of UP and LP obtained for 1s and 2s polaritons at 4 K are shown in Fig. 2h. The damping rate for bare 2s excitons and cavity photons are γexc = 6.5 meV and γcav = 19.0 meV, respectively (Fig. S3). Again, ΩRef is larger than (γexc + γcav)/2 = 12.7 meV, reflecting that a narrower excitonic linewidth of 2s exciton makes it easier to fulfill the criterion for a given cavity photon linewidth (γcav) compared with 1s exciton.

The Rabi splitting we measure for the coupling between the 2s exciton and the PhC mode is twice the value of the Rabi splitting measured for the 2s exciton coupling to a cavity mode in a vertical microcavity (ref. 17), where three layers of monolayer TMD were stacked to overcome the reduced oscillator strength. The increased Rabi splitting in our structure is explained by the enhanced overlap between the exciton and the electric field of the PhC mode, which is confined to a subwavelength thickness, as opposed to the vertical DBR structure, where the photonic mode extends over several wavelengths (Figs. S6, S7, Supplementary Note 1). Notably, by positioning the active layers close to the maximum of the electric field, our obtained ΩRef for both 1s and 2s polaritons are comparable with that measured in planar DBR-based microcavities, in which the monolayer is located close to an antinode of the electric field distribution (Table S2). As the Rabi splitting is proportional to the square-root of the oscillator strength and the overlap between the active material and the photonic field (Supplementary Note 1)1, a Rabi splitting ratio between 1s and 2s polaritons of 2.2 could be calculated, which matches well our experimental results, i.e., 35/16.7 = 2.1.

Numerical simulations of the corresponding dispersions (right panel in Fig. 1e), show a clear anti-crossing feature for 1s and 2s excitons coupled with the X-polarized mode 1. To make sure our designed structure exhibits prominent coupling with 1s and 2s excitons, we optimized the polaritonic dispersion by varying three geometric parameters of the PhC, i.e., thickness of the grating (h), period of the unit cell (p), and the filling factor (FF, width of the unetched grating portion over the unit cell dimension) utilizing FDTD (“Methods”, Figs. S8–10). Note that the as-designed 1D PhC cannot only support guided mode resonances but also a so-called bound state in the continuum (BIC) mode at a wavevector equal to zero (Fig. S11)3, although in our study it does not couple to either of the excitonic resonances. However, the photonic branches leading to the BIC mode indeed do couple, exhibiting a coupling strength Ω of 38 ± 3 meV with the 1s excitons at 4 K and at a non zero wavevector.

Temperature and magnetic field dependent behaviors of 2s exciton polaritons
To assess the robustness of the strong coupling regime of 2s excitons in device 1, we conduct temperature and magnetic field-dependent angle-resolved reflectance measurements. The temperature-dependent reflectivity spectra (Figs. 3a, S12) indicate that Rydberg 2s exciton polaritons survive up to 80 K: indeed, the Rabi splitting is larger than (γexc + γcav)/2 at each temperature and ensures that the system operates in the strong coupling regime (Figs. S13–15 and Supplementary Note 2). Further increasing the temperature will enhance phonon scattering and accelerate exciton dephasing, consequently reducing the Rabi splitting31. Besides, temperature-dependent angle-resolved reflectance spectra around 1s excitons from 15 K to 295 K (Figs. S16–18) were also explored. The Rabi splitting is roughly constant (32–35 meV) from low temperatures up to about 200 K, consistent with an almost constant excitonic broadening (7−9 meV). As temperature increases further, the Rabi splitting decreases from 35 ± 3 to 25 ± 2 meV, while the linewidth of the 1s exciton increases from 9 to 25 meV, due to enhanced electron-phonon coupling and an acceleration of the exciton dephasing49. The Rabi splitting is thus also larger than (γexc + γcav)/2 at each temperature, ensuring the robustness of the strong coupling. Herein, the larger intrinsic exciton dephasing of 2s excitons compared to 1s excitons is mainly due to their smaller exciton binding energy and larger Bohr radius (Eb(2) is 1/9 of Eb(1))50, indicating a higher sensitivity of the 2s excitons to the surrounding environment, in particular to phonons50.

Note that we have used Fano resonances31 to fit the polariton dispersions, but whenever the q value was lower than 0.1, we used instead two Lorentzian functions to simplify, as the Fano resonance reproduces a Lorentzian when q = 0. Overall, in the current work we employed Fano resonances to fit the temperature dependence of 2s and 1s bare excitons (Figs. S14, S17) as well as to fit 1s polaritons at low temperature (Fig. S18), while two Lorentzian functions were used to fit the 2s polaritons at low temperature as well as 1s polaritons at room temperature (Figs. 2 and 3) whenever extended studies were conducted at those temperatures. Herein, the smaller q values for the 2s exciton and the photonic mode coupled to it may be due to the reduced overlap between the background broad F-P mode and the exciton and the photonic modes at the 2s exciton energy, respectively. For clarity, we have explicitly indicated in each series of measurements what functional dependence was used to fit the reflectivity spectra.

To explore the stability of Rydberg polaritons, we analysed their magneto-optical properties, as shown in Figs. 3c, S19 for 2s Rydberg and 1s ground exciton polaritons, respectively. Based on perturbation theory within low magnetic fields, the Rabi splitting under non-zero magnetic field, ΩRef(B), is given by: ΩRef(B) ≈ ΩRef(0) + a · (μB · B/R*)2/4, with a a coefficient related to the exciton wavefunction and R* the effective Rydberg constant19. Compared to 1s ground states, 2s excitonic states exhibit relatively weaker binding energy and are thus more sensitive to external magnetic fields and suffer stronger modifications of their relative electron-hole wavefunction (Fig. S20) for the same magnetic field. Indeed, Fig. 3d indicates a faster quadratic increase with magnetic field of ΩRef(B) for 2s Rydberg polaritons than for 1s polaritons, illustrating the expected stronger modification of the coupling strength for 2s polaritons. Eventually, ΩRef(B) could be enhanced by 9% and by up to 14% for 1s polaritons and 2s Rydberg polaritons at 9 T, respectively. It should be noted that even though the magnetic field could induce a Zeeman splitting of the bare WS2 excitons and, therefore, a broadening of the excitonic and polaritonic linewidths and in turn a reduction of the Rabi splitting, 1s and 2s exciton polaritons in the 1D PhC do show a clear enhancement of the Rabi splitting with increasing magnetic field, which is promising for magneto-optical polaritonic devices.

Nonlinearity of PhC 1s and 2s exciton polaritons
To quantify the nonlinearity of 1s and 2s exciton polaritons in device 1 we utilize a picosecond supercontinuum white light source (see “Methods”), enabling the coverage of the whole spectral range defined by 1s and 2s polaritons. Herein, two different experimental configurations are employed to assess the nonlinear interactions between the two different excitonic species: 1) in the first configuration, 1s and 2s polaritons are simultaneously pumped with a broadband (BB) source and, therefore, we can expect nonlinear interaction between 1s-1s polaritons, 2s-2s polaritons as well as 1s-2s polaritons; 2) in the second configuration, 1s polaritons are resonantly pumped with a narrow band (NB) source, enabling the measurement of the 1s-1s nonlinear interaction independently of higher excited state excitonic contributions.

In the BB configuration, upon increasing the pump power density from 5.0 × 10−4 mW μm−2 to 2.8 mW μm−2, the Rabi splitting between the LP and UP gradually decreases, as shown in Figs. 4a, S21 for 2s Rydberg and 1s exciton polaritons. Figure S22 shows the pump power-dependent LP and UP mode energies, and Fig. 4b shows the corresponding power density dependent differential Rabi splitting, ΩRef(N)/ΩRef(0). It is observed that for 1s exciton polaritons, ΩRef(N) under 0.46 mW μm−2 pumping intensity is equal to 90% of ΩRef(0), while for 2s exciton polaritons a transition into the weak-coupling regime is observed above 0.18 mW μm−2 (ΩRef(N) = 0.83 ΩRef(0)), based on the criterion ΩRef < (γexc + γcav)/2.

Increasing the polariton density not only decreases the Rabi splitting (2ΔEsat), due to phase space filling, but also blueshifts both polariton modes (ΔEee), due to the exchange interaction of excitons17,44,51. In our configuration, we find that the LP (UP) blueshifts (redshifts) 1.4 meV (1.5 meV) for 2s exciton polaritons under 0.18 mW μm−2 pumping power density and LP (UP) blueshifts (redshifts) 1.7 meV (1.3 meV) for 1s exciton polaritons under 0.46 mW μm−2 pumping power density. At relatively low power densities (5.0 × 10−4 mW μm−2 to 0.18 mW μm−2), phase space filling is the predominant polariton-polariton interaction mechanism, which is verified by the equal shift of UP and LP in 2s polaritons. Such an effect inhibits the simultaneous occupation of the same state within the blockade radius (Rb) based on the Pauli exclusion principle, and it results in a reduction of the oscillator strength as well as of the coupling strength, as fewer polaritons can be generated effectively46,52–54. Further increasing the power density fills the phase space of polaritons within Rb and leads to saturation. As a result, the polariton interaction potential beyond Rb starts to become significant, causing an overall blueshift in the polariton modes (ΔEee), which is verified by the unequal shift of UP and LP in 1s polaritons. Herein, ΔEsat and ΔEee could be calculated as: ΔEsat = (ΔELP + ΔEUP)/2 and ΔEee = (ΔELP − ΔEUP)/243. Because exchange interaction becomes sizeable at much larger polariton densities (~10 times larger densities than when phase space filling effects become important, Fig. S23), we only discuss the ΔEsat term in the main text.

In the NB configuration, for which we solely pump 1s polaritons, the ΩRef(N) was reduced by 9.3% at the maximum pumping power density of 0.45 mW μm−2 (Fig. S22). Under this configuration, the UP branch of 1s exciton polaritons redshifts 1.1 meV while the LP branch blueshifts 1.5 meV. Note that as the pump power increases, under both BB and NB pumping conditions, the fitted q values for the three Fano contributions reproducing the dispersion of 1s exciton polaritons start to decrease (Fig. S22) at the onset of the resonances’ spectral shifts. Overall, the nonlinear evolution of the polariton energies is thus accompanied by a reduction of the oscillator strength of 1s excitons (q for 1s exciton reduces and FWHM increases), which reduces the Rabi splitting and is consistent with exciton-exciton induced dephasing as previously characterized in TMDs monolayers55.

To quantify the different polariton-polariton interaction strengths for phase-space filling effect (ΔEsat) and of the excitonic species involved therein (1s-1s, 2s-2s, and 1s-2s), we model our polariton system as follows. For a given particle density, we can express the energy shift of each polariton branch associated to 2s and 1s polaritons as a function of the exciton/photon fraction and of the polariton population-dependent phase space filling term43,56, considering an unpolarized pump laser that generates both Y-polarized excitons and X-polarized polaritons (Supplementary Note 3): C1s/2s = X1s/2s in Eqs. (1)-(4) are all 0.5. Eventually, the extracted nonlinearity constants are 49 ± 12, 13 ± 6, and 4.9 ± 0.8 μeV μm2 for gsat2s-2s, gsat2s-1s, and gsat1s-1s in device 1, which exhibit the record values among PhC-based polaritons (Table S3). According to previous literature, the polariton–polariton interaction strength is proportional to the square of the Bohr radius and the Rabi splitting17, i.e., gxx ∝ aB2 Ω. Therefore, the enhancement that one could expect by using 2s polaritons compared to 1s polaritons is given by aB2(2s) Ω2s / aB2(1s) Ω1s = (3.8)2 16.7 / 35 = 7, which is in agreement with the experimental results (about 10 times). A more accurate value can be obtained by pumping 1s and 2s polaritons independently under NB conditions, which enables to extract two independent values of gsat2s-2s and gsat1s-1s. This methodology was applied to a second hBN/WS2/PhC structure (device 2), with the results shown in Figs. S24, S25 in the Supplementary Materials. The analysis of the data provides thus a set of interaction constants, 48.2 ± 6.7 and 8.2 ± 0.9 μeV μm2 for gsat2s-2s and gsat1s-1s, respectively, under NB pumping conditions. These values are comparable to those in device 1. Combining these values with the energy shifts measured under BB pumping in the same structure, we obtain 14.4 ± 6.8 μeV μm2 for gsat2s-1s, which is in good agreement with the theoretical prediction. Overall, the values of nonlinearity enhancement between 2s and 1s polariton as well as the gsat1s-2s interaction constant in both devices are comparable considering the error bars.

While the values of gsat2s-2s and gsat1s-1s for our TMD monolayer are in good agreement with previous reports on TMD-based polaritons, the use of a PhC instead of a vertical microcavity modifies the actual picture. In order to determine the differences in terms of polariton-polariton interactions due to the differences in the electric field distribution between the vertical cavity case and the PhC, we simulated the value of the blueshift for the two via developing a non-linear Bloch-Maxwell FDTD algorithm (Methods), and assuming that gsat for 2s and 1s are 1.36 and 0.34 μeV μm2 in the vertical dual-DBR cavity configuration (using a nonlinearity ratio of 4 from ref. 17). We first quantify the polariton population and its spatial distribution in real space, as well as its dispersion in momentum space; then we obtain the energy shift of UP and LP polaritons in a vertical DBR-based microcavity as a function of pump power. The simulated polariton population as well as the energy shift of each UP and LP peak are used as the benchmarks for comparing the nonlinearity in PhC structures. Considering comparable polariton populations in the DBR-based and the PhC-based microcavities, our simulations indicate that saturation is 15% larger for the 2s excitons and 20% larger for the 1s excitons in the corresponding PhC structures due to the spatial localization of polaritons (Fig. 4d and Supplementary Note 4). As the non-linear coefficients in the Bloch equations are chosen arbitrarily, the ΔE values are only available to compare the ratio between different systems considered. Further systematic simulation of the PhC geometry would be necessary to optimize the nonlinearity enhancement.

Finally, we notice that the nonlinearity of the 2s polaritons in our WS2-PhC structure is slightly smaller than the one measured in a vertical microcavity using three WS2 monolayers (ref. 17). The discrepancy of the calculated nonlinearity may result from several factors: 1) the assumption in ref. 17 that the contribution of the 1s polariton-2s polariton interaction to the measured spectral shifts is negligible, which we have shown to be not; 2) the difference in cavity geometry, which can affect the estimation accuracy of the absorbed number of photons and, thus, of the actual particles density (our structure lacks a top mirror and we assume full absorption at the device surface, neglecting scattering losses by the PhC structure, overestimating the number of polaritons).

Methods
Sample preparation
The demonstrated SiN layer and 1.2 μm SiO2 were grown by low-pressure chemical vapor deposition onto Si substrate. The SiN photonic crystal is then fabricated via e-beam lithography and reactive ion etching. The thin hBN layer and monolayer WS2 were obtained by mechanical exfoliation of the bulk counterparts from 2D semiconductor and then dry transferred onto the as-fabricated 1D PhC via polycarbonate film. The 1D PhC grating parameters are listed in Fig. S1.

Optical experiments
RT angle-resolved PL and reflectance are obtained via a home-built 4f Fourier-space imaging into an Andor spectrometer (SR-500i). An 80× objective with a numerical aperture of 0.75 was used to construct the confocal system for excitation and collection. A 532 nm continuous-wave laser was utilized to excite the sample to obtain the PL. A linear polarizer and a half-waveplate were put into the collecting path to obtain the polarization-dependent angle-resolved spectra. Cryogenic temperature angle-resolved PL and reflectance are built based on Attodry 1000 (Attocube, Inc.), and the detailed schematic diagram for the setup is shown in Fig. S30. A 50× objective with a numerical aperture of 0.8 was used. The spectra are collected into a Princeton Instruments spectrometer with a cooled charge-coupled camera. For the nonlinearity, a supercontinuum pulsed light source (AnYang Laser, YSL, SC-PRO, repetition rate 0.01–80 MHz, pulse duration 100 ps) was utilized to realize two pump frameworks via long-pass and short-pass filters.

Fitting the reflectance spectra of the excitonic and polaritonic species
Due to the interference between the discrete exciton state and a continuum of states, a Fano resonance arises for the PhC cavity mode, which can be described by31:

F(E) = (q + (E − E0)/2γ)2 / 1 + ((E − E0)/2γ)2 (5)

with E the energy, E0 the exciton resonance energy, γ representing the resonance linewidth, and q the Fano factor, which describes the asymmetry of the lineshape. As the bare 1s exciton at RT and 2s exciton at 4 K show a symmetric lineshape with a close-to-zero q Fano parameter, we utilize two Lorentzian functions to obtain the polaritonic UP and LP peak positions. For 1s exciton and polariton at 4 K, we utilize one Fano lineshape to obtain the bare exciton positions and three Fano lineshapes, i.e., bare exciton background, UP and LP peaks, to fit the Fano-type polaritonic reflectance spectra (Fig. S13 and Supplementary Note 2). For the temperature-dependent analysis of the bare excitonic reflectance spectra (Figs. S14, S17), we have used Fano analysis for both 1s and 2s excitons, from which we know at which temperatures we can use two Lorentzian functions to fit the reflectance spectra. For the temperature-dependent analysis of the coupling strong regime, we have used Fano analysis for 1s polaritons and two Lorentzian functions for 2s polaritons.

Calculation of polariton/exciton density
For the nonlinearity measurements, we utilized a pulsed laser with a pulse duration of 100 ps, which is longer than the exciton lifetimes of 1s and excited 2s exciton. Therefore, the pumping source can be regarded as a continuous wave laser, and we utilized the peak energy power of each pulse. The corresponding incident pulse energy can be calculated as: Epol = Pτ / Rd, where P is the pump power, τ is set as 48/50.7 fs for the 1s/2s polaritons in WS2, regarding a photon lifetime of 35 fs, R is the repetition rate of the laser, and d is the pulse duration. Then the absorbed photon by the polariton branches of the super continuum pulsed light source was estimated via the R integral of the absorption of the pumped laser profile: E(k) = Eto Rt Aabsor(k) / Alaser(k), where Aabsor(k) is the absorbed dip of the spectra at different k-vectors and Alaser(k) is intrinsic pumped laser profile. Then we can calculate the polariton density as N = E(k)/Epol Abeam, where Abeam = πD2/4 with D roughly 2.5 μm and Epol is the energy of the polaritons. As we utilize unpolarized light (with a polarization degree 20% along x direction), Y-polarized excitons (in the weak coupling regime) will also be generated (Figs. S31, S32). According to Eqs. 1–4, we included the effect of Y-polarized excitons, considering the excitonic lifetime and the bare excitonic absorption, with measured excitonic linewidths of 6.5 and 7.9 meV for 2s and 1s excitons, respectively, at 4 K.

Numerical methods
All simulations are performed using the non-linear Bloch-Maxwell FDTD57. The system we consider is similar to the experimental sample. We make use of the periodicity of the system and consider one period under periodic boundary using Bloch theorem58 along the x-direction and perfectly matched layers (PML) along the y-direction (Fig. S33) for the calculations in the linear regime. In the non-linear regime, since different k-vectors are coupled due to the exciton-exciton scattering, the Bloch theorem is no longer applicable in the framework of our algorithm, and we therefore use PMLs along both x and y directions and consider 200 grating periods.

Simulation results obtained in Fig. 1 are obtained using the total field/scattered field method58, calculating the reflectivity spectrum for each k-vector independently. We used a pulsed plane wave source with a finite k-vector, with a pulse duration 1 fs in order to excite the system broadband. In order to fit the experiment, the refractive indices have been slightly adjusted between the 4 K and RT simulations. The exciton resonances and the light-matter coupling coefficients are also adjusted to fit the experiment59 (see Supplementary materials for detailed parameters).

Simulation results obtained in Fig. 4d are obtained in the framework of the non-linear Bloch-Maxwell FDTD. We pump the system using an X-polarized plane wave CW source resonant and measure the reflectivity at the anti-crossing. We then compare the obtained dispersion to the dispersion of the bare system to obtain the blueshift. We repeat this procedure on a microcavity polariton system in order to compare the effective interaction coefficient (Figs. S34, S35).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript reported the strong coupling between 2s state of excitons and a cavity photon on a SiN waveguide. Moreover, they identified the nonlinearity of the 2s Rydberg exciton polaritons. Benefiting from the broad range of photonic band, they can compare the nonlinearity with 1s exciton polaritons simultaneously. In the end, they show the propagation properties of exciton polaritons due to the finite wavevector. The evidence of strong coupling is clear, and the characterization of nonlinearity is solid. I would like to suggest the manuscript to be published in Nature Communications if the following questions can be addressed properly.\n1、I suggest the authors show the full dispersion of the photonic mode in the manuscript. considering the grating can support both guided mode resonances and the so called BIC mode.\n2、 With increasing temperature, the Rabi splitting of 1s state increase from 22 meV to 35 meV, which is quite surprising to me. Could the author show a temperature dependence of the Rabi splitting and elaborate more on it?\n3、The mode volume of DBR-metal cavity used in the reference 17 should be comparable with the waveguide mode used in the manuscript. Moreover, the monolayer stamped on the top of the waveguide will only interact with the evanescent wave of the mode. I am still confused how the interaction strength could be so strong.\n4、It seems like the author can reproduced the experiment with FDTD simulation. The parameters are shown in the supplemental materials. The light matter coupling is described by a coefficient Omega. I am wondering the relationship between the Omega and the Rabi splitting or the coupling strength. Please clarify why Omega of 1s state used for the figure 1c and figure1d is so different. And in figure 1d, the Omega for 2s state is more than 10 times smaller than that of 1s. However, the Rabi splitting for 2s state and 1s state are not that different, which are 15 meV and 35 meV respectively.\n5、The last section of the current manuscript discussed the propagation of 1s exciton polaritons, which does not seem to be relevant with the topic of the manuscript. I suggest the authors to move it to the supplemental materials.", "reviewer_2": "In this manuscript, Shang et al. demonstrate the first realization of on-chip Rydberg polaritons in monolayer WS2 coupled with a 1D photonic crystal (PhC) cavity structure. By leveraging the large Bohr radius of Rydberg excitons and PhC-induced field localization, the authors achieve a nonlinearity enhancement of 23 ± 10 times for 2s polaritons compared to ground-state 1s polaritons, with robust operation up to 80 K and magnetic field-tunable Rabi splitting.\nThe manuscript addresses a critical gap in on-chip polaritonic devices by combining monolayer TMDs with PhC engineering, yielding groundbreaking results in Rydberg polariton nonlinearity and propagation. The experimental and theoretical framework is rigorous, and the results are well-supported by angle-resolved spectroscopy and FDTD simulations. I think this manuscript is suitable for publication in Nature Communications after revision. The following comments should be addressed in detail to improve the manuscript:\n(1) The introduction mentions that Rydberg excitons have large Bohr radii and are promising for enhancing optical nonlinearities. However, it is not clear how the specific properties of Rydberg excitons in monolayer WS2 compare to other materials. Could you provide more details on this?\n(2) In Figure 1c and 1d, the strong coupling between the WS2 excitons and PhC modes is demonstrated. How do the specific geometric parameters of the PhC (period, width) influence the coupling strength and the resulting polariton dispersion?\n(3) The temperature-dependent measurements show that the 2s exciton polaritons survive up to 80 K. What specific mechanisms cause the degradation of the strong coupling regime above this temperature, and are there any strategies to mitigate these effects?\n(4) From Figure S3 in SI, it can be seen that the line widths of 1s and 2s excitons and cavity photons are different at room temperature and 4 K. Does this have an impact on the judgment of strong coupling? For example, does the narrower line width of the 2s exciton at 4 K mean a higher quality coupling to the photon mode?\n(5) The propagation decay constant of the PhC polaritons is measured to be around 6.3 μm. How does this value compare to other polariton systems, and what factors limit the propagation distance in this case?", "reviewer_3": "In the manuscript titled “Enhanced nonlinearity of photonic crystal Rydberg exciton polaritons in monolayer WS2”, the authors present a photonic crystal (PhC)-based on-chip polaritonic device demonstrating strong coupling between PhC modes and both the ground-state and Rydberg-state excitons of WS2. They conduct a detailed characterization of the polariton states through reflection measurements under varying magnetic field and temperature, as well as nonlinearity and propagation measurements. The key claim of the work is a 23-fold enhancement of the non-linearity associated with the Rydberg state polaritons relative to the ground state. They attribute the enhancement to the enhanced overlap of the PhC mode electric field with the excitons in the PhC geometry. The authors have also estimated the interaction constants and propagation decay constants for the ground state and excited state exciton-polaritons.\nThe manuscript is thorough and clearly written. The exploration of strong polaritonic effects, particularly large nonlinearity using PhCs instead of the more common Fabry-Perot cavities, is of potential interest. However, I have several comments that should be addressed before recommending publication:\n1. Reference 17, which probed nonlinearity of WSe2 embedded in a Fabry-Perot cavity, reported that the interaction for the 2s exciton-polaritons is 4.6 times stronger than for the 1s exciton-polaritons. While the current study finds a similar nonlinearity for 2s exciton-polaritons, it reports a 23-fold suppression for 1s exciton-polaritons. Direct comparison is difficult due to differences in geometry and photon absorption estimates, yet this discrepancy is notable. It is not clear how this difference (with respect to cavity geometry) can result from “the combination of an extended exciton radius due to the excited-state nature of the employed excitons, and to the PhC-induced spatial field localization.” Since this is a key point in the abstract, the authors should clarify their explanation.\n2. Building on the previous comment, the title is somewhat misleading – it seems to suggest that using a PhC (instead of a traditional Fabry-Perot cavity) enhances the nonlinearity. However, the authors themselves note that the extracted nonlinearity constants align with previous studies. I recommend revising the title to more accurately reflect the findings\n3. The focus of the paper is somewhat diluted. It is unclear how the propagation measurement supports the central claim of the paper.\n4. In the temperature-dependent measurements (Figure 2c), the coupling strength of 2s excitons decreases monotonically, while 1s excitons maintain a nearly constant coupling strength. Could the authors provide further insight into this monotonic decrease for 2s excitons?\n5. In the magnetic field-dependent measurements (Figure 2d), the coupling strength exhibits quadratic enhancement with the magnetic field, but this enhancement is not symmetric around zero field, particularly for 2s excitons. Could the authors clarify the origin of this asymmetry?\n6. In the nonlinearity measurements, the authors have performed experiments in two configurations, viz., the broadband (BB) pumping of both the 1s and 2s states, and a narrow band (NB) pumping of the 1s states alone. Therefore, the authors could effectively compare the interactions between the various types of excitonic states. To further probe the role of the Rydberg excitons, I suggest an additional experiment where 2s excitons are resonantly pumped, analogous to the narrowband 1s excitation. While phonon assisted relaxation to the 1s state could still occur, this measurement could provide valuable insights into the intrinsic 2s-2s interactions.\n7. Determining whether the system is in the strong or weak coupling regime requires accurate extraction of the bare exciton energy and linewidth. How do the authors obtain these values? Do they use the same device with different polarization of light?\n8. The authors use dashed guidelines to indicate the energy values of LP and UP in several figures. Accurate extraction of these values is crucial for assessing temperature dependence, magnetic field dependence, Rabi splitting, nonlinearity, etc. How are the LP and UP energies determined? Could the authors elaborate on their extraction/fitting method? That is important since in some cases, e.g., in Fig. S10, the energy shift is not prominent.\n9. Could the authors further elaborate on the rationale behind Equations (1–4) in the main text? For example, it is unclear why the 1s-2s terms contributing to the energy shift of 2s excitons depend on N_1s but not on N_2s.\n10. The propagation experiment methodology is unclear. Why do the authors block certain regions (white rectangles) in the angle-resolved spectra? With the full spectrum available, they could simply select the desired energy and angle range for integration. Additionally, how do they physically block specific energy ranges in the actual experiment?\n11. In Figure 4e, the PL intensity decay for the exciton polaritons at energies of 2.11 and 2.15 eV are plotted. However, as observed in Figure 1b and discussed towards the beginning of the Results section, the reflectance data of the bare cavity is dominated by Fabry-Perot modes corresponding to the underlying thick SiO2 layer between 2.10-2.14 eV. How did the authors distinguish the exciton-polariton features despite the strong Fabry-Perot oscillations?\n12. Previous studies have explored excitons in TMDCs using 1D photonic crystals (e.g., L. Zhang et al., Nat. Commun. 9, 713 (2018)). Could the authors comment on how their approach differs in any significant way? Additionally, could they include dispersion data for the bare cavity across the full range of measured angles in the SI?\n13. Unlike Fabre-Perot cavity, 1d PhCs has additional anisotropy. Could the authors comment whether this anisotropy affects exciton-polariton properties and their nonlinearity?", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_5": "The work reports on the experimental investigation and theoretical interpretation of Rydberg polaritons in one-dimensional (1D) photonic crystal lattices. In particular, a WS2 monolayer is deposited, with a 15 nm buffer of hBN, on top of a 220 nm thick 1D-grating in SiN sitting on a glass (SiO2) cladding. Optical excitations are probed in reflectance spectroscopy. This result is, per se, not novel. Exciton-polaritons have been reported in similar settings (actually, from the very same two-dimensional semiconductor, WS2, coupled to a SiN photonic crystal, with a two-dimensional lattice, instead of a simpler 1D one), see, e.g., Nano Letters vol. 20, 5292 (2020).\nThe Rabi splitting in the order of 20 meV reported in these works is compatible, which is good to begin with, but not sufficient to claim a significant innovation as a result. In addition, polariton linewidths are extremely large in the experiment reported here, roughly of the same magnitude as the Rabi splitting basically, which makes the strong coupling hardly distinguished from the measurements, in fact. This is actually much worse than what was shown in the quoted Nano Letters. The authors may want to give an interpretation on this. Looking at the spectra of Figs. 1 and 2, one might actually even doubt that the sample is in the strong coupling regime, was not for a clever lineshape fit.\nIn any case, going further than the linear polariton physics, the authors explore the coupling between excited Rydberg states from the WS2 monolayer excitonic response to the same photonic eigenmode of the 1D photonic lattice. The experimental evidence points towards an enhanced nonlinear response of the 2s-related exciton polariton as compared to the 1s-related one. In particular, the 2s-2s nonlinearity is quantified to be (with large exp error, though) more than 20 times larger as compared to the 1s-1s. This result might be considered as an interesting one, per se, since it justifies current interest in exploring Rydberg polariton physics for various applications. However, also in this case the result cannot be considered as completely unexpected or novel. In fact, as also acknowledged in the manuscript large nonlinearities for the 2s-related exciton polaritons were already reported in Nat. Comm. vol. 12, 2269 (2021), albeit in a vertical microcavity geometry.\nauthors shuold make sure that when exciting the system with the pulsed pump, any excitation of the exciton reservoir can be neglected, otherwose the estimat eof the nonlinear might be inaccurate\nHere the authors also report estimates for the polariton propagation constant, which is obtained from non-resonant excitation experiments in the few microns range. Also in this case, while it may be considered one of the first experiments showing Rydberg polariton propagation in WS2 monolayers, much longer polariton propagation distances have been already shown in alternative materials, so also in this case I feel very reluctant to recommend publication in Nature Communications, in the absence of a stronger argument justifying the novelty of this work as compared to the literature. Honestly, the linewidths of these polaritons are really broad, and thus losses must be huge. What are the reasons for that? The authors could elaborate a bit more on this aspect, and try to figure out mitigation strategies, if any. In fact, I would expect resonant modes from the underlying SiN grating to display much narrower lines (notice that a photon lifetime of 35 fs quoted in methods...this is really short for a dielectric grating!), thus affecting the overall polariton losses.\nIn conclusion, while the experimental results reported in this work, as well as the agreement with numerical simulations based on the nonlinear finite-difference time-domain method, seem to be well conceived and reported, the overall novelty and main claims of enhanced figure of merits as compared to the literature are not sufficiently sufficiently justified or groundbreaking to be published in Nature Communications. There are certain elements of novelty that deserve publication of these results, but I would still consider them as incremental and, as such, suited to a more specialized journal in condensed matter physics, not necessarily belonging to the Springer Nature group. As it stands, I would feel very reluctant to reconsider this work for publication here, in the absence of a truly game changing result based on these materials."}, "author_response": "Response to Reviewers (NCOMMS-25-13902A)\nReply to Reviewer 1\nGeneral Comment:\nThe manuscript reported the strong coupling between 2s state of excitons and a cavity photon on a SiN waveguide. Moreover, they identified the nonlinearity of the 2s Rydberg exciton polaritons. Benefiting from the broad range of photonic band, they can compare the nonlinearity with 1s exciton polaritons simultaneously. In the end, they show the propagation properties of exciton polaritons due to the finite wavevector. The evidence of strong coupling is clear, and the characterization of nonlinearity is solid. I would like to suggest the manuscript to be published in Nature Communications if the following questions can be addressed properly.\nResponse:\nWe sincerely thank the referee for the favorable evaluation of our work and for summarizing clearly the main achievements of our work. We wish to express our deep gratitude and respect for reviewing our manuscript.\nHerein, we would like to point out that, due to reviewers 3&4 and 5 questioning on the analysis of the reflectance spectra, to obtain the polaritons' spectral position, we have fitted all reflectance spectra again with Fano functions. Furthermore, due to reviewers 3&4 and 5 questioning on the effect of optical and excitonic anisotropies on the measured nonlinearities, we have revised equations 1-4 of the main text to take into account cross-polarized contributions to the polariton nonlinearity. Based on this new formulation, we have reanalyzed the data (device 1) shown in the first version of the manuscript (Figure 4) as well as a new series of measurements on a second identical structure (device 2) fabricated for answering some of the questions raised during the revision of the article (Figure S24 and S25). See details in comment 13 to reviewer 3 and comment 3 to reviewer 5. The two structures give consistent values for the Rabi splittings and nonlinearities.\nComment 1:\nI suggest the authors show the full dispersion of the photonic mode in the manuscript. considering the grating can support both guided mode resonances and the so called BIC mode.\nResponse:\nWe thank the reviewer for the suggestion on comprehensively clarifying the full dispersion of the bare photonic modes. As shown in Figure R1a and b, the X-polarized bound state in the continuum (BIC) mode appears at zero wavevector and at an energy of ~2.11 eV.\nGiven the large detuning between the BIC mode and the 1s exciton, the coupling happens rather between the excitonic mode and the photonic branches at a nonzero wavevector that end up forming the BIC mode at zero wavevector. For completeness, we have explored the coupling between these photonic branches and the 1s excitons. Figure R1b and c display a zoom-in into the polaritons' dispersion around the corresponding energies and wavevectors. The detailed angle-resolved reflectance spectra near the anti-crossing area are shown in Figure R1d. The angle-resolved reflectance spectra are fitted via a three-Fano lineshape function (see comment 2 for details) to obtain the peak positions of the upper polariton branch (UP) and lower polariton branch (LP). Further, the obtained peak positions are fitted with a coupled-oscillator model (Figure R1e), which indicates a coupling strength Ω of 38 ± 3 meV, larger than (gamma_exc + gamma_cav)/2 = (8.9+24)/2 = 16.5 meV, which ensures the system is in the strong coupling regime. Not surprisingly, this coupling strength is very close to the one observed for the photonic mode coupled with the 1s exciton at 4 K that is analyzed in the main manuscript.\nBased on the reviewer’s suggestion (as well as comment 12 of reviewer 3), we have updated Figure 1 in the main text and presented an overview of our cavity properties, as shown in Figure R2 below. To highlight the full dispersion of the photonic modes and indicate explicitly the position of the BIC mode, we added the corresponding polariton dispersions at 4 K (Figure R1) into the supplementary information, illustrating the formation of polaritons with these additional photonic modes.\nFigure R1. Full dispersion of the X-polarized photonic modes, including a BIC mode.\na, The full dispersion of the X-polarized photonic modes with monolayer WS and hBN 2 on top, displaying a BIC mode around 2.1 eV at null wavevector. b, Zoomed-in polariton dispersion at 4 K of 1s exciton coupled to the photonic branch leading to the BIC mode. c, Replotted b after subtracting the signal at k=0. d, The detailed angle-resolved reflectance spectra in c for X-polarized BIC near the anti-crossing area (white dashed boxes in b and c). e. Fitting of the UP and LP peak positions with a coupled-oscillator model indicates a coupling strength of 38 ± 3 meV. f, The corresponding photonic and excitonic fractions of UP and LP for X-polarized photonic mode coupling with 1s exciton whose dispersion is shown in (e).\nFigure R2. Demonstration of as-fabricated X-polarized photonic mode. a, A schematic diagram of PhC dressed Rydberg exciton polariton based on a hybrid hBN/WS /PhC structure. b, The experimental angle-resolved reflectance spectra for X-polarized modes of the bare PhC structure with hBN but without monolayer WS on top. Two black dashed lines indicate the energy for the 1s and 2s excitons at 4 K, while the red dashed circles indicate the regions around zero detuning. c, Middle two panels: the simulated electric-field profile in the y−z plane for X-polarized mode 1 at the targeted angles and energies (red circles in b) for the hBN/PhC structure, considering an hBN thickness of 15 nm. Side two panels: The corresponding integrated electric-field intensity across one unit cell for the 1s and 2s exciton wavelengths, indicating that the electric-field intensity at the WS position (black dashed lines) is 0.8 and 0.6 times that of the maximum electric field intensity across the whole structure for 1s and 2s excitons, respectively.\nRevision:\nLine 147, page 7, main text: added “Note that the as-designed 1D PhC can not only support guided mode resonances but also a so-called bound state in the continuum (BIC) mode at a wavevector equal to zero (Fig. S11), although in our study it does not couple to either of the excitonic resonances. However, the photonic branches leading to the BIC mode indeed do couple, with a coupling strength Ω of 38 ± 3 meV, with the 1s excitons at 4 K and at a nonzero wavevector.”\nRevised Fig. 1 as Fig. R2 in the main text (both figure caption and description in the main text).\nSupplementary materials: added Supplementary Fig. S11 to present the full dispersion of the X-polarized photonic mode and the BIC mode at k = 0.\nComment 2:\nWith increasing temperature, the Rabi splitting of 1s state increase from 22 meV to 35 meV, which is quite surprising to me. Could the author show a temperature dependence of the Rabi splitting and elaborate more on it?\nResponse:\nActually, the Rabi splitting for the 1s exciton polariton decreases when going from low temperature (35 meV) to room-temperature (25 meV). Our actual statement was: “In addition, the X-polarized mode 1 also couples to 1s exciton states with an energy separation of 35.0 ± 3.0 meV, larger than at RT due to the narrowing of the excitonic line”. Herein, we have followed the reviewer's suggestion and provided now a thorough study of the temperature-dependent strong coupling of 1s state exciton polaritons in WS . We have thus fabricated a new photonic crystal device (device 2) onto which we have transferred again hBN and WS and performed therein the temperature dependence dispersion measurements. Figure R3 presents the temperature dependent reflectance spectra around the 1s exciton energy from 295 K to 15 K. Since the bare exciton absorption line (accessible typically at large angles) can be distinguished in each spectrum, we could extract the bare exciton energy and linewidth of the 1s exciton for each temperature. The corresponding reflectance spectra of the 1s exciton for all temperatures are shown in Figure R4a. Each reflectance spectrum is fitted with an asymmetric Fano lineshape function due to the interference between the discrete 1s exciton state and a continuum of states (Nat. Commun. 9, 713 (2018)):\nF(E) = (q +(E -E0)/2gamma)^2 / 1+(E -E0)/2gamma)^2 (1)\nwith E the energy, E0 the exciton resonance energy, gamma representing the resonance linewidth, and q the Fano factor, which describes the asymmetry of the lineshape.\nFigure R3. Temperature dependent reflectance spectra around 1s exciton for device 2 from 295 K to 15 K.\nThe extracted peak energy and the linewidth for the 1s exciton versus temperature are shown in Figure R4b and c. As temperature decreases, the peak position blue shifts from 2.028 to 2.086 eV, which is due to the reduction in electron-phonon interactions and an increase in the band gap at lower temperatures (Nano Lett. 21, 6240-6246 (2024)). In the same temperature range, the linewidth decreases from 24.8 to 7.0 meV, which is ascribed to the decreased scattering of excitons with phonons (Nat. Commun. 7, 13279 (2016)).\nFigure R4. Temperature dependent reflectance spectra of 1s exciton in device 2 from 15 K to 295 K at large angles (a), the fitted peak position (b), and linewidth (c).\nFigure R5. Temperature dependent Rabi splitting for 1s polariton and (gamma_exc + gamma_cav)/2.\nSubsequently, we extracted the Rabi splitting for each temperature from Figure R3 and plotted it in Figure R5. The Rabi splitting is larger than (gamma_exc + gamma_cav)/2 at each temperature, ensuring that the system operates in the strong coupling regime from 15 K to 295 K. One can observe that the Rabi splitting is roughly constant (32-35 meV) from low temperatures up to about 200 K, consistent with an almost constant excitonic broadening (7-9 meV). As temperature increases further, the Rabi splitting decreases from 35 ± 3 to 25 ± 2 meV, while the linewidth of the 1s exciton increases from 9 to 25 meV due to enhanced electron phonon coupling and an acceleration of the exciton dephasing (Nat. Commun. 9, 713 (2018)). Interestingly, the two studied devices (the one analyzed in the main manuscript and the new one fabricated for answering the reviewers’ question) display the same quantitative values (~35 meV at low temperatures for both devices, and 25 meV at room-temperature), confirming the robustness of the photonic crystal fabrication and the quality and reproducibility of the hBN/WS transfer.\nBased on the reviewer’s suggestion, we have added a discussion where we present the temperature dependent Rabi splitting in the main text and put Figure R3-5 into the supplementary information to present the corresponding results.\nRevision:\nLine 154, page 7, main text: revised as “the Rabi splitting is larger than (gamma_exc + gamma_cav)/2 at each temperature and ensures that the system operates in the strong coupling regime (Fig. S13-15 and Supplementary Note 2). Further increasing the temperature will enhance phonon scattering and accelerate exciton dephasing, consequently reducing the Rabi splitting.”\nLine 156, page 7, main text: added “Besides, temperature-dependent angle-resolved reflectance spectra around 1s excitons from 15 K to 295 K (Fig. S16-18) were also explored. The Rabi splitting is roughly constant (32-35 meV) from low temperatures up to about 200 K, consistent with an almost constant excitonic broadening (7-9 meV). As temperature increases further, the Rabi splitting decreases from 35 ± 3 to 25 ± 2 meV, while the linewidth of the 1s exciton increases from 9 to 25 meV, due to enhanced electron-phonon coupling and an acceleration of the exciton dephasing (Nat. Commun. 9, 713 (2018)). The Rabi splitting is thus also larger than (gamma_exc + gamma_cav)/2 at each temperature, ensuring the robustness of the strong coupling.”\nLine 156, page 7, main text: added “Note that we have used Fano resonances to fit the polariton dispersions, but whenever the q value was lower than 0.1, we used instead two Lorentzian functions to simplify, as the Fano resonance reproduces a Lorentzian when q = 0. Overall, in the current work, we employed Fano resonances to fit the temperature dependence of 2s and 1s bare excitons (Fig. S14 and S17) as well as to fit 1s polaritons at low temperature (Fig. S18), while two Lorentzian functions could be used to fit the 2s polaritons at low temperature as well as 1s polaritons at room temperature (Fig. 2 and 3) whenever extended studies were conducted at those temperatures. For clarity, we have explicitly indicated in each series of measurements what functional dependence was used to fit the reflectivity spectra.”\nMethods, main text: added “Fitting the reflectance spectra of the excitonic and polaritonic species”\nSupplementary materials: added Supplementary Fig. S16-18 and Note 2 to demonstrate the temperature-dependent behaviors of 2s and 1s exciton polaritons and Fano-type excitonic and polaritonic reflectance spectra fitting.\nComment 3:\nThe mode volume of DBR-metal cavity used in the reference 17 should be comparable with the waveguide mode used in the manuscript. Moreover, the monolayer stamped on the top of the waveguide will only interact with the evanescent wave of the mode. I am still confused how the interaction strength could be so strong.\nResponse:\nWe thank the reviewer for raising this interesting question that will allow us to clarify the large interaction strength in our PhC configuration, which has been scarcely exploited compared to the standard vertical cavity configuration. If, for the moment, we obviate the difference in active material (WSe in Ref. 17 and WS in our case), the main difference in coupling strength will originate from the distinct exciton-electric field overlap between the two structures. Figure 4a of the Supplementary Material of Ref. 17 shows the calculated photonic field distribution along with the position of the WSe layers in the structure studied in Ref. 17. It should be noted that the WSe monolayers are not positioned at the antinodes of the electric field distribution, but rather between an antinode and a secondary maximum, leading to a coupling between the active material and the photonic field that does not attain its potential maximum, but rather just 40% of the potential maximum. This leads to a smaller Rabi splitting than what can be expected if the monolayers were located at the maximum of the photonic field. In our system, contrary to what one could initially think, the active layers are indeed positioned close to the maximum of the electric field, as can be seen in Figure R2c (now Figure 1c) of our main manuscript.\nTo provide a more accurate comparison, we would ideally need to compare our structure with a vertical microcavity with the same active layer as ours and in which the monolayers are located at the maximum of the electric field intensity distribution. To do so, we can compare, for instance, Ref 52 (Nat. Commun. 14, 1512 (2023)), where WS is also employed as an active region and located at the center of the cavity. In this case, the Rabi splitting for 1s excitons at room temperature becomes stronger, of the order of 32-35 meV, which matches consistently our Rabi splitting values.\nTo give a global overview of the Rabi energy of intralayer exciton-based exciton polaritons in TMDs, we present a table below where we make a comparison among representative TMD based microcavities. Within PhC based polaritons, our Rabi energy (35 meV at 4 K) is indeed comparable to previous works (17.6 to 32 meV). However, when looking into different TMD based configurations, e.g., DBR based cavity, Bloch surface wave waveguided polariton, and metal-based nanocavity, our Rabi energy at room temperature (22-25 meV) is far from being the maximum measured, even with the same active material.\nOverall, the interaction strength in our case is explained by the rather optimized overlap between the WS monolayer and the electric field. To reassure the reader that this is so, new Figure 1c of the main manuscript provides the intensity of the electric field at the position of the TMD monolayer. Besides, we have included Table R1 in the supplementary information to compare quantitatively our results with those previously published.\nTable R1. Comparison of Rabi energy among representative TMD based configurations.\nRevision:\nLine 111, page 5, main text: added “The electric field intensity at the position where monolayer WS will be located (black dashed lines in side panels) is 0.6 and 0.8 times that of the maximum electric field intensity across the entire photonic structure for 1s and 2s excitons, respectively.”\nLine 145, page 7, main text: added “Notably, by positioning the active layers close to the maximum of the electric field, our obtained Ω for both 1s and 2s polaritons are comparable with that measured in planar DBR-based microcavities, in which the monolayer is located close to an antinode of the electric field distribution (Table S2).”\nSupplementary materials: added Table S2 to compare the Rabi energy among representative TMD based configurations.\nComment 4:\nIt seems like the author can reproduced the experiment with FDTD simulation. The parameters are shown in the supplemental materials. The light matter coupling is described by a coefficient Omega. I am wondering the relationship between the Omega and the Rabi splitting or the coupling strength. Please clarify why Omega of 1s state used for the figure 1c and figure1d is so different. And in figure 1d, the Omega for 2s state is more than 10 times smaller than that of 1s. However, the Rabi splitting for 2s state and 1s state are not that different, which are 15 meV and 35 meV respectively.\nResponse:\nWe thank the reviewer for the constructive suggestion on clarifying the coupling coefficient. The light matter coupling coefficient, Omega’ (Ω’), that is used in the FDTD simulations, is completely independent of the distribution of the photon and exciton modes: it is a coefficient that depends solely on the exciton properties (proportional to the exciton oscillator strength). In our simulations, it is chosen as a fitting parameter (read further below), whose numerical values can be compared to those necessary to reproduce existing experimental results in literature, in particular the results of Ref 52.\nMore precisely, the Rabi splitting is proportional to the square root of the normalized overlap between the photonic and exciton modes, which is taking into account automatically in our FDTD calculations by the design of the simulated structure, times the light matter coupling coefficient, Ω_Ref ~(Ω’)2 *sigma_overlap (Phys. Rev. B 106, 115424 (2022); Condens. Matter Phys. 17, 23702 (2014); A. Kavokin & al. Microcavities). To illustrate this, we performed FDTD simulations of a vertical 1D microcavity where the only parameter that we modified was the value of Ω’. As shown in Figure R6, a 4-fold Ω’ increase induces a 2-fold increase of the Rabi energy, as expected.\nFigure R6. 1D FDTD simulation of double-6 pairs TiO /SiO based exciton-polariton microcavity for two different values of the light matter coupling coefficient.\nThe difference between the Ω’ values used in Figure 1c and 1d (52 meV at 4 K to 11.2 meV at RT) can be explained by the fact that our FDTD simulations do not take into account polariton-phonon coupling. Indeed, Figure 1c and 1d show that the Rabi splitting decreases from 35 (at 4 K) to 25 (at RT) meV, which is mainly due to the exciton dephasing resulting from phonon scattering as temperature increases (Nat. Commun. 9, 713 (2018)). As the only free parameters in our FDTD simulations are the spatially (and spectrally) varying refractive index distribution, which defines the photonic structure, and the excitonic resonance properties, defined by the exciton resonance energy and Ω’. To account for the reduction in Rabi splitting by polariton-phonon interaction, we need to modify Ω’ (for each temperature the exciton energy is taken from the experimental reflectivity measurements, and it is thus fixed). As explained in the previous paragraph, a reduction of Rabi splitting is necessarily accompanied in our model by a reduction of Ω’.\nFor the ratio between Ω’ for 1s and 2s states, a ratio of 3.8 for the square roots of the simulated omegas (52 meV for 1s and 3.7 meV for 2s) is found, which is comparable with previous works (2.3 - 3.2 in Nat. Commun. 12, 2269 (2021)). The reason why our ratio does not match the Rabi energy ratio (35 meV/16.7 meV) is because, as explained in the first paragraph of this response, the Rabi is also proportional to the normalized overlap between the photonic and exciton modes (Phys. Rev. B 106, 115424 (2022)), which is different for the two excitonic species.\nHerein, we calculate the overlap between the photonic and exciton modes via ∫Ψ_exc(r)Ψ_cav(r)d3r , with Ψ_exc(r) and Ψ_cav(r) the exciton and photon wavefunction, where 1s and 2s exciton wavefunctions could be expressed as below based on typical hydrogen-like wavefunctions (which is certainly an approximation for TMDs):\nΨ_1s(r) = 1/sqrt(pi*aB^3) * exp(-r/aB) (2)\nΨ_2s(r) = 1/sqrt(8*pi*aB^3) * (1-r/2aB) * exp(-r/2aB) (3)\nwhere aB are the exciton Bohr radii. Considering that within aB the photon field can be regarded as a constant (0.6 and 0.8 for 1s and 2s excitons, Figure R2c), we could estimate the overlap between the photonic and exciton modes as:\n1s: ∫ 0 aB Ψ_1s(r)Ψ_cav(r)d3r ≈ A * ∫ 0 aB 1/sqrt(pi*aB^3) * exp(-r/aB) d3r ≈ 0.91*(aB)^3/2\n2s: ∫ 0 aB Ψ_2s(r)Ψ_cav(r)d3r ≈ B * ∫ 0 aB 1/sqrt(8*pi*aB^3) * (1-r/2aB) * exp(-r/2aB) d3r ≈ 0.22*(aB)^3/2\nConsidering the Bohr radii of 1.8 nm for the 1 s state and 6.6 nm for the 2 s state, as in WSe (Nat. Commun. 12, 2269 (2021)), we could calculate the ratio between the overlap integrals overlap_1s/overlap_2s ≈ 0.58. Similarly, if we consider an overlap across a distance of 2aB, the ratio will be 0.89. Thus, depending on whether we consider aB or 2aB, the ratios are 2.2 or 3.4, similar to our calculated Rabi energy ratio (2.1). Overall, the obtained values of Ω’ in Figure 1c and 1d (which come from the fitting of the experimental data) are thus compatible with reported excitonic properties of WS .\nBased on the above discussion, we further revised the main text and the supplementary information to make it clearer for the reader.\nRevision:\nLine 145, page 7, main text: added “As the Rabi splitting is proportional to the square-root of the oscillator strength and the overlap between the active material and the photonic field (Supplementary Note 1), a Rabi splitting ratio between 1s and 2s polaritons of 2.2 could be calculated, which matches well our experimental results, i.e., 35/16.7 = 2.1.”\nSupplementary materials: added Supplementary Note 1 describing the calculation of the overlap between the active material and the photonic mode.\nComment 5:\nThe last section of the current manuscript discussed the propagation of 1s exciton polaritons, which does not seem to be relevant with the topic of the manuscript. I suggest the authors to move it to the supplemental materials.\nResponse:\nWe thank the reviewer for the constructive suggestion. Indeed, we just wanted to show that the studied polaritons possessed a certain in-plane velocity and, thus, that they are naturally propagating polaritons. To focus the attention on the distinct polariton-polariton interactions, we have moved the propagation of 1s exciton polaritons into the supplemental information.\nReply to Reviewer 2\nGeneral Comment:\nIn this manuscript, Shang et al. demonstrate the first realization of on-chip Rydberg polaritons in monolayer WS2 coupled with a 1D photonic crystal (PhC) cavity structure. By leveraging the large Bohr radius of Rydberg excitons and PhC-induced field localization, the authors achieve a nonlinearity enhancement of 23 ± 10 times for 2s polaritons compared to ground-state 1s polaritons, with robust operation up to 80 K and magnetic field-tunable Rabi splitting.\nThe manuscript addresses a critical gap in on-chip polaritonic devices by combining monolayer TMDs with PhC engineering, yielding groundbreaking results in Rydberg polariton nonlinearity and propagation. The experimental and theoretical framework is rigorous, and the results are well-supported by angle-resolved spectroscopy and FDTD simulations. I think this manuscript is suitable for publication in Nature Communications after revision\nResponse:\nWe sincerely thank the referee for the favorable evaluation of our work and its detailed summary. Besides, we believe that the statement “The manuscript addresses a critical gap in on-chip polaritonic devices by combining monolayer TMDs with PhC engineering” points out the main significance of our work.\nHerein, we would like to point out that, due to reviewers 3&4 and 5 questioning on the analysis of the reflectance spectra, to obtain the polaritons' spectral position, we have fitted all reflectance spectra again with Fano functions. Furthermore, due to reviewers 3&4 and 5 questioning on the effect of optical and excitonic anisotropies on the measured nonlinearities, we have revised equations 1-4 of the main text to take into account cross-polarized contributions to the polariton nonlinearity. Based on this new formulation, we have reanalyzed the data (device 1) shown in the first version of the manuscript (Figure 4) as well as a new series of measurements on a second identical structure (device 2) fabricated for answering some of the questions raised during the revision of the article (Figure S24 and S25). See details in comment 13 to reviewer 3 and comment 3 to reviewer 5. The two structures give consistent values for the Rabi splittings and nonlinearities.\nComment 1:\nThe introduction mentions that Rydberg excitons have large Bohr radii and are promising for enhancing optical nonlinearities. However, it is not clear how the specific properties of Rydberg excitons in monolayer WS2 compare to other materials. Could you provide more details on this?\nResponse:\nWe thank the reviewer for the suggestion of putting into context Rydberg excitons in monolayer WS . As the exciton-exciton interaction g depends quadratically on the Bohr radius (aB) (Sci. Adv. 8, eadd8857 (2022)), Rydberg excitons, which exhibit a larger Bohr radius compared to the ground state, are promising for enhancing optical nonlinearities. In Table R2, we present the specific properties of Rydberg exciton polaritons for representative semiconductors in the strong coupling regime, namely the measured Rabi energy and nonlinearity. Note that the 2s exciton Bohr radius indicated in the table was either obtained from literature or estimated taking into account the n2 scaling, i.e., the 2s exciton Bohr radius corresponds to 4 times the 1s exciton Bohr radius.\nTable R2. Comparison of Rydberg excitons among representative semiconductors.\nBased on the reviewer’s suggestion and to put into perspective our work, we have added Table R2 into the supplementary information.\nRevision:\nLine 61, page 3, main text: added “, among which the TMD based Rydberg polaritons show a record nonlinearity (Table S1).”\nSupplementary materials: added Table S1 to compare Rydberg excitons among representative semiconductors.\nComment 2:\nIn Figure 1c and 1d, the strong coupling between the WS excitons and PhC modes is demonstrated. How do the specific geometric parameters of the PhC (period, width) influence the coupling strength and the resulting polariton dispersion?\nResponse:\nWe thank the reviewer for requesting us to further discuss the impact of geometric parameters on the polariton dispersion. To answer this request, we have conducted additional numerical simulations (shown below). Herein, we consider three geometric parameters defining the PhC, namely the thickness of the grating (h), the period of the unit cell (p), and the filling factor (FF, width of the unetched grating portion over the unit cell dimension). Note that the experiment results reported in the main manuscript are acquired in a cavity whose parameters are h = 220 nm, p = 380 nm, and FF = 0.51.\nFigure R7. Comparison of reflectivity spectra calculated using FDTD for different grating thicknesses with p = 380 nm and FF = 0.51.\nFigure R8. Comparison of reflectivity spectra calculated using FDTD for different grating pitches with h = 220 nm and FF = 0.51.\nFigure R9. Comparison of reflectivity spectra calculated using FDTD for different filling factors with h = 220 nm and p = 380 nm.\nTo illustrate the effect of varying geometric parameters, we thus simulated three series of structures where only one parameter is changed at a time (the thickness, the period of the unit cell, and the filling factor of the grating), while all other parameters reproduce those of the fabricated samples. The results are displayed in Figure R7-9.\n1) Increasing the thickness localizes the photonic mode inside the grating, thereby reducing the overlap with the TMDs excitons and, therefore, decreasing continuously the Rabi splitting. The loss of the strong coupling regime is observed for a thickness of about 280 nm. 2) Increasing the period redshifts the photonic mode and modifies the wavevector (angle) at which anti-crossing occurs. Otherwise, the Rabis are mostly unaffected. 3) Increasing the filling factor degrades the linewidth of the photonic modes (i.e., their quality factor) and thereby weakens the Rabi splitting.\nBased on the reviewer’s suggestion and the above simulations, we added a discussion on PhC design optimization and included Figures R7-9 into the supplementary information.\nRevision:\nLine 147, page 7, main text: added “To make sure our designed structure exhibits prominent coupling with 1s and 2s excitons, we optimized the polaritonic dispersion by varying three geometric parameters of the PhC, i.e., thickness of the grating (h), period of the unit cell (p), and the filling factor (FF, width of the unetched grating portion over the unit cell dimension) utilizing FDTD (Methods, Fig. S8-10).”\nSupplementary materials: added Fig. S8-10 to discuss the PhC design optimization.\nComment 3:\nThe temperature-dependent measurements show that the 2s exciton polaritons survive up to 80 K. What specific mechanisms cause the degradation of the strong coupling regime above this temperature, and are there any strategies to mitigate these effects?\nResponse:\nWe thank the reviewer for the constructive suggestion to explore the mechanisms causing the degradation of the strong coupling as temperature increases, which is complementary to a similar question raised by the first reviewer concerning the temperature dependence of 1s/2s polaritons.\nLet us first remind the criterion for strong coupling, i.e., Ω > (gamma_exc + gamma_cav)/2. To analyze its dependence on temperature, one must thus monitor the homogeneous exciton broadening as a function of temperature (assuming that the refractive index for the cavity is temperature independent). The temperature dependence of the 2s exciton linewidth is shown in Figure R10. The reflectance spectrum is fitted with an asymmetric Fano line shape function due to the interference between the discrete 2s exciton state and a continuum of states (Nat. Commun. 9, 713 (2018)):\nF(E) = (q +(E -E0)/2gamma)^2 / 1+(E -E0)/2gamma)^2 (4)\nwith E the energy, E0 the exciton resonance energy, gamma representing the resonance linewidth, and q the Fano factor, which describes the asymmetry of the lineshape. The fitted q values are shown in Figure R10b, which are all below 0.08, indicating 2s excitons exhibit a symmetric lineshape close to a Lorentz oscillator.\nFigure R10. The temperature dependent reflectance spectra (a), the fitted q values (b), the peak position (c), and linewidth (d) of 2s exciton from 100 K to 4 K.\nThe extracted peak energy and the linewidth for the 2s exciton versus temperature are shown in Figure R10c and d. As temperature increases, the peak position redshifts from 2.261 to 2.257 eV, which is due to the increase in electron-phonon interactions and a reduction of the bandgap at higher temperatures (Nano Lett. 21, 6240-6246 (2024)). Concomitantly, the linewidth increases from 7.5 to 15.0 meV, which is ascribed to the enlarged exciton scattering.\nCombining the measured 2s exciton broadening (Figure R10d) with the fitting of the reflectance spectra as a function of temperature shown in Figure 3a of the main text (Figure R11a), which provides the Rabi splitting as a function of temperature, we can now plot all the quantities involved and verify the Ω > (gamma_exc + gamma_cav)/2 criterion as a function of temperature. As shown in Figure R11b, the Rabi splitting is larger than (gamma_exc + gamma_cav)/2 up to about 80 K. Further increasing temperature prevents distinguishing LP and UP (120 K for example) due to exciton dephasing and broadening of the 2s exciton linewidth.\nOverall, to mitigate this effect and sustain the strong-coupling with 2s exciton up to higher temperatures, one could either 1) reduce the linewidth of monolayer WS 2s excitons by minimizing their broadening; people usually utilize hBN encapsulation or annealing to improve the excitonic linewidth (Phys. Rev. Applied 14, 021002 (2020)); 2) optimize the linewidth of the cavity mode via refining the fabrication process of the cavity or by using photonic modes displaying intrinsically larger quality factors, such as BIC (Nature 605, 447–452 (2022)), which in principle have an infinite lifetime.\nBased on the reviewer’s suggestion, we added a discussion on the degradation of the strong coupling of 2s excitons as the temperature increases in the main text.\nFigure R11 The temperature dependent Rabi splitting for X-polarized 2s polaritons. a, The fitted temperature dependent angle-resolved reflectance spectra for 2s polaritons at the anti-crossing angle from 4 K to 120 K. b, Temperature dependent Rabi splitting and (gamma_exc + gamma_cav)/2 for 2s exciton polaritons.\nRevision:\nLine 154, page 7, main text: revised as “the Rabi splitting is larger than (gamma_exc + gamma_cav)/2 at each temperature and ensures that the system operates in the strong coupling regime (Fig. S13-15 and Supplementary Note 2). Further increasing the temperature will enhance phonon scattering and accelerate exciton dephasing, consequently reducing the Rabi splitting.”\nSupplementary materials: added Fig. S14 and S15 to discuss the temperature dependent Rabi energy for X-polarized 2s polariton.\nComment 4:\nFrom Figure S3 in SI, it can be seen that the line widths of 1s and 2s excitons and cavity photons are different at room temperature and 4 K. Does this have an impact on the judgment of strong coupling? For example, does the narrower line width of the 2s exciton at 4 K mean a higher quality coupling to the photon mode?\nResponse:\nWe thank the reviewer for this question, which is intimately related to the previous one. As discussed in detail in the previous question, the excitonic linewidth enters directly into the criterion determining whether strong coupling can be sustained or not: Ω > (gamma_exc + gamma_cav)/2 (Solid State Commun. 93, 733-739 (1995)). Thus, the narrower the excitonic linewidth, the easier to fulfill the previous criterion for a given cavity photon linewidth (gamma_cav). If we assume, as a first approximation, that the linewidth of Rydberg excitons in TMDs follows the inverse cubic law of n (Nature 514, 343–347 (2014)), then the larger the principal quantum number the narrower the excitonic linewidth and, in turn, the smaller the right-hand side of the above criterion. However, the principal quantum number also affects the oscillator strength and, thus, the left-hand side of the criterion.\nMoreover, since the coupling strength between excitons and photonic modes also depends on the overlap between the excitonic wavefunction and the electric field distribution, one should also take into account the effect of potentially unequal overlaps (which is our case). This overlap is given by ∫Ψ_exc(r)Ψ_cav(r)d3r, with Ψ_exc(r) and Ψ_cav(r) the exciton and photon wavefunction. If we assume 1s and 2s exciton wavefunctions based on hydrogen-like wavefunctions (which are certainly an approximation for TMD-based excitons):\nΨ_1s(r) = 1/sqrt(pi*aB^3) * exp(-r/aB) (5)\nΨ_2s(r) = 1/sqrt(8*pi*aB^3) * (1-r/2aB) * exp(-r/2aB) (6)\na being the exciton Bohr radii. Taking into account that within aB the photon field can be considered as a constant (0.6 and 0.8 for 1s and 2s excitons, Figure R2c), we could calculate the overlap between the photonic and exciton modes as:\n1s: ∫ 0 aB Ψ_1s(r)Ψ_cav(r)d3r ≈ A * ∫ 0 aB 1/sqrt(pi*aB^3) * exp(-r/aB) d3r ≈ 0.91*(aB)^3/2\n2s: ∫ 0 aB Ψ_2s(r)Ψ_cav(r)d3r ≈ B * ∫ 0 aB 1/sqrt(8*pi*aB^3) * (1-r/2aB) * exp(-r/2aB) d3r ≈ 0.22*(aB)^3/2\nConsidering the Bohr radii of 1.8 nm for the 1 s state and 6.6 nm for the 2 s state, as in WSe (Nat. Commun. 12, 2269 (2021)), we could calculate the ratio between the overlap integrals overlap_1s/overlap_2s ≈ 0.58. Similarly, if we consider an overlap across a distance of 2aB, the ratio will be 0.89. Thus, depending on whether we consider aB or 2aB for the spatial extension of the overlap, the ratios are 2.2 or 3.4, similar to our calculated and measured Rabi energies. In this sense, the coupling strength of 2s excitons is “extrinsically” enhanced compared to that of 1s excitons due to a better exciton-photonic mode overlap at the energy of the 2s exciton. This enhancement reinforces the effect of the larger spatial extension of the 2s exciton wavefunction.\nBased on the above discussion, we further developed the section discussing the criterion for the strong coupling regime in the main text as well as its temperature dependence, which is mostly governed by the temperature dependence of the excitonic linewidths.\nRevision:\nLine 111, page 5, main text: added “The electric field intensity at the position where monolayer WS will be located (black dashed lines in side panels) is 0.6 and 0.8 times that of the maximum electric field intensity across the entire photonic structure for 1s and 2s excitons, respectively.”\nLine 138, page 6, main text: added “reflecting that a narrower excitonic linewidth of 2s exciton makes it easier to fulfill the criterion for a given cavity photon linewidth (gamma_cav) compared with 1s exciton.”\nLine 145, page 7, main text: added “As the Rabi splitting is proportional to the square-root of the oscillator strength and the overlap between the active material and the photonic field (Supplementary Note 1), a Rabi splitting ratio between 1s and 2s polariton of 2.2 could be calculated, which matches well the experimental results, i.e., 35/16.7 = 2.1.”\nLine 154, page 7, main text: revised as “the Rabi splitting is larger than (gamma_exc + gamma_cav)/2 at each temperature and ensures that the system operates in the strong coupling regime (Fig. S13-15 and Supplementary Note 2). Further increasing the temperature will enhance phonon scattering and accelerate exciton dephasing, consequently reducing the Rabi splitting.”\nLine 156, page 7, main text: added “Besides, temperature-dependent angle-resolved reflectance spectra around 1s excitons from 15 K to 295 K (Fig. S16-18) were also explored. The Rabi splitting is roughly constant (32-35 meV) from low temperatures up to about 200 K, consistent with an almost constant excitonic broadening (7-9 meV). As temperature increases further, the Rabi splitting decreases from 35 ± 3 to 25 ± 2 meV, while the linewidth of the 1s exciton increases from 9 to 25 meV, due to enhanced electron-phonon coupling and an acceleration of the exciton dephasing. The Rabi splitting is thus also larger than (gamma_exc + gamma_cav)/2 at each temperature, ensuring the robustness of the strong coupling.”\nSupplementary materials: added Supplementary Note 1 describing the calculation of the overlap between the active material and the photonic field.\nSupplementary materials: added Supplementary Note 2 to demonstrate the temperature-dependent behaviors of 2s and 1s exciton polaritons.\nComment 5:\nThe propagation decay constant of the PhC polaritons is measured to be around 6.3 μm. How does this value compare to other polariton systems, and what factors limit the propagation distance in this case?\nResponse:\nWe thank the reviewer’s interest for their interest in the propagation of polaritons. Let us remind first that the propagation decay constant (D) is proportional to the group velocity (v) and the polariton lifetime (tau_pol), which in turn depends on the excitonic/photonic fraction of the considered polaritons i.e., the position along the polariton dispersion (Nat. Nanotechnol. 13, 906-909 (2018)). v can be calculated from the experimental dispersions as v = dE/dk*hbar, while tau_pol can be calculated from 1/tau_pol = X/tau_EX + C/tau_P, if both the exciton lifetime (tau_EX) and the photon lifetime (tau_P) are known. Since in most polariton systems the excitonic lifetime is much larger than the photonic lifetime, the polariton lifetime is mostly limited by the photonic lifetime, which would push to exploit more excitonic-like polaritons. However, the larger the excitonic fraction, the smaller the group velocity and, thus, for each material system and cavity structure, a compromise between group velocity and polariton lifetime needs to be found. In general, longer propagation lengths are found for more photonic-like polaritons, showing that larger group velocities over larger polariton lifetimes are favored. In this context, maximizing the photonic lifetime is thus a must, as indicated explicitly in the main text.\nThe propagation decay constant (6.3 m) measured in our work is comparable to previous reported TMD based polariton systems, as listed in the Table R3.\nTable R3. Comparison of propagation decay constants among representative polariton systems.\nBased on the first and third reviewer’s suggestions and the above discussion, we have included Figure 4 and Table R3 in the supplemental information.\nRevision:\nSupplementary materials: added Supplementary Table 4 to compare the propagation decay constant among representative polariton systems.\nReply to Reviewer 3\nGeneral Comment:\nIn the manuscript titled “Enhanced nonlinearity of photonic crystal Rydberg exciton polaritons in monolayer WS2”, the authors present a photonic crystal (PhC)-based on-chip polaritonic device demonstrating strong coupling between PhC modes and both the ground-state and Rydberg-state excitons of WS2. They conduct a detailed characterization of the polariton states through reflection measurements under varying magnetic field and temperature, as well as nonlinearity and propagation measurements. The key claim of the work is a 23-fold enhancement of the non-linearity associated with the Rydberg state polaritons relative to the ground state. They attribute the enhancement to the enhanced overlap of the PhC mode electric field with the excitons in the PhC geometry. The authors have also estimated the interaction constants and propagation decay constants for the ground state and excited state exciton-polaritons.\nThe manuscript is thorough and clearly written. The exploration of strong polaritonic effects, particularly large nonlinearity using PhCs instead of the more common Fabry-Perot cavities, is of potential interest. However, I have several comments that should be addressed before recommending publication:\nResponse:\nWe sincerely thank the referee for the favorable evaluation of our work. We also agree with the reviewer that exploration of strong polaritonic effects using PhCs is of potential interest, which we believe is one of the main conclusions of our work, given the engineering flexibility of PhC compared to other types of cavities.\nHerein, we would like to point out that, due to reviewers 3&4 and 5 questioning on the analysis of the reflectance spectra, to obtain the polaritons' spectral position, we have fitted all reflectance spectra again with Fano functions. Furthermore, due to reviewers 3&4 and 5 questioning on the effect of optical and excitonic anisotropies on the measured nonlinearities, we have revised equations 1-4 of the main text to take into account cross-polarized contributions to the polariton nonlinearity. Based on this new formulation, we have reanalyzed the data (device 1) shown in the first version of the manuscript (Figure 4), as well as a new series of measurements on a second identical structure (device 2), fabricated for answering some of the questions raised during the revision of the article (Figure S24 and S25). See details in comment 13 to current reviewer and comment 3 to reviewer 5. The two structures give consistent values for the Rabi splittings and nonlinearities.\nComment 1:\nReference 17, which probed nonlinearity of WSe2 embedded in a Fabry-Perot cavity, reported that the interaction for the 2s exciton-polaritons is 4.6 times stronger than for the 1s exciton-polaritons. While the current study finds a similar nonlinearity for 2s exciton-polaritons, it reports a 23-fold suppression for 1s exciton-polaritons. Direct comparison is difficult due to differences in geometry and photon absorption estimates, yet this discrepancy is notable. It is not clear how this difference (with respect to cavity geometry) can result from “the combination of an extended exciton radius due to the excited-state nature of the employed excitons, and to the PhC-induced spatial field localization.” Since this is a key point in the abstract, the authors should clarify their explanation.\nResponse:\nAs the reviewer points out, this is an extremely important point that deserves a deeper analysis. Please note that upon reanalysis of the data shown in the main text, based on the reviewer’s comment 8 (devoted to data fitting), the actual interaction for the 2s exciton-polaritons is 8.0 times stronger than for the 1s exciton-polaritons, and not 23 times larger, as announced in the first version of the manuscript. This difference in the extracted nonlinearities is due to: 1) the new determination of the polaritons' spectral positions thanks to the use of Fano resonances, which result in different splittings compared to our initial analysis, especially at high pumping powers. 2) the fabrication of a second, nominally identical, PhC structure on which we have also measured the polariton-polariton interaction constants, with the reported value being the average of the two devices. Thus, the 8.0-times enhancement is obtained upon analyzing all the available data but using the methodology described in the following paragraphs.\nBefore discussing the actual 2s/1s nonlinearity ratio, let us explain in detail the methodology we have followed to reanalyze all the raw data: the new analysis has been applied to the data previously (and still now) shown in the main text, as well as on the data measured on a new sample requested by the first and second reviewers in order to assess the temperature dependence of the Rabi splitting (and that gives the same results as the first sample within the error bar). In the next three paragraphs, we will thus explain step by step how we have analyzed the reflectance spectra on our samples (answering thereby comment 8 too):\n1. Obtaining the Fano q values for the bare cavity mode and bare exciton resonances\nThe reflectance spectra of the bare 1s exciton resonance and the bare cavity mode (at energies close to the 1s resonance) are fitted with an asymmetric Fano lineshape function that is necessary due to the interference between the discrete states and a continuum of states (Nat. Commun. 9, 713 (2018)):\nF(E) = (q +(E -E0)/2gamma)^2 / 1+(E -E0)/2gamma)^2 (10)\nwith E the energy, E0 the resonance energy, gamma representing the resonance linewidth, and q the Fano factor, which describes the asymmetry of the lineshape. The fitting results (shown in Figure R12) indicate that the cavity mode around 1s exciton and the bare 1s exciton exhibit an asymmetric lineshape with a q factor of 0.2 and 0.31, respectively. Meanwhile, the 2s exciton and the bare cavity mode around the 2s exciton energy show a much smaller q value, pointing to a symmetric lineshape that approaches a Lorentzian one. Therefore, to fit and extract the actual polariton position, we considered Fano-type polaritonic reflectance spectra for 1s polaritons, consisting of three Fano contributions, i.e., bare exciton background, UP and LP peaks, and two Lorentz functions for 2s exciton polaritons (UP and LP peaks) at low temperature. Meanwhile, for the temperature-dependent study of bare excitons (see comment 4), we performed the analysis of 1s and 2s exciton species with Fano resonances. Indeed, this study shows the temperature range on which the employed Fano functions approach Lorentz functions.\nFigure R12. The fitted Fano q values for the bare cavity mode at 1s (a) and 2s (c) exciton energy as well as the bare 1s (b) and 2s exciton (d).\n2. Fitting of angle-resolved reflectance spectra for 1s and 2s exciton polaritons\nBased on the functional dependences presented in step 1, we fitted again all angle-resolved reflectance spectra for 1s and 2s exciton polaritons, as shown in Figure R13 for the pumping power intensity dependence (and the same methodology has been followed to analyze the temperature and magnetic-field dependences).\nFigure R13. Angle-resolved reflectance spectra for 1s and 2s exciton polaritons for the structure (device 1) presented in the main text (same raw data as in the initial manuscript).\nThe corresponding fitting parameters (i.e., q for the three Fano lineshapes, FWHM of 1s exciton, and corresponding LP and UP peak positions for 1s and 2s polaritons) are presented in Figure R14. Note that as the pump power increases, the q values for the three Fano contributions reproducing the dispersion of 1s exciton polaritons decrease (Figure R14a and d). This reduction of the q values starts at the onset of the peak spectral shifts, irrespective of broadband (BB) or narrow band (NB) pumping conditions. The nonlinear evolution of the polariton energies is accompanied by a reduction of the oscillator strength of 1s excitons (q for 1s exciton reduces (Figure R14a) and FWHM increases (Figure R14b)), which reduces the Rabi splitting (Figure R14c and e). Figure R14f shows the corresponding polariton positions for 2s polaritons as a function of pumping power intensity.\nFigure R14. Fitting parameters extracted from the new fitting procedure (q for three Fano contributions, FWHM of 1s exciton, and corresponding LP and UP peak positions for 1s and 2s polaritons). Power density dependent fitted q values for the three Fano contributions to reflectance spectra around 1s polaritons under BB (a) and NB pumping conditions (d). Power density dependent FWHM of 1s exciton (b) in a. Power density dependent LP and UP peak positions for 1s polariton under BB (c) and NB (e) pumping conditions. Power density dependent LP and UP peak positions for 2s polaritons under BB pumping conditions (f).\n3. Fitted nonlinearity for 1s and 2s polaritons\nBased on the spectral positions accurately determined thanks to the fitting described in step 2, we updated Figure 3a-c in the main text as shown in Figure R15; from these new spectral shifts ΔEsat data, we recalculated the nonlinearity constants for device 1 that amount to 49.3 ± 12.0, 13.3 ± 6.3, and 4.9 ± 0.8 eV m2 for g_2s-2s^sat, g_1s-2s^sat, and g_1s-1s^sat. The corresponding enhancement factor of the 2s polariton interactions with respect to the 1s polaritons interactions is now 10.0 ± 3.0, which is a factor 2.2 times larger than the previously reported one in reference 17, where a stack of 3 monolayers were employed. For the replotting of Figure R15c, please see comment 13 in detail.\nFigure R15. Nonlinear behaviors of 1s and 2s exciton polaritons. (a) Angle-resolved X-polarized reflectance spectra for 2s exciton polaritons at the anti-crossing under a pulsed supercontinuum white light excitation and a magnetic field of 9 T, with a power density from 5.0×10-4 mW m-2 to 2.8 mW m-2. Two black dashed lines indicate the UP and LP polaritons. The spectra are fitted with two Lorentzian functions. At a pumping density of 0.18 mW m-2 (indicated with its power density) the UP and LP can no longer be distinguished (dashed dots), indicating the transition to the weak coupling regime for the 2s exciton-PhC photon coupling, based on the Ω < (gamma_exc + gamma_cav)/2 criterion. (b) Power density dependent differential Rabi splitting Ω(N)/Ω(0) for 1s and 2s exciton polaritons. (c) Extracted total blueshift ∆E for 2s and 1s polaritons under BB pumping conditions and ∆E for 1s polaritons under NB pumping conditions. Circles: experimental results. Solid curves: linear fitting of the experimental results.\nWe can now compare the nonlinearity extracted from our experimental data, as well as that reported in reference 17, with the expected nonlinearity enhancement due exclusively to the unequal Bohr radii of different Rydberg polaritons. In the simplest picture, the polariton–polariton interaction strength is proportional to the square of the Bohr radius and the Rabi splitting, i.e., g_xx ∝ aB^2 * Omega. Therefore, the enhancement one could expect by using 2s polaritons compared to 1s polaritons is given by (aB_2s^2 * Omega_2s) / (aB_1s^2 * Omega_1s) = (3.66/3.8)^2 * 15/35 = 5.7/6.5, considering a Bohr radii ratio of 3.66 (Nat. Commun. 12, 2269 (2021)) or 3.8 (Nano Lett. 20, 7635−7641 (2020)), respectively.\nNext, we can add into our considerations the enhancement of polariton-polariton nonlinear interactions due to the unequal field distribution in PhC and vertical microcavities (formed by two DBRs). For the comparison, we use perfectly matched layers boundary along both x and y directions and consider a 200 period PhC structure in the nonlinear Bloch-Maxwell FDTD simulation. We then pump the system using a planewave CW source and measure the reflectivity at the anticrossing area. We further quantify the polariton population and its spatial distribution by monitoring the photonic component of the polaritons (i.e., the real space distribution of the photonic electric-field profile at the energies of UP and LP for both 1s and 2s polaritons). By varying the pump power of the light source and comparing the obtained dispersion to the dispersion of the system in the linear regime, we obtain the energy shift of UP and LP polaritons, from which we deduce the nonlinearity strength (i.e., phase space filling interaction in the main text).\nTo compare with the planar case, we apply the same simulation procedure to a planar microcavity containing WS too and we assume that g^sat for 2s and 1s are 1.36 and 0.34 eV m2 (using a nonlinearity ratio of 4, close to the 4.6 value from Nat. Commun. 12, 2269 (2021), i.e., from reference 17). Compared to reference 17, the only difference is that in our simulation of the planar microcavity, we place the WS at the maximum of the cavity mode electric field. Figure R16 shows the polariton distribution (obtained from its photonic component) for the UP and LP polaritons associated with 1s and 2s excitons for the highest simulated pump power. As expected in a planar microcavity, the polariton density peaks at the centre of the dual-DBR cavity and is uniform along the y-axis. The simulated polariton population (integrated at the WS position along the y-axis) as well as the energy shift of each UP and LP peak are used as the benchmarks for the nonlinearity calculation in PhC structures. This calculation was repeated for eight pump powers (Figure 4d in the main text).\nFigure R16. Dual-DBR cavity: Polariton population (photonic EM field intensity) in real space for the UP and LP polaritons associated to 1s and 2s excitons, under the highest pump power.\nFigure R17 shows the results for the polariton population under the highest pump power in the PhC structure studied in our simulation. Notably, one can see that, compared to the planar cavity, in this case, the polariton population is inhomogeneously distributed along the PhC y direction. This effective field and population confinement can be more clearly seen in the 1D cuts shown in the right panels. The as-simulated populations (integral population at the WS position along the y-axis) and energy shifts of each UP and LP polariton were thus compared with those from the dual-DBR cavity (Figure 4d in the main text) by tuning the pump power to have similar polariton populations.\nAs shown in Figure 4d of the main text, we could obtain that the phase space filling interaction is 15% larger for the 2s excitons and 20% larger for the 1s excitons in PhC structures compared with these magnitudes in a dual-DBR cavity. Such enhancement is qualitatively similar to the situation encountered in micropillar cavities (Light: Sci. Appl. 12, 220 (2023); Phys. Rev. B 73, 193306 (2006)), where the confinement of the photonic wavefunction induces an effective increase of the exciton phase space filling effect. This numerical comparison shows that, indeed, the field confinement induced by the PhC structure can contribute to an enhancement of polariton-polariton interactions.\nFigure R17. The polariton population in the real space from its photonic component for the UP and LP of 1s and 2s polaritons under the highest pump power in PhC structures.\nTo enable the interested readers to reproduce accurately our results, in the revised version, we present a detailed description of the fitting procedure in the supplementary information and have also included Figures R12-R14. Furthermore, to complete the discussion on the PhC-induced nonlinearity enhancement, we also include now in the supplementary information Figures R16-R17.\nRevision:\nLine 30, page 2, main text: revised as “This enhancement results mostly from the extended exciton radius of the 2s excitons compared to the 1s excitons. Additionally, the fabricated PhC induces a 15-20% enhancement in the polaritonic nonlinearity due to a stronger spatially-localized electric field distribution compared to the dual-DBR geometry.”\nLine 88, page 2, main text: revised as “Additionally, PhC-induced confinement of polaritons in real space enables a 15-20% enhancement of nonlinearity compared with dual-DBR geometry. Furthermore, the simultaneous quenching of the oscillator strength (accounted for by the Fano q parameter) and broadening of the Fano resonances' linewidth indicates the onset of the nonlinearity.”\nLine 111, page 5, main text: added “The electric field intensity at the position where monolayer WS will be located (black dashed lines in side panels) is 0.6 and 0.8 times that of the maximum electric field intensity across the entire photonic structure for 1s and 2s excitons, respectively.”\nLine 145, page 7, main text: added “(Fig. S6 and S7, Supplementary Note 1). Notably, by positioning the active layers close to the maximum of the electric field, our obtained Ω for both 1s and 2s polaritons are comparable with that measured in planar DBR-based microcavities, in which the monolayer is located close to an antinode of the electric field distribution (Table S2). As the Rabi splitting is proportional to the square-root of the oscillator strength and the overlap between the active material and the photonic field (Supplementary Note 1), a Rabi splitting ratio between 1s and 2s polaritons of 2.2 could be calculated, which matches well our experimental results, i.e., 35/16.7 = 2.1.”\nLine 156, page 7, main text: added “Note that we have used Fano resonances to fit the polariton dispersions, but whenever the q value was lower than 0.1, we used instead two Lorentzian functions to simplify, as the Fano resonance reproduces a Lorentzian when q = 0. Overall, in the current work, we employed Fano resonances to fit the temperature dependence of 2s and 1s bare excitons (Fig. S14 and S17) as well as to fit 1s polaritons at low temperature (Fig. S18), while two Lorentzian functions could be used to fit the 2s polaritons at low temperature as well as 1s polaritons at room temperature (Fig. 2 and 3) whenever extended studies were conducted at those temperatures. For clarity, we have explicitly indicated in each series of measurements what functional dependence was used to fit the reflectivity spectra.”\nLine 212, page 10, main text: added “Note that as the pump power increases, under both BB and NB pumping conditions, the fitted q values for the three Fano contributions reproducing the dispersion of 1s exciton polaritons start to decrease (Fig. S22) at the onset of the resonances' spectral shifts. Overall, the nonlinear evolution of the polariton energies is thus accompanied by a reduction of the oscillator strength of 1s excitons (q for 1s exciton reduces and FWHM increases), which reduces the Rabi splitting and is consistent with exciton-exciton induced dephasing as previously characterized in TMDs monolayers.”\nLine 238, page 11, main text: added “According to previous literature, the polariton–polariton interaction strength is proportional to the square of the Bohr radius and the Rabi splitting, i.e., g_xx ∝ aB^2 * Omega. Therefore, the enhancement that one could expect by using 2s polaritons compared to 1s polaritons is given by (aB_2s^2 * Omega_2s) / (aB_1s^2 * Omega_1s) = 7, which agrees well with the experimental results (10 times).”\nLine 246, page 11, main text: added “We first quantify the polariton population and its spatial distribution in real space, as well as its dispersion in momentum space; then we obtain the energy shift of UP and LP polaritons in a vertical DBR-based microcavity as a function of pump power. The simulated polariton population as well as the energy shift of each UP and LP peak are used as the benchmarks for comparing the nonlinearity in PhC structures. Considering comparable polariton populations in the DBR-based and the PhC-based microcavities, our simulations indicate that saturation is 15% larger for the 2s excitons and 20% larger for the 1s excitons in the corresponding PhC structures due to the spatial localization of polaritons (Fig. 4d and Supplementary Note 4).”\nMethods, main text: added “Fitting the reflectance spectra of the excitonic and polaritonic species”\nSupplementary materials: added Supplementary Fig. S6, S7, and Supplementary Note 1 to present the confined electric field in PhC structure.\nSupplementary materials: added Supplementary Fig. S13, S22, and Supplementary Note 2 to introduce the Fano fitting procedure.\nComment 2:\nBuilding on the previous comment, the title is somewhat misleading - it seems to suggest that using a PhC (instead of a traditional Fabry-Perot cavity) enhances the nonlinearity. However, the authors themselves note that the extracted nonlinearity constants align with previous studies. I recommend revising the title to more accurately reflect the findings.\nResponse:\nAs discussed in the paragraph above, our numerical simulations suggest that using PhC indeed enhances the overall nonlinearities (of both 1s- and 2s-based polaritons), but “only” by a modest ~20%. Thus, we agree with the reviewer that the title should be revised to avoid being misleading. We propose “On-chip photonic crystal dressed Rydberg exciton polaritons with enhanced nonlinearity in monolayer WS2”.\nComment 3:\nThe focus of the paper is somewhat diluted. It is unclear how the propagation measurement supports the central claim of the paper.\nResponse:\nWe thank the reviewer for the constructive suggestion, which aligns with that of other reviewers. We have moved the propagation of exciton polaritons into the supplemental information, just to show that these polaritons possess a non-zero in-plane velocity and that, therefore, they do naturally propagate in-plane.\nComment 4:\nIn the temperature-dependent measurements (Figure 2c), the coupling strength of 2s excitons decreases monotonically, while 1s excitons maintain a nearly constant coupling strength. Could the authors provide further insight into this monotonic decrease for 2s excitons?\nResponse:\nWe thank the reviewer for the constructive suggestion on further exploration on the temperature dependent coupling strength of 2s excitons, which complements the request of reviewers 1&2. In the following, we provide further insight into the two aspects raised by this comment/question:\n1.Why the Rabi energy for 1s excitons maintain a nearly constant coupling strength?\nTo answer one of reviewer 1&2 questions, we fabricated a new PhC structure with the same PhC geometric parameters and the same active region, based on hBN/WS , and performed a thorough temperature-dependence reflectivity study to complement the results shown in the initial manuscript. The reflectance measurements on this new sample as a function of temperature, from 15 K to 300 K, are shown in Figure R18. Interestingly, the bare exciton absorption line (accessible typically at large angles) can be distinguished in each spectrum. Therefore, we could extract out the bare (i.e., with negligible coupling to any photonic mode) exciton energy and linewidth of 1s exciton for each temperature. The corresponding reflectance spectra around the 1s exciton for each temperature are shown in Figure R19a. As discussed in detail in the answer to comment 1, the reflectance spectra were fitted with an asymmetric Fano function. The extracted peak energy and the linewidth for the 1s exciton versus temperature are shown in Figure R19b and c.\nAs temperature decreases, the peak position blue shifts from 2.028 to 2.086 eV, which is due to the reduction in electron-phonon interactions and an increase in the band gap at lower temperatures (Nano Lett. 21, 6240-6246 (2024)). Concomitantly, the linewidth decreases from 24.8 to 7.0 meV, which is ascribed to the decreased scattering of excitons with phonons (Nat. Commun. 7, 13279 (2016)).\nFigure R18. The temperature dependent reflectance spectra around 1s exciton from 295 K to 15 K.\nFigure R19. The temperature dependent reflectance spectra (a), the peak position (b), and linewidth (c) of 1s exciton from 295 K to 15 K.\nSubsequently, we extracted the Rabi splitting for each temperature and plotted it in Figure R20. One can observe that the Rabi splitting is roughly constant (32-35 meV) from low temperatures up to about 200 K, consistent with an almost constant excitonic broadening (7-9 meV). As temperature increases further, the Rabi splitting decreases from 35 ± 3 to 25 ± 2 meV, again consistent with the linewidth of the 1s exciton increasing from 9 to 25 meV due to enhanced electron-phonon coupling and an acceleration of the exciton dephasing (Nat. Commun. 9, 713 (2018)). Thus, the excitonic dephasing induced by coupling to phonons explains the behavior of the coupling strength over the whole temperature range studied for 1s polaritons.\nInterestingly, the two studied devices (the one analyzed in the main manuscript and the new one fabricated for answering the reviewers’ comments) display the same quantitative values (~35 meV Rabi splitting at low temperatures for both devices, and 25 meV at room temperature), confirming the robustness of the photonic crystal fabrication and the quality of the hBN/WS transfer.\n2. Further insight into the monotonic decrease of Rabi energy for 2s excitons?\nSimilarly, we repeated the analysis for 2s polaritons as a function of temperature, as shown in Figure R21. The fitted reflectivity spectra via Fano functions and the corresponding q values are shown in Figure R21a and b. In all cases, the q values are below 0.08, confirming that 2s excitons exhibit a symmetric lineshape approaching a Lorentz oscillator. The extracted peak energy and the linewidth for the 2s exciton versus temperature are shown in Figure R21c and d. As temperature decreases, the peak position blue shifts from 2.257 to 2.261 eV, and the linewidth decreases from 15.0 to 7.5 meV.\nFigure R21. Temperature dependent reflectance spectra (a), the fitted q values (b), the peak position (c), and linewidth (d) of 2s excitons from 100 K to 4 K.\nFigure R22. Temperature dependent Rabi splitting for X-polarized 2s polaritons. a, Temperature dependent angle-resolved reflectance spectra for 2s polaritons at the anti-crossing angle from 4 K to 120 K and their Fano fitting. b, Temperature dependent Rabi splitting and (gamma_exc + gamma_cav)/2 for 2s exciton polaritons.\nBased on the fitting procedure described in the response to comment 1, we fitted the temperature dependent angle-resolved reflectance spectra for 2s polaritons around the anti-crossing angle from 4 K to 120 K. These results are shown in Figure 2a and are reproduced in Figure R22a. As shown in Figure R22b, the Rabi splitting is larger than (gamma_exc + gamma_cav)/2 and remains roughly constant for temperatures up to 80 K, similarly to 1s exciton. However, due to a stronger exciton dephasing and broadening of the 2s exciton linewidth (Figure R22b), increasing the temperature further results in the loss of the strong-coupling regime above roughly 80 K, as attested by undistinguishable LP and UP resonances in the reflectance spectrum measured at 120 K. The larger intrinsic exciton dephasing of 2s excitons compared to 1s excitons is mainly due to their smaller exciton binding energy and larger Bohr radius (E_b^2 is 1/9 times that of E_b^1), indicating a higher sensitivity of the excitons to the surrounding environment, in particular to phonons.\nBased on the second and third reviewers’ suggestions and the associated discussions, we added a discussion on the degradation of the strong coupling as the temperature increases, where we present the temperature dependent Rabi splitting in the main text and include Figure R18-22 in the supplementary information.\nRevision:\nLine 154, page 7, main text: revised as “the Rabi splitting is larger than (gamma_exc + gamma_cav)/2 at each temperature and ensures that the system operates in the strong coupling regime (Fig. S13-15 and Supplementary Note 2). Further increasing the temperature will enhance phonon scattering and accelerate exciton dephasing, consequently reducing the Rabi splitting.”\nLine 156, page 7, main text: added “Besides, temperature-dependent angle-resolved reflectance spectra around 1s excitons from 15 K to 295 K (Fig. S16-18) were also explored. The Rabi splitting is roughly constant (32-35 meV) from low temperatures up to about 200 K, consistent with an almost constant excitonic broadening (7-9 meV). As temperature increases further, the Rabi splitting decreases from 35 ± 3 to 25 ± 2 meV, while the linewidth of the 1s exciton increases from 9 to 25 meV, due to enhanced electron-phonon coupling and an acceleration of the exciton dephasing49. The Rabi splitting is thus also larger than (gamma_exc + gamma_cav)/2 at each temperature, ensuring the robustness of the strong coupling. Herein, the larger intrinsic exciton dephasing of 2s excitons compared to 1s excitons is mainly due to their smaller exciton binding energy and larger Bohr radius (E_b^2 is 1/9 of E_b^1), indicating a higher sensitivity of the 2s excitons to the surrounding environment, in particular to phonons.”\nSupplementary materials: added Fig. S14-18 to discuss the temperature dependent Rabi energy for X-polarized 1s and 2s polaritons.\nComment 5:\nIn the magnetic field-dependent measurements (Figure 2d), the coupling strength exhibits quadratic enhancement with the magnetic field, but this enhancement is not symmetric around zero field, particularly for 2s excitons. Could the authors clarify the origin of this asymmetry?\nResponse:\nWe thank the reviewer for the constructive suggestion on clarifying the origin of this asymmetry. According to comment 1 and comment 8, and the associated explanations and discussions, we refitted all the experimental spectra to obtain the Rabi energy utilizing three Fano contributions for 1s exciton polaritons and two Lorentz functions (i.e., Fano resonances with very low q values) for 2s exciton polaritons. Herein, we plot the refitted results in Figure R23 and the corresponding differential Rabi energy from the fitting results. As expected, we clearly recover the quadratic and symmetric coupling strength enhancement with magnetic field, both for 1s and 2s polaritons. Accordingly, we revised the Figures in the main text. As mentioned in the response to comment 1, the difference between the initial Figure 2d and the new one (Figure R23c) stems just from the fitting procedure and the precision with which polariton resonances are now determined.\nFigure R23 Magnetic field dependent angle-resolved reflectance spectra for X-polarized mode coupling with 1s and 2s excitons. a. Magnetic field dependent angle-resolved reflectance spectra for X-polarized mode coupling with 1s exciton at the anti-crossing angle from -9 T to 9 T. Two dashed curves indicate the UP and LP peak positions. b. Magnetic field dependent angle-resolved reflectance spectra for X-polarized mode coupling with 2s exciton at the anti-crossing angle from -9 T to 9 T. Two dashed curves indicate the UP and LP peak positions. c. Summary of magnetic field dependent differential Rabi splitting Ω(B)/Ω(0) for 1s and 2s exciton polaritons.\nComment 6:\nIn the nonlinearity measurements, the authors have performed experiments in two configurations, viz., the broadband (BB) pumping of both the 1s and 2s states, and a narrow band (NB) pumping of the 1s states alone. Therefore, the authors could effectively compare the interactions between the various types of excitonic states. To further probe the role of the Rydberg excitons, I suggest an additional experiment where 2s excitons are resonantly pumped, analogous to the narrowband 1s excitation. While phonon assisted relaxation to the 1s state could still occur, this measurement could provide valuable insights into the intrinsic 2s-2s interactions.\nResponse:\nWe thank the reviewer for pushing us to perform an experiment that we initially considered but did not perform. Indeed, we didn’t conduct it in the first version of the article because, as suggested by the referee, we thought that even if we pumped (a narrow band) around the 2s energy, we would end up creating polaritons at the 1s exciton level due to phonon-assisted relaxation. As detailed below, the experiment was worth being conducted.\nIn the following, we present the experimental results on a new structure (device 2, i.e., the same used for the new temperature-dependent measurements) under BB and NB pumping conditions, both around 1s and 2s polaritons (Figure R24). Again, we fitted all the polaritonic reflectance spectra employing the same methodology discussed in comment 1: three Fano functions for 1s exciton polariton and two Lorentzian functions for 2s exciton polaritons. The analysis of the data provides thus a set of “new” interaction constants, 48.2 ± 6.7, and 8.2 ± 0.9 eV m2 for g_2s-2s^sat and g_1s-1s^sat, respectively, under NB pumping conditions (Figure R25). Combining these interaction constants with the energy shifts measured under BB pumping conditions on the same device, we can obtain two independent determinations of g_1s-2s^sat, which amount to 17.1 ± 8.0 and 13.0 ± 6.0 eV m2, respectively, depending on which NB interaction constant we use (for 2s or for 1s). Note that these two values agree well, and even if when determining the g_2s-2s^sat interaction constant, thanks to NB pumping around the 2s resonance, we certainly create some 1s polaritons. However, as the referee suggested, the density of 1s polaritons when pumping resonantly 2s polaritons seems to be low enough.\nFigure R24. Angle-resolved reflectance spectra and fittings for 1s and 2s exciton polaritons under BB and NB pumping conditions for the new sample (device 2).\nFigure R25. The extracted total ∆Esat for 2s and 1s polaritons under BB pumping conditions and ∆Esat for 1s and 2s under NB pumping conditions. Circles: experimental results. Solid curves: linear fitting of the experimental results.\nBased on the reviewer’s suggestion and the above discussion, we now present the data of device 2 (Figure R24 and 25) in the supplementary information and show the corresponding results based on the combination of the BB and the two NB pumping configurations.\nRevision:\nLine 238, page 11, main text: added “An alternative determination of the interaction constants can be done by pumping 1s and 2s polaritons independently under NB conditions, which enables to extract two independent values of g_2s-2s^sat and g_1s-1s^sat. This methodology was applied to a second hBN/WS /PhC structure (device 2), with the results shown in Fig. S24 and S25 in the Supplementary Materials. The analysis of the data provides thus a set of interaction constants, 48.2 ± 6.7 and 8.2 ± 0.9 eV m2 for g_2s-2s^sat and g_1s-1s^sat, respectively, under NB pumping conditions. These values are comparable to those in device 1. Combining these values with the energy shifts measured under BB pumping in the same structure, we can obtain two independent determinations of g_1s-2s^sat, which amount to 17.1 ± 8.0 and 12.9 ± 6.0 eV m2, depending on which NB interaction constant we use (2s or 1s). Overall, the values of nonlinearity enhancement between 2s and 1s polariton as well as the g_1s-2s^sat interaction constant in both devices are comparableconsidering the error bars.”\nSupplementary materials: added Fig. S24 and S25 to present the nonlinear measurements for device 2.\nComment 7:\nDetermining whether the system is in the strong or weak coupling regime requires accurate extraction of the bare exciton energy and linewidth. How do the authors obtain these values? Do they use the same device with different polarization of light?\nResponse:\nWe thank the reviewer for raising this question, which is especially important when working with TMD-based active regions given that even within the same flake, the exciton energy (and linewidth) can fluctuate depending on the actual environment (e.g., depending on the local strain). Indeed, within the studied devices (1 and 2), we could observe slightly different bare exciton energies and linewidths by changing the measurement position, as illustrated in Figure R26. Thus, for each set of measurements (e.g., as a function of magnetic field), we measured all spectra at the same location and, for each location (i.e., experiment series), we determined the actual exciton bare energy and linewidth.\nFigure R26. The X-polarized angle-resolved reflectance spectra from as-demonstrated configuration under several spots. White lines: the 1s exciton positions.\nThankfully, the absorption of bare 1s and 2s excitons could be easily measured far from any photonic mode at 4 K, and up to room-temperature for the 1s excitons. Thus, for each experiment, we could always determine the bare exciton energies and linewidths. Moreover, we also performed the measurement of polaritons at low temperature in the perpendicular polarization (Y-polarization), as shown in Figure R27. As discussed in the main text, we found that in the Y-polarization the system operates in the weak-coupling regime, consistent with the results in Nat. Commun. 9, 713 (2018). For completeness, and using the fitting procedure described in comment 1, we have analyzed again the previous data, as shown in Figure R27c. After fitting each spectrum with a Fano lineshape, we could get a bare exciton energy of 2.076 eV and a linewidth of 0.008 eV in the X-polarization and corresponding 2.076 eV and 0.009 eV in the Y-polarization. This comparison confirms that the measurement using perpendicular polarization is an alternative option to get reliable values for the bare exciton properties. Note that in the fitting results shown in Figure R27c, we have left one more digit to show that, indeed, the two curves are slightly different, but that this difference is much smaller than the error bars.\nBased on the reviewer’s suggestion and the above discussion, we added the description of how we extracted the bare exciton energy and linewidth in the main text. We further added Figures R26 and 27 into the supplementary information to illustrate the spatial variation of exciton energy within one flake, and the quantitative comparison of both polarizations.\nFigure R27. Angle-resolved X-polarized (a) and Y-polarized (b) reflectivity spectra around 1s polaritons and the corresponding Fano fittings at an angle of -30 degrees.\nRevision:\nLine 124, page 6, main text: added “Notably, given that within the same flake the bare exciton energy and gamma_exc of WS can fluctuate depending on the actual environment (Fig. S4), the excitonic values used for the corresponding modelling of the polaritonic dispersions are those measured at the same location but at larger angles, typically ~30 degrees, where the exciton is not coupled to any photonic mode. Alternatively, such bare exciton properties could also be obtained from measurements in Y-polarization, for which the system operates in the weak coupling regime (Fig. S5).”\nSupplementary materials: added Fig. S4 and S5 to present the typical spatial variations in our devices and the results of Y-polarized reflectance spectra around 1s exciton.\nComment 8:\nThe authors use dashed guidelines to indicate the energy values of LP and UP in several figures. Accurate extraction of these values is crucial for assessing temperature dependence, magnetic field dependence, Rabi splitting, nonlinearity, etc. How are the LP and UP energies determined? Could the authors elaborate on their extraction/fitting method? That is important since in some cases, e.g., in Fig. S10, the energy shift is not prominent.\nResponse:\nWe thank the reviewer for the great suggestion on accurate extraction of LP and UP peak positions, which has really helped us a lot in reevaluating the initial raw data and providing more accurate estimations of both Rabi splittings and, especially, of nonlinear interaction coefficients. As described in detail in the answer to comment 1, we fitted again all the initial temperature dependence, magnetic field dependence, and power dependence reflectance spectra, and extracted “new” Rabi splittings and nonlinear interaction coefficients based on this new analysis.\nThese data and fittings have already been shown in Figures R13, 15, 22, and 23, which we will not show again. Just to illustrate here the accuracy of the new fitting procedure, we would like to show the definitive analysis of the strong-coupling for 2s excitons-cavity photons at 4 K, as summarized in Figure R28 and where we see that the experimental reflectivity curves can be accurately reproduced.\nFigure R28. The refitted angle-resolved reflectance spectra for 2s polariton at 4 K.\nRevision:\nLine 117, page 6, main text: added “The spectra are fitted with two Lorentzian functions (solid curves in 2b) to obtain the upper polariton branch (UP) and the lower polariton branch (LP) peak positions, shown as dots in Fig. 2c (Methods). The obtained peak positions are further fitted with a coupled-oscillator model (solid curves in 2c) with a Rabi splitting in reflectivity (Ω_Ref) of 25.0 ± 2.0 meV. The corresponding exciton (X) and photon fractions (C) of UP and LP obtained are shown in Fig. 2d.”\nLine 135, page 6, main text: added “Fig. 2e. The spectra are fitted with two Lorentzian functions (solid curves in 2f) to obtain UP and LP peak positions (Methods), shown as dots in Fig. 2g. The obtained peak positions are further fitted with a coupled-oscillator model (solid curves in 2g) with a Rabi splitting in reflectivity (Ω_Ref) of 16.7 ± 1.5 meV. The corresponding X and C of UP and LP obtained for 1s and 2s polaritons at 4 K are shown in Fig. 2h.”\nLine 400, page 18, main text: added “c, Fitting of the UP and LP peak positions with a coupled-oscillator model (solid curves) indicates a coupling strength of 25 ± 2 meV. Dashed line: exciton dispersion. Solid line: photon dispersion. d, The corresponding photonic and excitonic fractions of UP and LP from c for X-polarized mode 1 coupling with 1s exciton.”\nLine 402, page 18, main text: added “g, Fitting of the UP and LP peak positions for 2s polariton with a coupled-oscillator model (solid curves) indicates a coupling strength of 16.7 ± 1.5 meV. Dashed line: exciton dispersion. Solid line: photon dispersion. h, The corresponding photonic and excitonic fractions of UP and LP from g for X-polarized photonic mode 1 coupling with 1s and 2s excitons.”\nComment 9:\nCould the authors further elaborate on the rationale behind Equations (1–4) in the main text? For example, it is unclear why the 1s-2s terms contributing to the energy shift of 2s excitons depend on N_1s but not on N_2s.\nResponse:\nWe thank the reviewer for giving us the possibility to further explain our theoretical model. Equations (1-4) describe the blueshift of the four polariton branches depending on the population of 1s and 2s particles, and on the different light-matter fractions. These equations contain similar terms. The first term describes the “self” Coulomb interaction induced blueshift: the exciton is blueshifted by interacting with the other excitons of the same kind. The second term describes the blueshift induced by the Coulomb interaction between excitons of different kinds, e.g., the blueshift of the 1s exciton due to the interaction with 2s excitons, and therefore this term only depends on N_1s, since the blueshift due to N_2s is already included in the first term. The third term describes the “self” blueshift due to phase-space filling, i.e., quenching of the Rabi splitting due to Pauli blocking. The last term corresponds to the quenching of the Rabi splitting, but this time due to the occupancy of states by the other species of excitons. A more detailed description of how to derive these phenomenological equations can be found in Phys. Rev. B 110, 115304 (2024).\nFinally, note that upon taking into account the effect of cross-polarized excitons (to answer comment 13 of the current referee), the equations 1-4 have been further generalized to explicitly separate the contributions of co- and cross-polarized excitons (or exciton-polaritons).\nComment 10:\nThe propagation experiment methodology is unclear. Why do the authors block certain regions (white rectangles) in the angle-resolved spectra? With the full spectrum available, they could simply select the desired energy and angle range for integration. Additionally, how do they physically block specific energy ranges in the actual experiment?\nResponse:\nSorry for the misunderstanding: the signal is not really blocked by any physical optical element. We collected the full spectrum for all data (Figure R29a), but to highlight the polaritons within the energy range between 2.1 to 2.17 eV, we needed to set the intensity scale from (0, 550) to (0, 180), as shown in Figure R29b. To remove the attention of the audience on the intense two peaks at about 2.18 eV and 2.03 eV (which are now saturated), we added the two white boxes to block them out, as shown in Figure R29c. To clarify this misunderstanding, we have revised the description of this section. Besides, following reviewers 1 and 2's suggestions, we have moved the whole propagation section to the Supplementary Information.\nFigure R29. Angle-resolved X-polarized mode PL spectra for 1s exciton polaritons.\nRevision:\nSupplementary materials: revised Fig. 4 as Fig. S27.\nComment 11:\nIn Figure 4e, the PL intensity decay for the exciton polaritons at energies of 2.11 and 2.15 eV are plotted. However, as observed in Figure 1b and discussed towards the beginning of the Results section, the reflectance data of the bare cavity is dominated by Fabry-Perot modes corresponding to the underlying thick SiO2 layer between 2.10-2.14 eV. How did the authors distinguish the exciton-polariton features despite the strong Fabry-Perot oscillations?\nResponse:\nWe thank the reviewer for this interesting question. As the reviewer said, we do find that there is a background from about 2.11 to 2.2 eV in the reflectance spectra, arising from the Fabry-Perot modes established in the underlying thick SiO layer. One of these broad modes is marked by a white dashed curve in Figure R30a. The important point to note is that the EM field intensity of this mode is strongly confined inside the SiO layer, as shown in Figure 30c. There is only 20% of the maximum intensity at the position of the TMD (Figure 30d). Therefore, the PL intensity will not be modified by this Fabry-Perot modes, as confirmed by Figure R30b, where we don’t observe any feature associated with such modes.\nFigure R30. Characterization of Fabry-Perot modes established in the underlying thick SiO layer. The angle-resolved X-polarized mode reflectance (a) and PL (b) spectra of 1s exciton polariton. The white dashed curves indicate the Fabry-Perot modes corresponding to the underlying thick SiO layer. The black dashed lines indicate the 1s exciton peak. c, The electric field distribution of the Fabry-Perot mode at the energy and angle indicated in b. c, The electric field intensity along the z axis; the dashed line indicates the location of WS .\nRevision:\nLine 109, page 5, main text: added “These lossy Fabry-Perot modes display a negligible electric field intensity at the WS location and, therefore, barely modify the photoluminescence (PL) of the excitons and polaritons (Fig. S1).”\nSupplementary materials: added Figure R30 into Fig. S1.\nComment 12:\nPrevious studies have explored excitons in TMDCs using 1D photonic crystals (e.g., L. Zhang et al., Nat. Commun. 9, 713 (2018)). Could the authors comment on how their approach differs in any significant way? Additionally, could they include dispersion data for the bare cavity across the full range of measured angles in the SI?\nResponse:\nWe thank the reviewer for the constructive suggestion on further explaining the differences with previous papers, in particular with that of L. Zhang et al., Nat. Commun. 9, 713 (2018)).\nFirst, we totally agree that some previous studies have also explored exciton polaritons in TMDs utilizing 1D photonic crystals:\n1. L. Zhang et al. demonstrated the first PhC dressed exciton-polariton based on TMD materials.\n2. V. Kravtsov et al. demonstrated the first 1D BIC coupled exciton-polariton based on TMD materials (Light: Sci.& Appl. 9, 56 (2020)).\n3. E. Maggiolini reported similar results as the previous work with a different BIC design, which maximizes the field overlap (Nat. Mater. 22, 964-969 (2023)).\n4. O. Koksal et al. demonstrated that trions can sustain the strong coupling regime and implemented them in a 1D PhC waveguide BIC (Phys. Rev. Research 3, 033064 (2021)).\nSo we fully convey, as already indicated in the main manuscript, that 1D photonic crystals are not novel in the field of TMD based polaritonic.\nHowever, our work differs in several aspects that we are convinced will make it appealing to the community:\n1) The main point of our work is to study the nonlinearity of Rydberg excitons when strongly-coupled to a 1D PhC, which has not been reported so far (in any material system). To our knowledge the sole previous study devoted to Rydberg polariton nonlinearities is from J. Gu et al., who reported the first ever Rydberg exciton polaritons in TMDs; in their work, the authors used a vertical Fabry-Perot DBR cavity with 3 stacked WSe monolayers (Nat. Commun. 12, 2269 (2021)), where 2s-based polaritons exhibited 4.6 times larger nonlinearity than ground-state polaritons. In Table R4 below, we summarize all previous works reporting Rydberg polaritons, where we can see that all of them employ vertical microcavities, which restrain their applicability in photonic integrated technologies. To ease the integration and compactness of polaritonic devices, we chose PhC structures, which are compatible with TMDs (as they can be easily exfoliated and transferred onto the PhC) to realize the first on-chip Rydberg polaritons. We believe this point is most significant, and differentiates already our paper from previous realizations (also with L. Zhang et al., Nat. Commun. 9, 713 (2018)), but it is certainly not the only originality.\n2) Second, the measured nonlinearity is not the largest measured in the polaritonic field, where J. Gu et. al reported a nonlinearity of g = 46.4 eV m2 for Rydberg polaritons (Nat. Commun. 12, 2269 (2021)), and L. Zhang reported a nonlinearity of g close to 100 eV m2 in a vertical microcavity containing Moire exciton-polaritons thanks to the use of a MoSe -WS heterobilayer (Nature 591, 61-65 (2021)). Nonetheless, to our knowledge, it is the largest reported among PhC-based polaritons (Table R5).\n3) Third, apart from the enhancement of the nonlinearity due to the increased Bohr radii of 2s excitons, we have identified that the electric-field localization, induced by the PhC, could contribute to further enhance (by 15-20%) exciton-polariton nonlinearities, as discussed in the comment 1. This shows that PhC structures are not only promising for on-chip compact devices but also allow for further enhancement of the nonlinearity compared with planar microcavities.\n4) Fourth, the coexistence of 2s polaritons with 1s polaritons enables us to experimentally quantify their interaction strength, i.e., 13.3 ± 6.3 eV m2 for device 1, 17.1 ± 8.0 and 12.9 ± 6.0 eV m2 for device 2, for the first time. This aspect was overlooked in the report by J. Gu et al., (Nat. Commun. 12, 2269 (2021)), who indeed neglected it explicitly.\nMeanwhile, the dispersion data for the bare cavity across the full range are present in Figure S11.\nRevision:\nLine 61, page 3, main text: added “, among which the TMD based Rydberg polaritons show a record nonlinearity (Table S1).”\nLine 238, page 11, main text: added “which exhibits the record values among PhC-based polaritons (Table S3)”\nSupplementary materials: added Table S1 to compare Rydberg excitons among representative semiconductors.\nSupplementary materials: revised Table S3 to compare the Rabi energy among representative TMD based configurations.\nSupplementary materials: added Supplementary Fig. S11 to present dispersion data for the bare cavity across the full range.\nComment 13:\nUnlike Fabre-Perot cavity, 1d PhCs has additional anisotropy. Could the authors comment whether this anisotropy affects exciton-polariton properties and their nonlinearity.\nResponse:\nWe thank the reviewer for inquiring about the anisotropy of the PhC cavity, which has pushed us to refine our theoretical modelling, as described below.\nThe geometry of the PhC indeed breaks the symmetry between the X and Y directions and creates an effective anisotropy (Nat. Commun. 9, 713 (2018)). Regarding the exciton-polariton properties, which we already presented in the first version of the manuscript, one polarization displays the strong coupling (parallel to the PhC grooves) while the perpendicular one displays weak coupling, both at RT (Figure R31a) and at 4 K (Figure R31b). The main reason is that in this polarization, the maximum intensity of the electric field is located at the center of the SiN (along the vertical direction) while almost no field intensity is found at the position of the TMD (Figure R31c).\nRegarding the nonlinearity, it is indeed important to consider how uncoupled Y-polarized excitons can affect the measured nonlinearity of X-polarized polaritons. First, as shown in Figure R32, the excitonic absorption under Y-polarization (weak coupling) is almost 15% that of the polaritonic absorption under X-polarized (strong coupling). Besides, we excited the system using an unpolarized light with a polarization degree of 20% along x direction.\nHerein, we now calculate the phase filling energy shifts considering the coexistence of X-polarized (in the strong coupling regime) and Y-polarized (in the weak coupling regime) excitons. The second quantization Hamiltonian describing the phase space filling written in the exciton basis can be expressed as (Rev. Mod. Phys. 85, 299 (2013)):\nH = g_sat * sum(b_sigma^dagger * b_sigma^dagger * b_sigma * a_sigma) + h.c. (11)\nwhere sigma is the pseudospin, b is the exciton bosonic operator, and a is the photonic operator. We have made the assumption that the phase space filling interaction is spin diagonal. In order to switch to the basis of X and Y polarized excitons, we write:\nb_± = 1/sqrt(2) * (b_X ± i*b_Y), b_±^dagger = 1/sqrt(2) * (b_X^dagger ∓ i*b_Y^dagger) (12)\na_± = 1/sqrt(2) * (a_X ± i*a_Y), a_±^dagger = 1/sqrt(2) * (a_X^dagger ∓ i*a_Y^dagger) (13)\nInserting (12) and (13) into (11) and simplifying we get:\nH_PSF = 1/2 * (b_X^dagger * b_X^dagger * b_X * a_X + b_Y^dagger * b_Y^dagger * b_Y * a_Y - b_X^dagger * b_X^dagger * b_Y * a_Y + b_Y^dagger * b_Y^dagger * b_X * a_X + b_X^dagger * b_Y^dagger * b_X * a_Y + b_Y^dagger * b_X^dagger * b_Y * a_X + b_X^dagger * b_Y^dagger * b_Y * a_X - b_Y^dagger * b_X^dagger * b_X * a_Y) + h.c. (14)\nFor macroscopic populations we get that:\nH_PSF = 1/2 * ((N_X + 2n_Y) * b_X^dagger * a_X + (2N_X + N_Y) * b_Y^dagger * a_Y - b_X^dagger * b_X^dagger * b_X * a_X - b_Y^dagger * b_Y^dagger * b_Y * a_Y) + h.c. (15)\nTherefore, writing the Heisenberg equation and keeping the terms contributing to the energy shift, we get for the 1s LPB:\nΔE_1s_LP = g_1s-1s^sat * sqrt(X_1s * C_1s) * (X_1s * N_1s^X + 2n_1s^Y) + g_1s-2s^sat * sqrt(X_1s * C_1s) * (X_2s * N_2s^X + 2n_2s^Y) (16)\nwhere N_1s/2s^X is the population of 1s/2s polaritons with X polarization and n_1s/2s^Y is the population of 1s/2s excitons with Y polarization under a certain pump power; C/X are the corresponding excitonic and photonic fractions for polaritons on the LP with a population of N_1s/2s^X. Therefore, the population of Y-polarized excitons created under unpolarized pumping does contribute to the nonlinearity of the X-polarized polaritons.\nTo experimentally explore how Y-polarized excitons affect the nonlinearity of the X-polarized polariton/exciton, we excited our system with a linearly polarized laser (and used NB configuration). As shown in Figure R33a (a half-waveplate is utilized to realize X or Y-polarized pumping), the polarization degree in this experiment attains 98.7% (along Y polarization). Figure R33b and c present the experimental results of pump power dependent X-polarized reflectance under X or Y-polarized laser excitation under NB pumping conditions (device 2). The calculated energy shifts for the phase space filling effect are plotted in Figure R33d (the absorbance differentials are already considered for X and Y-polarized particles). Employing equation 16, it indicates a comparable g_1s-1s^sat (interaction between excitons) of 12.4 (XX configuration) and 13.7 eV m2 (YX configuration), respectively. Herein, compared with the calculation of polariton density, the exciton density is calculated considering the Y-polarized excitonic absorption along with the excitonic lifetime.\nFurthermore, we monitored the pure exciton nonlinearity (exchange interaction here) when using Y-polarized laser excitation and NB pumping conditions (Figure R34). Employing the equation for the energy shift due to exchange interaction, we could calculate the g_1s-1s^ee of 7.2 (YY configuration) and 6.4 eV m2 (XX configuration).\nThus, the anisotropy of the PhC cavity indeed affects the exciton-polariton properties and their nonlinearity. Based on the reviewer’s suggestion, we added comments on such anisotropy when we discussed the weak coupling of the exciton with the Y-polarized cavity mode and revised the expressions of the energy shift of each polariton for 1s and 2s excitons in the main text (equations 1-4), considering the contribution of the uncoupled Y-polarized exciton. Furthermore, the corresponding data for calculating g_1s-1s^sat are replotted with the effective exciton density (N_eff), i.e., (X_1s/2s * N_1s/2s^X + 2n_1s/2s^Y) in equation 16 (as shown in Figure R15c).\nRevision:\nLine 218, page 10, main text: added “considering an unpolarized pump laser that generates both Y-polarized excitons and X-polarized poalritons (Supplementary Note 3).”\nLine 219, page 10, main text: revised as “\nΔE_2s-LP^sat = g_2s-2s^sat * sqrt(X_2s * C_2s) * (X_2s * N_2s^X + 2n_2s^Y) + g_1s-2s^sat * sqrt(X_2s * C_2s) * (X_1s * N_1s^X + 2n_1s^Y) (1)\nΔE_2s-UP^sat = g_2s-2s^sat * sqrt(X_2s * C_2s) * (C_2s * N_2s^X + 2n_2s^Y) + g_1s-2s^sat * sqrt(X_2s * C_2s) * (X_1s * N_1s^X + 2n_1s^Y) (2)\nΔE_1s-LP^sat = g_1s-1s^sat * sqrt(X_1s * C_1s) * (X_1s * N_1s^X + 2n_1s^Y) + g_1s-2s^sat * sqrt(X_1s * C_1s) * (X_2s * N_2s^X + 2n_2s^Y) (3)\nΔE_1s-UP^sat = g_1s-1s^sat * sqrt(X_1s * C_1s) * (C_1s * N_1s^X + 2n_1s^Y) + g_1s-2s^sat * sqrt(X_1s * C_1s) * (X_2s * N_2s^X + 2n_2s^Y) (4)”\nLine 228, page 10, main text: added “N_1s/2s^X is the population of 1s/2s polaritons with X polarization and n_1s/2s^Y is the population of 1s/2s excitons with Y polarization”.\nLine 343, page 15, main text: added “tau_pol is set as 48/50.7 fs for the 1s/2s polaritons”.\nLine 343, page 15, main text: added “As we utilize unpolarized light (with a polarization degree 20% along x direction), Y-polarized excitons (in the weak coupling regime) will also be generated (Fig. S31 and S32). According to equations 1-4, we included the effect of Y-polarized excitons, considering the excitonic lifetime and the bare excitonic absorption, with measured excitonic linewidths of 6.5 and 7.9 meV for 2s and 1s excitons, respectively, at 4 K.”\nSupplementary materials: added Fig. S31 and S32 to present Y-polarization reflectance spectra and the nonlinearity calculation under YX and XX configurations.\nSupplementary materials: added Supplementary Note 3 to calculate the phase space filling energy shifts terms for X and Y polarized excitons.\nReply to Reviewer 4\nComment 1:\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.\nResponse:\nWe sincerely thank the referee for the co-review of this manuscript.\nReply to Reviewer 5\nComment 1:\n1, The work reports on the experimental investigation and theoretical interpretation of Rydberg polaritons in one-dimensional (1D) photonic crystal lattices. In particular, a WS2 monolayer is deposited, with a 15 nm buffer of hBN, on top of a 220 nm thick 1D-grating in SiN sitting on a glass (SiO2) cladding. Optical excitations are probed in reflectance spectroscopy. This result is, per se, not novel. Exciton-polaritons have been reported in similar settings (actually, from the very same two-dimensional semiconductor, WS2, coupled to a SiN photonic crystal, with a two-dimensional lattice, instead of a simpler 1D one), see, e.g., Nano Letters vol. 20, 5292 (2020).\n2, Here the authors also report estimates for the polariton propagation constant, which is obtained from non-resonant excitation experiments in the few microns range. Also in this case, while it may be considered one of the first experiments showing Rydberg polariton propagation in WS2 monolayers, much longer polariton propagation distances have been already shown in alternative materials, so also in this case I feel very reluctant to recommend publication in Nature Communications, in the absence of a stronger argument justifying the novelty of this work as compared to the literature.\nResponse:\nWe sincerely thank the reviewer for suggesting us to give a clearer demonstration of the novelty of the results.\nFirst, we totally agree that some previous studies have explored exciton polaritons in TMDs utilizing 1D or 2D photonic crystals:\n1. L. Zhang et al. demonstrated the first PhC dressed exciton-polariton based on TMD materials (Nat. Commun. 9, 713 (2018)).\n2. V. Kravtsov et al. demonstrated the first 1D BIC coupled exciton-polariton based on TMD materials (Light: Sci.& Appl. 9, 56 (2020)).\n3. E. Maggiolini et al. reported similar results to V. Katsov and collaborators but with a different BIC design, which maximizes the field overlap (Nat. Mater. 22, 964–969 (2023)).\n4. O. Koksal et al. demonstrated that trions can sustain the strong coupling regime and implemented it in a 1D PhC waveguide BIC (Phys. Rev. Research 3, 033064 (2021)).\nIn parallel:\n5. W. Liu et al. demonstrated helical topological exciton-polaritons utilizing 2D PhC and developed an analogous quantum spin Hall system (Science 370, 600-604 (2020)).\n6. Y. Chen et al. also presented 2D metasurfaces integrated with monolayer exciton polariton and realized 2D subwavelength-scale engineering of the optical meta-atoms in the same year (Nano Lett. 20, 5292–5300 (2020)).\n7. M. Li et al. showed experimental observation of topological Z exciton-polaritons (Nat. Commun. 12, 4425 (2021)).\n8. X. Xie et al. demonstrated an order-of-magnitude enhancement in transport length of polariton compared with bare TMD exciton based on 2D photonic crystals (Sci. Adv. 11, eads0231 (2025)).\nSo we fully convey, as we already indicated in the main manuscript, that combining 1D (or 2D) photonic crystals with TMD monolayers to generate polaritons is not new.\nIndeed, as correctly identified by the other referees, the novelty does not stem from the fact of creating PhC-based polaritons nor from the polariton propagation that, following the other referees’ suggestions, we have moved to the supplementary information.\nInstead, our work differs in several essential aspects that we are convinced make it appealing for the community:\n1) The main point of our work is to study the nonlinearity of Rydberg excitons when strongly-coupled to a 1D PhC, which has not been reported so far (in any material system). To our knowledge the sole previous study devoted to Rydberg polariton nonlinearities is from J. Gu et al., who reported the first ever Rydberg exciton polaritons in TMDs; in their work, the authors used a vertical Fabry-Perot DBR cavity with 3 stacked WSe monolayers (Nat. Commun. 12, 2269 (2021)), where 2s-based polaritons exhibited 4.6 times larger nonlinearity than ground-state polaritons. In Table R6 below, we summarize all previous works reporting Rydberg polaritons, where we can see that of all of them employed vertical microcavities, which restrain their applicability in photonic integrated technologies. To ease the integration and compactness of polaritonic devices, we chose PhC structures, which are compatible with TMDs (as they can be easily exfoliated and transferred onto the PhC) to realize the first on-chip Rydberg polaritons. We believe this point is most significant and differentiates our paper from previous realizations, but it is certainly not the only originality.\n2) Second, the measured nonlinearity is not the largest reported in the field of polaritonic, where J. Gu et. al reported a nonlinearity of g = 46.4 eV m2 for Rydberg polaritons (Nat. Commun. 12, 2269 (2021)), and L. Zhang reported a nonlinearity g close to 100 eV m2 in a vertical microcavity containing Moire exciton-polaritons thanks to the use of a MoSe -WS heterolayer (Nature 591, 61-65 (2021)). Nonetheless, to our knowledge, it is the largest reported among PhC-based polaritons (Table R7).\n3) Third, apart from the enhancement of the nonlinearity due to the increased Bohr radii of 2s excitons, we have identified that the electric-field localization, induced by the PhC, could contribute to further enhance (by 15-20%) exciton-polariton nonlinearities. This shows that PhC structures are not only promising for on-chip compact devices but also allow for further enhancement of the nonlinearity compared with planar microcavities.\n4) Fourth, the coexistence of 2s polaritons with 1s polaritons enables to experimentally quantify their interaction strength, i.e., 13.3 ± 6.3 eV m2 for device 1, 17.1 ± 8.0 and 12.9 ± 6.0 eV m2 for device 2, for the first time. This aspect was overlooked in the report by J. Gu et al., (Nat. Commun. 12, 2269 (2021)), who indeed neglected it explicitly.\nFinally, we also need to emphasize that, based on reviewers 3&4 and 5’ suggestions on considering the anisotropy of the X-polarized and Y-polarized cavity modes as well as the nonlinear interaction between X-polarized polaritons and Y-polarized exciton, we have extended the nonlinear equations 1-4 in the main text and the corresponding calculations of the nonlinearity (Figure 4) to take into account explicitly the contribution of co- and cross-polarized excitons to the measured nonlinearity, which is also an original contribution to the field. See details in the comment 13 to reviewer 3 and comment 3 to reviewer 5.\nRevision:\nLine 61, page 3, main text: added “, among which the TMD based Rydberg polariton shows a record nonlinearity (Table S1).”\nLine 238, page 11, main text: added “which exhibits the record values among PhC-based polaritons (Table S3)”\nSupplementary materials: added Table S1 to compare Rydberg excitons among representative semiconductors.\nSupplementary materials: revised Table S3 to compare the Rabi energy among representative TMD based configurations.\nComment 2:\n1, The Rabi splitting in the order of 20 meV reported in these works is compatible, which is good to begin with, but not sufficient to claim a significant innovation as a result. In addition, polariton linewidths are extremely large in the experiment reported here, roughly of the same magnitude as the Rabi splitting basically, which makes the strong coupling hardly distinguished from the measurements, in fact. This is actually much worse than what was shown in the quoted Nano Letters. The authors may want to give an interpretation on this. Looking at the spectra of Figs. 1 and 2, one might actually even doubt that the sample is in the strong coupling regime, was not for a clever lineshape fit.\n2, Honestly, the linewidths of these polaritons are really broad, and thus losses must be huge. What are the reasons for that? The authors could elaborate a bit more on this aspect, and try to figure out mitigation strategies, if any. In fact, I would expect resonant modes from the underlying SiN grating to display much narrower lines (notice that a photon lifetime of 35 fs quoted in methods...this is really short for a dielectric grating!), thus affecting the overall polariton losses.\nResponse:\nWe thank the reviewer for suggesting a clearer demonstration of Rabi splitting and further discussion on the polariton linewidths. We agree with the reviewer that the Rabi splittings, i.e., 17-20 meV for 2s and 35 meV for 1s at 4 K, are not large compared to other materials, which can show Rabi splitting as large as hundreds of meV (e.g., organic materials), and indeed we never claimed or meant to claim that this was an important aspect. At the same time, these Rabi splittings are much larger than those found in GaAs-based microcavities that are, nonetheless, the most studied material system in the polaritonic community, and that clearly shows that large or small Rabi splittings can not be considered “worse or better” per se.\nSecond, as discussed below in detail, the measured photonic and excitonic broadenings combined with the 2s-photon interaction strength are sufficient to unambiguously prove the strong-coupling for 2s Rydberg polaritons, following the criteria established 30 years ago by V. Savona and collaborators (Solid State Communications 93 (9), 733-739), and sufficient too for assessing the strong-coupling between 1s excitons and the PhC mode from low temperature up to room temperature.\nBecause the broadening of the 1s excitons and the photonic mode at room temperature and the broadening of the 2s excitons and the photonic modes at low temperatures are similar to the respective Rabi splittings, care must be taken to assess unambiguously the coupling regime. In response to a request by referees 3&4, we have reanalyzed all the raw data of the initial manuscript, as well as additional measurements on a new structure replicating the first one (which we have fabricated in response to a comment by reviewers 1&2), by considering Fano resonances, which accurately reproduce the spectral dependence of all measured reflectance spectra in our work (comment 1 to reviewer 3).\nWe start first to analyze the strong-coupling of 1s excitons with the PhC mode from low to room-temperature. The reflectance measurements on the new sample, which replicates the first one discussed in the main text, as a function of temperature from 15 K to 300 K, are shown in Figure R35. Interestingly, the bare exciton absorption line (accessible typically at large angles) can be distinguished in each spectrum. Therefore, we could extract out the bare (i.e., with negligible coupling to any photonic mode) exciton energy and linewidth of 1s exciton for each temperature. The corresponding reflectance spectra around the 1s exciton for each temperature are shown in Figure R36. After being fitted with an asymmetric Fano function, the extracted peak energy and the linewidth for the 1s exciton versus temperature are shown in Figure R36b and c.\nAs temperature decreases, the peak position blue shifts from 2.028 to 2.086 eV, which is due to the reduction in electron-phonon interactions and an increase in the band gap at lower temperatures (Nano Lett. 21, 6240-6246 (2024)). Concomitantly, the linewidth decreases from 24.8 to 7.0 meV, which is ascribed to the decreased scattering of excitons with phonons (Nat. Commun. 7, 13279 (2016)).\nFigure R35. The temperature dependent reflectance spectra around 1s exciton from 295 K to 15 K.\nFigure R36. The temperature dependent reflectance spectra (a), the peak position (b), and linewidth (c) of 1s exciton from 295 K to 15 K.\nSubsequently, we extracted the Rabi splitting for each temperature and plotted it in Figure R37. One can observe that the Rabi splitting is roughly constant (32-35 meV) from low temperatures up to about 200 K, consistent with an almost constant excitonic broadening (7-9 meV). As the temperature increases further, the Rabi splitting decreases from 35 ± 3 to 25 ± 2 meV, again consistent with the linewidth of the 1s exciton increasing from 9 to 25 meV, due to enhanced electron-phonon coupling and an acceleration of the exciton dephasing (Nat. Commun. 9, 713 (2018)). Thus, the excitonic dephasing induced by coupling to phonons explains the behavior of the coupling strength over the whole temperature range studied for 1s polaritons.\nInterestingly, the two studied devices (the one analyzed in the main manuscript and the new one fabricated for answering the reviewers comments) display exactly the same quantitative values (~35 meV at low temperatures for both devices, and 25 meV at room-temperature), confirming the robustness of the photonic crystal fabrication and the quality of the hBN/WS transfer thereon.\nIn the following lines, we discuss the strong-coupling regime for the 2s excitons and the PhC mode. We repeated the same analysis for 2s polaritons as a function of temperature, as shown in Figure R38. The fitted reflectance spectra via Fano functions and the corresponding q values are shown in Figure R38a and b. In all cases, the q values are below 0.08, confirming that 2s excitons exhibit a symmetric lineshape approaching a Lorentz oscillator. The extracted peak energy and the linewidth for the 2s exciton versus temperature are shown in Figure R38c and d. As temperature decreases, the peak position blue shifts from 2.257 to 2.261 eV, and the linewidth decreases from 15.0 to 7.5 meV.\nFigure R38. Temperature dependent reflectance spectra (a), the fitted q values (b), the peak position (c), and linewidth (d) of 2s excitons from 100 K to 4 K.\nFigure R39. Temperature dependent Rabi energy for X-polarized 2s polaritons. a, Reflectance spectra for different angles around the anti-crossing region of 2s polaritons. b, 2s polaritons energies as a function of angle extracted from the fitting of the reflectance spectra in a. c, Temperature-dependent angle-resolved reflectance spectra for 2s polariton at the anti-crossing angle from 4 K to 120 K. d, Temperature dependent Rabi energy and (gamma_exc + gamma_cav)/2 for 2s exciton polaritons.\nBased on the fitting procedure described in the response to comment 1 of referee 3, we fitted the temperature dependent angle-resolved reflectance spectra for 2s polaritons around the anti-crossing angle from 4 K to 120 K, shown in Figure 3a of the main text utilizing two Lorentzian functions (which correspond to Fano resonances with low, i.e., close to zero, q Fano parameters), and reproduced in Figure R39. As an illustration, in Figure R39a, we display the reflectance spectra of 2s polaritons at 4 K for different angles around the anti-crossing region, and we plot in Figure R39b the polariton energies as a function of angle, showing a clear anti-crossing, which is the fingerprint of the strong coupling. We have done this analysis for different temperatures and summarized the results in Figures R39c and d. As shown in Figure R39d, the Rabi energy is larger than (gamma_exc + gamma_cav)/2 and remains roughly constant for temperatures up to 80 K, similarly to 1s polaritons. However, due to a stronger exciton dephasing and broadening of the 2s exciton linewidth (Figure R39d), increasing the temperature further results in the loss of the strong-coupling regime above roughly 80 K, as attested by undistinguishable LP and UP resonances in the reflectance spectrum measured at 120 K (Figure R39c). The larger intrinsic exciton dephasing of 2s excitons compared to 1s excitons is mainly due to their smaller exciton binding energy and larger Bohr radius (E_b^2 is 1/9 times that of E_b^1), indicating a higher sensitivity of exciton to the surrounding environment, in particular to phonons.\nThe above discussion and analysis prove unambiguously that, indeed, the strong-coupling can be assessed reliably in these two samples. Thus, based on the second, third, and fifth reviewers’ suggestions and the associated discussions, we added a discussion on the evolution of the strong coupling as the temperature increases, where we present the temperature-dependent Rabi splitting in the main text and include Figure R36-39 into the supplementary information.\nFinally, let us respond to the referee's opinion on the PhC quality factor (“rather large broadening”). The measured linewidths correspond to quality factors that are in the order of Q~100-200, depending on the PhC mode that we consider. While, once again, these Qs are not the record ones, typically obtained in GaAs-based microcavities, they are comparable to those obtained on vertical microcavities based on one dielectric DBR and a top metallic mirror, as often used in TMDs-based microcavities (see for example references, Nat. Commun. 9, 713 (2018)) and Nat. Commun. 12, 2269 (2021)). Since these Q values enable us to sustain the strong-coupling for both 1s and 2s excitons, and as we have demonstrated that the loss of the strong-coupling is associated with the intrinsic exciton-phonon dephasing, improving the Q value would not have enabled us to improve any further the temperature resilience. Thus, we are convinced that to demonstrate the main results of this article (i.e., strong-coupling of 2s excitons with PhC modes, the demonstration of stronger interactions between 2s-2s polaritons than between 1s-1s polaritons, the effect of field localization in the PhC on the interactions, and the contribution of co- and cross-polarized excitons to the measured nonlinearity), the current Qs are sufficient.\nRevision:\nLine 117, page 6, main text: added “The spectra are fitted with two Lorentzian functions (solid curves in 2b) to obtain the upper polariton branch (UP) and the lower polariton branch (LP) peak positions, shown as dots in Fig. 2c (Methods). The obtained peak positions are further fitted with a coupled-oscillator model (solid curves in 2c) with a Rabi splitting in reflectivity (Ω_Ref) of 25.0 ± 2.0 meV. The corresponding exciton (X) and photon fractions (C) of UP and LP obtained are shown in Fig. 2d.”\nLine 135, page 6, main text: added “Fig. 2e. The spectra are fitted with two Lorentzian functions (solid curves in 2f) to obtain UP and LP peak positions (Methods), shown as dots in Fig. 2g. The obtained peak positions are further fitted with a coupled-oscillator model (solid curves in 2g) with a Rabi splitting in reflectivity (Ω_Ref) of 16.7 ± 1.5 meV. The corresponding X and C of UP and LP obtained for 1s and 2s polaritons at 4 K are shown in Fig. 2h.”\nLine 154, page 7, main text: added “where the Rabi energy is larger than (gamma_exc + gamma_cav)/2 at each temperature and ensures that the system operates in the strong coupling regime (Fig. S13-15 and Supplementary Note 2).”\nLine 156, page 7, main text: added “Besides, temperature-dependent angle-resolved reflectance spectra around 1s excitons from 15 K to 295 K (Fig. S16-18) were also explored. The Rabi splitting is roughly constant (32-35 meV) from low temperatures up to about 200 K, consistent with an almost constant excitonic broadening (7-9 meV). As temperature increases further, the Rabi splitting decreases from 35 ± 3 to 25 ± 2 meV, while the linewidth of the 1s exciton increases from 9 to 25 meV, due to enhanced electron-phonon coupling and an acceleration of the exciton dephasing. The Rabi splitting is thus also larger than (gamma_exc + gamma_cav)/2 at each temperature, ensuring the robustness of the strong coupling. Herein, the larger intrinsic exciton dephasing of 2s excitons compared to 1s excitons is mainly due to their smaller exciton binding energy and larger Bohr radius (E_b^2 is 1/9 of E_b^1), indicating a higher sensitivity of the 2s excitons to the surrounding environment, in particular to phonons.”\nLine 156, page 7, main text: added “Note that we have used Fano resonances to fit the polariton dispersions, but whenever the q value was lower than 0.1, we used instead two Lorentzian functions to simplify, as the Fano resonance reproduces a Lorentzian when q = 0. Overall, in the current work, we employed Fano resonances to fit the temperature dependence of 2s and 1s bare excitons (Fig. S14 and S17) as well as to fit 1s polaritons at low temperature (Fig. S18), while two Lorentzian functions could be used to fit the 2s polaritons at low temperature as well as 1s polaritons at room temperature (Fig. 2 and 3) whenever extended studies were conducted at those temperatures. For clarity, we have explicitly indicated in each series of measurements what functional dependence was used to fit the reflectivity spectra.”\nLine 400, page 18, main text: added “c, Fitting of the UP and LP peak positions with a coupled-oscillator model (solid curves) indicates a coupling strength of 25 ± 2 meV. Dashed line: exciton dispersion. Solid line: photon dispersion. d, The corresponding photonic and excitonic fractions of UP and LP from c for X-polarized mode 1 coupling with 1s exciton.”\nLine 402, page 18, main text: added “g, Fitting of the UP and LP peak positions for 2s polariton with a coupled-oscillator model (solid curves) indicates a coupling strength of 16.7 ± 1.5 meV. Dashed line: exciton dispersion. Solid line: photon dispersion. h, The corresponding photonic and excitonic fractions of UP and LP from g for X-polarized photonic mode 1 coupling with 1s and 2s excitons.”\nMethods, main text: added “Fitting the reflectance spectra of the excitonic and polaritonic species”\nSupplementary materials: added Supplementary Fig. S16-18 and Note 2 to demonstrate the temperature-dependent behaviors of 2s and 1s exciton polaritons and Fano-type excitonic and polaritonic reflectance spectra fitting.\nComment 3:\nIn any case, going further than the linear polariton physics, the authors explore the coupling between excited Rydberg states from the WS2 monolayer excitonic response to the same photonic eigenmode of the 1D photonic lattice. The experimental evidence points towards an enhanced nonlinear response of the 2s-related exciton polariton as compared to the 1s-related one. In particular, the 2s-2s nonlinearity is quantified to be (with large exp error, though) more than 20 times larger as compared to the 1s-1s. This result might be considered as an interesting one, per se, since it justifies current interest in exploring Rydberg polariton physics for various applications. However, also in this case the result cannot be considered as completely unexpected or novel. In fact, as also acknowledged in the manuscript large nonlinearities for the 2s-related exciton polaritons were already reported in Nat. Comm. vol. 12, 2269 (2021), albeit in a vertical microcavity geometry. Authors should make sure that when exciting the system with the pulsed pump, any excitation of the exciton reservoir can be neglected, otherwise the estimate of the nonlinear might be inaccurate.\nResponse:\nWe agree with the referee that exploiting Rydberg polaritons with larger interaction strengths is of relevance and might be exploited in various applications, in particular to approach the quantum polariton blockade regime. First of all, let us state that upon reanalyzing the raw data based on the methodology proposed by reviewers 3&4, the enhancement factor of the 2s-2s nonlinear interaction with respect to the 1s-1s interaction is 10 ± 3.0 for the previous structure (device 1) and 5.9 ± 1.1 for the newly fabricated structure (device 2, which displays nominally the same structure). Though the enhancement factor is smaller than previously announced, it is still among the highest values for nonlinearity in TMD based polaritons and is still the record value among PhC-based polaritons (Table R7).\nAs indicated by the referee, J. Gu et. al reported the nonlinearity of Rydberg polaritons in a vertical microcavity containing WSe . However, in their case, because of the smaller Rabi splitting compared to our PhC structure due to a worse photonic mode-exciton overlap, they needed to stack three monolayers of WSe to achieve a sizeable Rabi splitting and to be able to study the 2s-2s nonlinearity (Note: their Rabi splitting for the 2s exciton-microcavity photon coupling is roughly half of our Rabi energy, confirming that our PhC structures provide not only a qualitative but also a quantitative improvement). It is well known that while increasing the number of monolayers will increase the Rabi splitting (proportional to the square root of the number of monolayers), it will also reduce the nonlinear interaction (O. Bleu et al., Phys. Rev. Res. 2, 043185 (2020)). This is an additional aspect differentiating our work and theirs and highlights the usefulness of coupling TMDs to a PhC with a better exciton-electric field overlap.\nOne last issue concerns the effect of the excitonic reservoir and its role in determining the measured nonlinearities. Note that, as suggested by reviewers 3&4, we have performed new measurements in a configuration where we also pumped resonantly (narrow-band pumping conditions) 2s polaritons, so as to determine “ideally” 2s-2s polaritons interactions without exciting 1s polaritons.\nThe first aspect to note is that when using what we call in the main manuscript narrow-band (NB) pumping conditions, we are exciting only polaritons at either side of the exciton energy, i.e., LP and UP polaritons, in the same manner as J. Gu et al. (Nat. Comm. vol. 12, 2269 (2021)) and similar to P. Stepanov et al. (Phys. Rev. Lett. 126, 167401 (2021)), and L. Zhang et al. (Nature 591, 61 (2021)). In these two last reports, the spectral width of the probe is much narrower than ours but still large enough to probe LPB and UPB and, thus, larger enough to pump across the excitonic resonance, as in our NB configuration.\nOne concern, already mentioned by reviewers 3&4, is whether the Y-polarized bare exciton (let us recall that Y-polarized excitons are in the weak coupling regime) will affect the measured nonlinearity of the X-polarized polaritons. As discussed in comment 13 of reviewer 3, we have modified our theoretical model to account for this contribution and experimentally verified that the Y-polarized bare excitons indeed contribute to the nonlinearity of X-polarized exciton-polaritons. Therefore, we have added comments on such anisotropic effect when we discussed the weak coupling of the exciton with the Y-polarized cavity mode and revised the expressions of the energy shift of each polariton for 1s and 2s excitons in the main text (equations 1-4), as well as the corresponding calculations of the nonlinearity (Figure 4) considering the contribution of the uncoupled Y-polarized exciton.\nRevision:\nLine 218, page 10, main text: added “considering an unpolarized pump laser that generates both Y-polarized excitons and X-polarized poalritons (Supplementary Note 3).”\nLine 219, page 10, main text: revised as “\nΔE_2s-LP^sat = g_2s-2s^sat * sqrt(X_2s * C_2s) * (X_2s * N_2s^X + 2n_2s^Y) + g_1s-2s^sat * sqrt(X_2s * C_2s) * (X_1s * N_1s^X + 2n_1s^Y) (1)\nΔE_2s-UP^sat = g_2s-2s^sat * sqrt(X_2s * C_2s) * (C_2s * N_2s^X + 2n_2s^Y) + g_1s-2s^sat * sqrt(X_2s * C_2s) * (X_1s * N_1s^X + 2n_1s^Y) (2)\nΔE_1s-LP^sat = g_1s-1s^sat * sqrt(X_1s * C_1s) * (X_1s * N_1s^X + 2n_1s^Y) + g_1s-2s^sat * sqrt(X_1s * C_1s) * (X_2s * N_2s^X + 2n_2s^Y) (3)\nΔE_1s-UP^sat = g_1s-1s^sat * sqrt(X_1s * C_1s) * (C_1s * N_1s^X + 2n_1s^Y) + g_1s-2s^sat * sqrt(X_1s * C_1s) * (X_2s * N_2s^X + 2n_2s^Y) (4)”\nLine 228, page 10, main text: added “N_1s/2s^X is the population of 1s/2s polaritons with X polarization and n_1s/2s^Y is the population of 1s/2s excitons with Y polarization”.\nLine 343, page 15, main text: added “tau_pol is set as 48/50.7 fs for the 1s/2s polaritons”.\nLine 343, page 15, main text: added “As we utilize unpolarized light (with a polarization degree 20% along x direction), Y-polarized excitons (in the weak coupling regime) will also be generated (Fig. S31 and S32). According to equations 1-4, we included the effect of Y-polarized excitons, considering the excitonic lifetime and the bare excitonic absorption, with measured excitonic linewidths of 6.5 and 7.9 meV for 2s and 1s excitons, respectively, at 4 K.”\nSupplementary materials: added Fig. S31 and S32 to present Y-polarization reflectance spectra and the nonlinearity calculation under YX and XX configurations.\nSupplementary materials: added Supplementary Note 3 to calculate the phase space filling energy shifts terms for X and Y polarized excitons.\nComment 4:\nIn conclusion, while the experimental results reported in this work, as well as the agreement with numerical simulations based on the nonlinear finite-difference time-domain method, seem to be well conceived and reported, the overall novelty and main claims of enhanced figure of merits as compared to the literature are not sufficiently justified or groundbreaking to be published in Nature Communications. There are certain elements of novelty that deserve publication of these results, but I would still consider them as incremental and, as such, suited to a more specialized journal in condensed matter physics, not necessarily belonging to the Springer Nature group. As it stands, I would feel very reluctant to reconsider this work for publication here, in the absence of a truly game changing result based on these materials.\nResponse:\nWe firstly thank the reviewer for agreeing that our experimental results and numerical simulations seem to be well conceived and reported, which encouraged us to give a more solid and clear explanation of our work's originality (and provide quantitative comparisons between our figures of merit and those published previously). Apart from the response to comment 1, herein we would like to give a conclusion about the main revisions of our manuscript:\n1, Thanks to the suggestions of reviewers, we have precisely analyzed the Fano features of our reflectance spectra and provided, thanks to these accurate fittings, the Rabi splittings of 1s and 2s polaritons as a function of temperature (devices 1&2), establishing clearly the range of existence of 2s polaritons. All the detailed fitting results are presented either in the main text or in the supplementary information.\n2, We fabricated a completely new device (device 2) showing the same Rabi splittings and comparable nonlinearity enhancements, proving further the robustness of the fabrication process and the reliability of the experimental results obtained in device 1.\n3, Based on reviewers 3&4 and 5 suggestions on considering the anisotropy of the X-polarized and Y-polarized cavity modes, as well as the nonlinear interaction between X-polarized polaritons and Y-polarized excitons, we have extended the theoretical treatment of the nonlinear interactions, i.e., equations 1-4 in the main text, and exploited them to assess quantitatively the nonlinearity of 1s-1s, 2s-2s, and 1s-2s interactions (Figure 4). Such a strategy provides a deeper comprehension of polaritonic nonlinearity in an anisotropy cavity and provides, for the first time, interaction constants between different excitonic species (i.e., the 1s-2s interaction constants).\n4, Herein, the abstract is revised as: “Rydberg excitons, exhibiting large Bohr radii, hold significant promise for quantum optoelectronic devices exploiting their strong optical nonlinearities, both at the excitonic and polaritonic levels. Conventional polaritonic devices consist of optical microcavities made by vertically stacking distributed Bragg reflectors (DBR) and/or metallic mirrors together with excitonic active regions. This geometry may restrain their applicability in photonic integrated technologies. Here, we demonstrate the first on-chip photonic crystal (PhC) Rydberg polaritonic states formed by the strong-coupling of both ground state (1s) and excited state (2s) excitons in monolayer WS2 with PhC modes. The PhC Rydberg (2s) polaritons survive up to 80 K, and their robustness is enhanced by the application of a magnetic field. Most importantly, they exhibit a nonlinearity 8.0 ± 2.3 times larger than that of the ground polaritonic states. This enhancement results mostly from the extended exciton radius of the 2s excitons compared to the 1s excitons. Additionally, the fabricated PhC induces a 15-20% enhancement in the polaritonic nonlinearity due to a stronger spatially-localized electric field distribution compared to the dual-DBR geometry. Furthermore, the coexistence of excited 2s and ground 1s polaritonic states enables us to experimentally quantify their inter-interaction constant (14.4 ± 6.8 eV m2) for the first time. The first realization of on-chip Rydberg polaritons represents a solid step towards polaritonic-based integrated circuits.”\nResponse to Reviewers (NCOMMS-25-13902A)\nReply to Reviewer 1\nGeneral Comment:\nI appreciate the authors' careful and detailed response to my questions and those of the other reviewers. All of my concerns have been addressed thoroughly, and the revised manuscript is significantly improved. I recommend that the paper be published in its current form.\nResponse:\nWe sincerely thank the referee for the favorable evaluation of our work and for providing, in the first review round, numerous constructive comments that have improved our manuscript.\nReply to Reviewer 2\nGeneral Comment:\nThe authors have thoroughly addressed all previous concerns and significantly strengthened the manuscript. The revised version is now suitable for publication and I recommend acceptance without further revision.\nResponse:\nWe sincerely thank the referee for the favorable evaluation of our work and for providing numerous constructive comments that have improved our manuscript.\nReply to Reviewer 3\nGeneral Comment:\nThe revised manuscript by Shang et al. is a significantly improved version compared to the one previously submitted. The authors have now provided a tighter and more meticulous data analysis with a better estimate of the reported non-linearities. The additional experiments with Device 2 have greatly complemented the findings reported. The authors have also provided satisfactory and detailed responses to all comments by the different reviewers. I recommend publication of the manuscript in Nature Communications.\nWe sincerely thank the referee for the favorable evaluation of our work, for recognizing the interest of having a second device, and for providing constructive comments and suggestions that have improved our manuscript. In particular, the reviewer’s previous suggestion concerning the fitting methodology of the raw data (as Fano resonances) has enabled us to perform a reliable analysis of the numerous measurements under different temperature, magnetic-field, and detuning conditions. The ensemble of data/analysis on two different devices has certainly rendered our claims much more solid.\nComment 1:\nI have one comment with the present manuscript. While I appreciate the use of the Fano function for fitting the excitonic/polaritonic and the cavity modes, the rationale behind the absence of the Fano asymmetries in the case of the 2s exciton (and the bare cavity mode around the 2s exciton energy) has not been addressed in the manuscript. Why does the interference effect from the background continuum of FP cavity modes wash out for the 2s excitons?\nResponse:\nWe agree with the referee that both the 2s exciton and the photonic-crystal mode that couples to it display low q values and, thus, their spectral form approaches that of a pure Lorentzian. As the reviewer points out, the Fano profile for the exciton is due to the interference effect between the exciton emission in the TMD material and the low quality factor Fabry-Perot resonance arising from the SiN/SiO /Si substrate stack (Phys. Rev. Lett. 120, 037402 (2018)), and similarly for the photonic-crystal mode. From the simulation of the dispersions in Fig. 2e in the main text, we can notice that the broad (both spectrally and angularly) Fabry-Perot modes overlap with the 1s excitons but much less with the 2s excitons. The simulated integral overlaps amount to 57% and 42% for the 1s and 2s excitons, respectively. This reduced overlap with the low quality factor vertical Fabry-Perot modes could explain the obtained q values, which amount to less than 0.1 for 2s excitons (and the photonic crystal mode at the 2s exciton energy). Currently, we are explicitly investigating, both numerically and experimentally, how the overlap modifies the reflectivity profile. Preliminary results indicate that when the exciton resonance is coupled to low-Q Fabry-Perot modes, its exact spectral form is determined by the combination of the actual overlap and the relative phases of the mode at the exciton resonance. This is illustrated in the Figure below, where we have included a large number of excitonic resonances at different energies in order to monitor the spectral form under different phase/overlap conditions.\nBased on the reviewer’s suggestion, we have added a comment on the effect of the overlap, which is smaller for the 2s exciton, but we are not yet ready to ascertain the actual role of the relative phases.\nRevision:\nLine 211, page 9: add “Herein, the smaller q values for the 2s exciton and the photonic mode coupled to it may be due to the reduced overlap between the background broad F-P mode and the exciton and the photonic modes at the 2s exciton energy, respectively.”\nReply to Reviewer 4\nGeneral Comment:\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.\nResponse:\nWe sincerely thank the referee for the favorable evaluation of our work and for providing constructive comments to improve our manuscript.\nReply to Reviewer 5\nGeneral Comment:\nI have considered the Authors' answer to my previous concerns, and I think I had overlooked the potential novelty of this work. In particulare, I now understand that the key message is related to the detailed analysis on the effects of nonlinearity in Rydberg excitons when strongly coupled to 1D grating modes. As the authors' themselves state, the reported estimates of polariton nonlinearity in their system is not the largest reported, as compared to existing literature. So, the main novelty is related to the fact that this is 'the largest reported among PhC-based polaritons'. This per se would not be enough, in my opinion, to justify publication in Nat. Comm. However, the authors also point out the suggested evidence that a local increase of electric field intensity in the PhC intersitials actually increases the nonlinearity by 15-20%. The latter, in addition to the reported co-existence of 1s and 2s polaritons and the quantification of their nonlinear behavior, makes a compelling case for this work to be accepted for publication, in my opinion.\nResponse:\nWe sincerely thank the referees (4 and 5) for their favorable evaluation of our work and for reconsidering their initial evaluation after we provided a more reliable and solid analysis of the initial data, together with the measurement of a second device that provided results consistent with the first one. We wish to express our deep gratitude and respect for reviewing our manuscript."}]

Decision letter