Nontrivial band geometry in an optically active system

Optical activity, also called circular birefringence, is known for two hundred years, but its applications for topological photonics remain unexplored. Unlike the Faraday effect, the optical activity provokes rotation of the linear polarization of light without magnetic effects, thus preserving the time-reversal symmetry. In this work, we report a direct measurement of the Berry curvature and quantum metric of the photonic modes of a planar cavity, containing a birefringent organic microcrystal (perylene) and exhibiting emergent optical activity. This experiment, performed at room temperature and at visible wavelength, establishes the potential of organic materials for implementing non-magnetic and low-cost topological photonic devices.

The manuscript is well written and deserves publication once the comments and questions below are addressed.
1) The optical activity is described as a function of kx and it is further discussed at the end of the Method section. However, it is not clear why there is no dependence on ky. This fact should be properly commented and addressed.
2) In the introduction, the authors discuss the quantum anomalous Hall effect (QAHE) in photonics, where a Faraday effect (thus a magnetic field) is required. Immediately after, they move their discussion to the optical activity. The current presentation seems to suggest that optical activity is another way to obtain QAHE, which is not correct since it does not breaks time reversal symmetry (indeed the total Chern number of the Hamiltonian Eq.4 is zero). The authors should make more clear this distinction to the reader. Moreover, I would suggest to cite some relevant papers on the QAHE, as for example -The authors mention a non-abelian gauge field in the abstract and in the introduction related to their results. However, no other reference to non-abelian gauge fields is made in the rest of the manuscript. To me it appears that all the results are for an abelian description instead. The author should clarify and eventually remove this claim.
-In Eq.5, the authors represent the "magnetic field in momentum space" that is typically used to represent 2x2 hamiltonians. I would suggest the authors to write the Hamiltonian as H = \Omega . \sigma in order to explain the notation, namely the meaning of \Omega, which is currently only explained in words.
-It would be useful to plot the energy dispersion corresponding to the spectrum of hamiltonian in eq.4, assuming that the presented results concern a single band of the two.
-In Fig. 3D and 3E, the experimental results show a systematic squeezing of Berry curvature and quantum metric compared to the theory plot. Can the authors explain the origin of this phenomenon. Is some term missing in their modeling? -Concerning the Hamiltonian in Eq.4, the authors could also comment on possible topological transitions of this model or change in the topological features (e.g. the moving/merging of the Dirac points). Which parameters can be tuned in order to see any of these properties? Is it feasible? -At pag.9 there should be a typo -> "red \beta" -Since the goal of this work is the reconstruction of topological and geometrical band properties (as the curvature or the metric), the authors should dedicate a part of the introduction to a description of the accomplishment already obtained in this direction in cold atoms and photonics, and they could take the opportunity to call the attention on what is different in their system and methodology compared to other methods or platforms. I provide below a list of some experimental and theoretical papers that the authors should cite, but the authors could also add other relevant ones. In this document, we provide a point-by-point response to all comments of all reviewers. The changes to the manuscript are marked in red.

Measurement of topological invariants in
Reviewer #1 (Remarks to the Author): Reviewer writes: In the manuscript "Nontrivial band geometry in an optically active system" the authors measured dispersion relation of perylene crystal embedded in a metallic-mirror microcavity. They observed an anti-crossing of photonic modes and interpreted these results as the presence of a giant optical activity (OA) in perylene. They proposed an effective Hamiltonian describing observed phenomena using some fitting parameters found from the experiment.

Authors reply:
We thank the reviewer for providing an interpretation of our results, which clearly demonstrates that the reviewer has paid a particular attention to some aspects of our multidisciplinary work while overlooking others, which actually appear more central for us. Our manuscript is entitled "Nontrivial band geometry in an optically active system", and the central result of the manuscript is the measurement of this nontrivial geometry.
We would like to stress that this central result remains valid whatever the origin of optical activity of the system as a whole, and the presence of the latter is an undeniable experimental fact, evidenced by the circular polarization of the eigenmodes (as seen in Fig. 3(c), for example). However, we have excluded all possible origins of this activity, except the natural optical activity of the excitons in perylene, as we discuss in the manuscript and in our reply to the remarks below.

Reviewer writes:
Although these results are interesting for the perspective of topological optics and the interest of scientific community in the field of spin-orbit interaction of light this paper needs to be improved since the main hypothesis -the role of OA (circular birefringence) -wasn't proved and all results could be explained in terms of birefringence.

Authors reply:
We thank the reviewer for finding our results interesting for topological optics and spin-orbit interaction of light. We also thank the referee for pointing several alternative explanations, which for us were excluded from the start, but which are important to explicitly comment in the manuscript, which we do in the revised version.
Reviewer writes: 1. Origin of giant OA of perylene.
The hypothesis of a giant OA due to the presence of perylene "excimers" should be clarified and justified, because such a giant optical activity was never observed in a pure perylene. Especially, OA of the order of 14000 deg/mm is typical for cholesteric liquid crystals (or "chiral stacks of 2D materials" pg. 20) but not for nonchiral symmetric molecules! The authors claim that the symmetry in perylene crystal is broken due to breaking of glide plane, which allows the OA. This effect potentially can lead to optical rotation, however such perturbation would result in an effect at least 2-3 orders of magnitudes weaker than reported in the manuscript. Moreover the authors cite Ref. 33 "Long-lying excited states in crystalline perylene" [Rangel et al. (2018) PNAS], where Ranguel and co-workers showed that in the same crystallographic structure the direction of dipole moments (excitons) was oriented along the long molecular axis [Ref 33,Fig. 1D], and not perpendicularly to perylene "excimers" as proposed in the manuscript.
Authors reply: First, we never used the word "excimers", contrary to the statement of the reviewer, we were using the word "excitons". We have double checked the previously submitted file. Second, we do explain in the manuscript why the optical activity was never observed in a pure perylene: it is only present close to the exciton resonance and it is impossible to detect because of the linear birefringence. On the contrary, in a microcavity where this linear birefringence is compensated by the cavity TE-TM splitting, it is possible to observe a circular polarization of the modes due to the optical activity. Third, Fig. 1 As to the amplitude of the effect, once the glide plane is broken, the system effectively becomes a chiral stack of 2D layers, with the exciton polarization changing between the layers, and therefore, contrary to the estimate of the referee, the effect can potentially be as large as in the stacks. We have added this comment to the corresponding part of the Methods section. It reads: Indeed, once the glide plane is broken, the system effectively becomes a chiral stack of 2D layers, except that what is changing between the layers is not the orientation of the lattice vectors, but the orientation of the exciton polarization. Therefore the effect can potentially be as large as in the stacks.

Reviewer writes:
I understand that the role of the optical activity is the crucial point of this manuscript: it provides an approach different than a birefringence in https: //arxiv.org/abs/1912.09684 by A. Fieramosca, G. Malpuech, D. Solnyshkov et al. But I found no arguments that OA of such magnitude can be present in perylene crystal.

Authors reply:
Indeed, the optical activity, which manifests itself via the circular polarization of the eigenmodes at the anticrossing is a crucial difference with the arxiv by A. Fieramosca et al (where there is no anticrossing) and with Ref. 24 of the manuscript, where the anticrossing is produced by an applied magnetic field, breaking the TR symmetry.
Our arguments in favor of its presence in perylene crystal are given in the manuscript, they are also listed above. We also exclude all other explanations for the circular polarization of the optical modes, including the versions suggested by the referee (see below).

Interpretation of experimental data as the optical activity instead as the birefringence.
Alpha-perylene crystalizes in an "alpha" monoclinic phase of p21/c symmetry [Pick A, et al. (2015) "Polymorph-selective preparation and structural characterizationof perylene single crystals" Cryst Growth Des 15: 5495-5504. (a= 10.24 A,b= 10.79 A,c= 11.13 A, alpha=90 deg, beta= 100.92 deg, and gamma=90 deg)] and thus it is birefringent (probably it would be even biaxial). In supplementary materials to the publication of Pick (2015) there is Fig. S1 where the crystal structure of alpha-perylene is drawn. One can notice that a* axis is tilted by 11 deg from the direction perpendicular to the (bc)plane of perylene crystal. This suggests that also the optical axis of perylene crystal investigated in the manuscript wasn't perpendicular to the cavity plane. Such small tilt of the optical axis is sufficient to form "gapped Dirac cones" from "diabolic points" discussed in a similar paper https://arxiv.org/abs/1912.09684 by A. Fieramosca, G. Malpuech, D. Solnyshkov et al. for a birefringent medium. This tilt can also explain asymmetries of Stokes parameters in Fig. 3.

The authors should carefully exclude the role of birefringence before claiming the leading role of optical activity. Small tilt of optical axis is sufficient to mix linearly polarized optical TE-TM modes.
Authors reply: We agree that a small tilt of the optical axis can mix the TE-TM optical modes for a given direction, and that if without the tilt that direction would correspond to a crossing of the branches, one would instead observe the anticrossing for this particular direction. However, we respectfully disagree that this could explain our experimental data. First of all, this tilt cannot explain the observation of any non-zero circular polarization degree (except in the very particular configuration of the point 3 of the referee, see below): the modes of the birefringent crystal are linear polarized, the TE-TM cavity modes are also linear polarized, and the modes of the global structure remain linear polarized, whatever the tilt of the optical axis. For a given direction in the reciprocal space, the tilt of the optical axis affects the modes of the structure and changes their polarization, for example, from horizontal to diagonal, but not to circular. As to the suggestion that this tilt can explain the observation of the "gapped Dirac cones": the tilt of the optical axis can explain that for a given direction the TE-TM and the birefringence do not compensate each other anymore. But there are necessarily two directions (different from the case with zero tilt) where the TE-TM and the linear birefringence do compensate each other. In other words, each diabolical point simply shifts in the reciprocal space. This is what the referee seems to suggest. However, since we performed the tomography of the whole 2D reciprocal space, we have actually obtained the dispersion of the modes (and their polarization, as shown in Fig. 3) in all directions of the reciprocal space, and not only for a single cross-section. These complete experimental results confirm that the anticrossing of the modes actually takes place. We comment on this in the text: We note that Fig. 2(e) corresponds to the direction with the smallest gap, k x. We have mapped the whole reciprocal space and we can therefore exclude the possibility that the observed anticrossing is simply due to a tilt of an optical axis which could shift the crossing point away.

Reviewer writes:
"The fact that the OA is linked with the exciton resonance is confirmed by the inversion of the sign of the effect in Fig. 1

(B) of the main text" (pg. 19) can also be interpreted in terms of tilted optical axis.
Authors reply: The change of the sign in Fig 1(b) is simply a confirmation that the effect is linked with the exciton resonance. As to the tilted optical axis, this suggestion does not explain the anticrossing of the modes and their circular polarization, and therefore is not sufficient to describe our data.
It was recently shown that photonic modes of different parities in a microcavity filled with birefringent medium can "interact" leading to "anticrossing" of eigenstates (Rechinska et al. "Photonic Engineering of Spin-Orbit Synthetic Hamiltonians in Liquid Crystal" Science (2019), 366, 727-730). According to this paper (suplementary inf.) the sign of TE-TM splitting depends on the difference of ordinaryextraordinary refractive indexes values (uniaxial, biaxial etc.). In Fig. 2E the authors marked E0 ("light mass" with minimum at 550nm) and E1 ("Heavy mass", 525 nm) eigenstates, which suggests that they have the same party. However they can also mark E0 as "heavy" 600nm and E1 as "light" 550 nm, thus this pair can also have the same parity. In this manuscript I couldn't find arguments about the order of this splitting: is "light" above or below "heavy" eigenstate?

Due to the large TE-TM splitting in thick birefringent microcavity subsequent odd and even eigenstates can overlap at high k-vectors leading to "anticrossing" reported in the manuscript, which origin could be similar to what was observed by Rechinska et al. The authors should verify also this hypothesis. Again -small tilt of the optical axis mentioned in point 2 should favor anticrossing of modes.
Authors reply: We thank the referee for this very good remark. Indeed, it is important to prove that our data cannot be interpreted as an anticrossing of the modes of different parity. This is particularly difficult for the sample studied in the manuscript, since, as correctly noted by the referee, the choice of the polarization doublets of the same mode number seems arbitrary.
However, the measurements performed on a thicker cavity, where a higher number of modes are observed at the same time (including two entire anticrossing doublets), confirm that these are the modes of the polarization doublet of the same mode number which exhibit the anticrossing. The figure shown below is now included in the manuscript as a Supplemental Figure 5.
The text added in the manuscript (in the Methods section) reads: A very particular configuration is that of the crossing of the modes of opposite polarization and different parity [61]. In this case, chirality occurs not on the scale of individual molecules, but at the scale of the whole cavity. To exclude this interpretation of our data, we provide the results of additional measurements performed on a different sample, shown in Fig. S5. This sample is thicker, and its dispersion exhibits several pairs of modes of opposite polarization, at least three of which are visible in the figure. We mark two pairs of modes by their mode numbers n and n+1 and their polarization H and V (at k=0).
The polarization splitting (black arrows) at k=0 between the modes supposed to be of the same number (En,H-En,V) is smaller than in the sample studied in the main text, and the fact that the doublets are formed by the modes of the same number n or n+1 is confirmed by the fact that the inverse effective mass of the modes changes by the same factor as the polarization splitting (En+1,H-En+1,V)/(En,H-En,V)≈2, when approaching the exciton resonance. Indeed, the strong light-matter interaction in vicinity of the exciton resonance leads to the mixing of the excitonic and photonic modes. Closer to the resonance, the excitonic fraction of the modes increases, which increases their effective mass (excitons have a very large mass with respect to photons) and at the same time decreases the polarization splitting at k=0 (excitons exhibit no splitting, whereas photonic modes are split due to linear birefringence). The strong interaction of the excitonic and photonic resonances is also confirmed by the deviation of the observed modes from parabolicity visible in the figure, especially in panel a). Indeed, the observed dispersions exhibit a higher effective mass at higher wave vectors, whereas for purely photonic modes in a cavity the opposite would be expected.
The relation between the effective masses and the polarization splittings does not hold any more if one assumes that the modes are grouped differently. If the modes for which the anticrossing is observed were of the different parity (as in Ref. [61]), then the polarization splitting of the modes of the same parity would be given by the red arrows. Both of these correspond to approximately 200 meV without any notable difference. It is completely impossible for the polarization splitting to remain constant in spite of the strong overall change of the refractive index, as manifested by the change of the effective mass between the bands and along them. This hypothesis can therefore be rejected. We conclude that the optical activity that we observe does not arise from the anticrossing of the modes of different parity.
Reviewer writes: Fig. 3. It would be useful for a reader to see all the data measured for Fig. 2DE and Fig. 3

. (even as Supplementary Figures). What were the polarizations of the eigenstates observed in transmission measurement and how did they fit to the theoretical model? For instance: does the polarization of measured dispersion resemble supplementary Fig S1 AB? How the polarization of the transmission through perylene looks in kx-ky plane in the vicinity of "gapped Dirac cones"? Theoretical calculations should be compared with measurements of all Stokes vector components.
In my opinion such comparison would help to exclude "OA approach" from "birefringence approach" or "odd-even approach".

Authors reply:
There is probably some misunderstanding. The results of the measurements are presented in Fig.  3(a,b,c), which show precisely the polarization of the eigenstates observed in the measurements. And indeed, they do resemble the supplementary Figure S1 (a,b), which can be seen from the color in Fig.  3(c). The polarization of the reflectivity (not the transmission, as the reviewer mentioned) in the whole kx-ky plane, including the vicinity of gapped Dirac cones, is again precisely what is shown in Fig 3(a,b,c). The caption of this figure reads "Measured Stokes parameters of the mode E0".
It is this measurement which allows to exclude the birefringence interpretation, whereas the odd-even hypothesis is excluded by the analysis of the alternative sample, also exhibiting the mode anticrossing.
We are sorry that the reviewer did not understand that the data she/he requested were already shown in the manuscript. We now explain the whole measurement and treatment procedure better in the Methods section, together with providing more experimental data. We also provide an explicit comparison of all 3 measured pseudospin components with the theoretical calculations. Because of this, we had to split the Figure 3 of the main text of the previous version into 2 figures: Fig. 3 and Fig.  4. Moreover, we have replaced the theoretical figure S1 by an experimental one, in order to stress the qualitative features of the behavior of the eigenstates at the anticrossing points and at the same time provide more experimental data.

The description of the two figures has been modified accordingly:
The validity of the effective Hamiltonian is confirmed by the measured 2D wave vector maps of the Stokes vector components of the lower branch, shown in Fig. 3(a-c) compared with theoretical predictions shown in panels (d-f). We note that the experimentally measured Stokes components are zero outside an elliptic region where the detection is efficient. Inside this region (marked with a white dashed line), the experiment and the theory exhibit a good agreement.

Abelian gauge field
The authors claim the presence of non-Abelian gauge field. However TE-TM splitting and one linear term in momentum doesn't give an additional gauge field, see Shelykh (2018), Phys. Rev. B 98, 155428 "Optical analog of Rashba spin-orbit interaction in asymmetric polariton waveguides". The gauge field derived by authors is Abelian.

Authors reply:
We respectfully disagree with the statement of the referee: we do not claim that TE-TM together with a linear term give a gauge field. Our claim (based on Ref. 47 and several works in other systems, now cited in the text as new Refs. 44,45) is that the TE-TM plus a constant term give rise to a gauge field in the vicinity of the crossing points (at non-zero k). Then, the optical activity close to these points can be considered as a constant Zeeman splitting (of the opposite signs for the two cones), which can be also incorporated into the non-Abelian Yang-Mills Hamiltonian (see Ref. 45). Our claim of the non-Abelian gauge field is therefore correct.
The added text reads: In vicinity of the points k x=±k0, where the linear birefringence of the material is compensated by the TE-TM splitting, the Hamiltonian (4) can be written as a Rashba Hamiltonian with a constant Zeeman splitting, which was shown to be equivalent to a Hamiltonian of a nonrelativistic quantum particle coupled to a non-Abelian Yang-Mills field [44,45].
However, in the outlook of the paper we consider the limiting case of small wave vectors (not in the vicinity of the anticrossing, but in the vicinity of k=0) of a fictitious system with zero birefringence. In this case, the system indeed cannot be considered as a non-Abelian gauge field (and we never claimed it), but it is rather an Abelian electromagnetic-like Hamiltonian with opposite magnetic fields for opposite spins (which is what we state). We now point this aspect more clearly in the text, when discussing Eq. (6). The new text reads: We note that this field is Abelian, in spite of being spin-dependent.
Reviewer writes: (Fig 1 DE), of different range (Fig 1 DE), etc. There should be somewhere definition of axes (x,y,z)

in laboratory frame and corresponding axis directions of screw (if exist), birefringence, polarization of light etc.
Authors reply: We have modified all figures according to the comments of the reviewer and to the standards of Nature Communications: all figures are now plotted with the same axes, showing wave vector and energy, which is the most qualitatively clear representation. As to the panels (d,e) (the reviewer is probably speaking of Fig. 2, not of Fig. 1) which had different scales, we note, however, that the two panels represent two different samples, and thus are not intended to be directly comparable. The panel d with the empty microcavity is simply provided to show the dispersion with the only ingredient being the TE-TM splitting.

Although this paper presents an interesting phenomena in my opinion the hypothesis of the leading role of the optical activity hasn't been proven yet. The theory is interesting, but the authors should find another system for investigations, where a large optical rotation will be present without doubt.
Authors reply: The central point of our work is the extraction of the quantum geometry of an optically active system, and this is accomplished beyond any doubt, since the optical activity of the whole system (the circular polarization of the modes) is explicitly measured experimentally and has not been doubted even by the reviewer. We hope that with the new arguments provided in the reply and in the text, the reviewer will agree that since no other explanations are possible, the origin of the optical activity that we suggest must be the correct one.

Reviewer writes:
Therefore I don't recommend this article for publication in Nature Communications without serious changes in the text, which practically requires rewriting this article or doing the experiment again using chiral system.

Authors reply:
We hope that now that we have pointed out the misunderstandings of the reviewer (concerning the presence of experimental results in the manuscript) and provided more details, experimental data, and theoretical arguments, the reviewer will recommend our manuscript for publication in Nature Communications.

The manuscript presents the reconstruction of the Berry curvature and of the quantum metric in an optically active material embedded in a microcavity. The strategy exploits a combination of effects on the polarization of light (TE-TM splitting, linear and circular birefringence), which allows the authors to model the response of system with a 2x2 model for the two light polarizations that displays topological features as a function of the incident wavevector.
Authors reply: We thank the reviewer for correctly summarizing our results.

Reviewer writes:
The manuscript is well written and deserves publication once the comments and questions below are addressed.
Authors reply: We thank the reviewer for the positive appreciation. We have replied all comments and questions and corrected the manuscript accordingly. See details below.

1) The optical activity is described as a function of kx and it is further discussed at the end of the Method section. However, it is not clear why there is no dependence on ky. This fact should be properly commented and addressed.
Authors reply: We thank the reviewer for this remark. For symmetry reasons, the optical activity must change sign at k=0 (see Fig 1B, top part). Therefore, there is necessarily a line in the reciprocal space along which the corresponding term is zero. Because of the symmetry of the crystal structure, this line also has to be parallel to one of the two axes of the linear birefringence. This is why it is zero along k y. The new text added in the Methods section in order to comment on this reads: For symmetry reasons, the optical activity must change sign at k=0 (see Fig 1(b), top part). Therefore, there is necessarily a line in the reciprocal space, passing through k=0, along which the OA is zero. Because of the symmetry of the crystal structure, this line also has to be parallel to one of the two axes of the linear birefringence. This is why the leading OA term is linear in k_x and there is no dependence on k_y. This term can be deduced as follows.

Reviewer writes:
2) In the introduction, the authors discuss the quantum anomalous Hall effect (QAHE) in photonics, where a Faraday effect (thus a magnetic field) is required. Immediately after, they move their discussion to the optical activity. The current presentation seems to suggest that optical activity is another way to obtain QAHE, which is not correct since it does not breaks time reversal symmetry (indeed the total Chern number of the Hamiltonian Eq.4 is zero). The authors should make more clear this distinction to the reader. Moreover, I would suggest to cite some relevant papers on the QAHE, as for example Authors reply: We completely agree that the optical activity does not allow to obtain QAHE, and we now stress this clearly in the text (page 4). We have added both references suggested by the referee to the manuscript.
The new text reads: Of course, QAHE is not limited to photonics: it has been originally proposed [12] and recently demonstrated in electronics [13,14] and atomic lattices [15]. Authors reply: The reviewer is completely right, the polarization, and therefore the Berry curvature, of the second band is opposite with respect to the first one. We now discuss the selection of the eigenstates more in details in the Methods section. We provide more experimental data as well, as requested also by Reviewer 1. In particular, we have replaced the theoretical Supplemental figure 1 (showing dispersion and pseudospin) by an experimental figure, where, for one direction in the reciprocal space passing through the anticrossing point, the extracted dispersion is clearly visible, together with the orientation of the extracted pseudospin, which is indeed opposite for the two bands. The new text reads: We consider a reflectivity spectrum measured under white light excitation for total intensity, such as shown in Fig. S7 (black circles). We first determine the wavelength λ0 and the energy E0 corresponding to the particular mode, by fitting the total reflection spectrum with Lorentzian-broadened resonances over an approximately linear background (red solid line). We then fit the individual intensity components to determine the relative weight of resonance (taking into account the magnitude and the width of the peak) in each of the 6 polarizations (blue and cyan triangles in Fig. S7 for experimental circular polarization and red dashed and dash-dotted lines for theory), which allows finally to determine the 3 components of the Stokes vector. In our example, we show only two polarization projections of the six (to avoid overloading the figure). We note that the positions of the reflectivity minima detected in two polarizations under a non-polarized excitation do not necessarily correspond to the positions of the modes: their position depends on the linewidth and the polarization degree, and the maximal deviation can be of the order of the linewidth. As an example, the reflectance in the two circular polarizations is given by:

Reviewer writes:
-The authors mention a non-abelian gauge field in the abstract and in the introduction related to their results. However, no other reference to non-abelian gauge fields is made in the rest of the manuscript.
To me it appears that all the results are for an abelian description instead. The author should clarify and eventually remove this claim.

Authors reply:
We now discuss this point better, adding several references, as requested also by Reviewer 1 (see above for the inserted text appearing in pages 10 and 17). The essential idea is that the Hamiltonian (4) in vicinitiy of an anticrossing or a crossing can be considered as a Hamiltonian of a massive particle in a non-Abelian Yang-Mills field. This is not the central point of the manuscript, but it is worth being mentioned, since it is one of the first observations of such configurations in photonics.

Reviewer writes:
-In Eq.5, the authors represent the "magnetic field in momentum space" that is typically used to represent 2x2 hamiltonians. I would suggest the authors to write the Hamiltonian as H = \Omega . \sigma in order to explain the notation, namely the meaning of \Omega, which is currently only explained in words.

Authors reply:
We thank the reviewer for this remark. We have now provided an explicit reformulation of the Hamiltonian according to the advice of the reviewer. The new text reads: it is a linear combination of Pauli matrices that can be physically interpreted as an effective magnetic field acting on the Stokes vector, that is, Hk=Ωσ, where σ is a vector of Pauli matrices and Ω is the effective field

Reviewer writes:
-It would be useful to plot the energy dispersion corresponding to the spectrum of hamiltonian in eq.4, assuming that the presented results concern a single band of the two.

Authors reply:
This dispersion is shown in Fig 2(e) by dashed lines. We now stress it more clearly in the text (this was only written in the figure caption). The figure has been replotted as E(k), so that it really corresponds directly to the Hamiltonian (4).

Reviewer writes:
-In Fig. 3D and 3E, the experimental results show a systematic squeezing of Berry curvature and quantum metric compared to the theory plot. Can the authors explain the origin of this phenomenon. Is some term missing in their modeling?
Authors reply: We thank the reviewer for this interesting remark. The reason of this squeezing might be the nonparabolicity of the dispersion in vicinity of the exciton resonance, which is indeed neglected in the simples 2x2 Hamiltonian, but also the precision of the measurements. We now comment on this in the text.
The new text reads: The distortion of the maxima of the Berry curvature in experiment might be explained by the non-parabolicity of the dispersion in vicinity of the exciton resonance and by the difference of the experimental resolution in the two directions.

Reviewer writes:
-Concerning the Hamiltonian in Eq.4, the authors could also comment on possible topological transitions of this model or change in the topological features (e.g. the moving/merging of the Dirac points). Which parameters can be tuned in order to see any of these properties? Is it feasible?
Authors reply: We thank the reviewer for this interesting remark! Although no topological transitions in the strict sense are possible because the Chern number of each band always remains zero, the band geometry can indeed be strongly modified by tuning the effective Hamiltonian parameters. We now discuss it better in the text. The new text reads: Tuning the coefficients of the Hamiltonian by choosing different materials would allow to deeply modify the band geometry. In Ref.
[24], a system described by a similar effective Hamiltonian was studied, except that the OA was replaced by an effective Zeeman splitting. The bands were showing two split Dirac cones, like in the present work, but with the same sign for the Berry curvature in a given band. By tuning the linear birefringence β 0 to zero in Eq. (4), the dispersion bands would exhibit a crossing at k=0, but the reciprocal space nevertheless remains split into two "valleys" of opposite Berry curvature because of the OA. Instead of being concentrated at the anticrossing points, the Berry curvature exhibits a crescent shape in this case. If both OA and birefringence are set to zero, which the case of an empty cavity shown in Fig. 2(d), the dispersion represents two touching parabola, which are characterized by two Berry monopoles of opposite charges at k=0.

Reviewer writes:
-At pag.9 there should be a typo -> "red \beta" Authors reply: We thank the reviewer! Typo corrected.

Reviewer writes:
-Since the goal of this work is the reconstruction of topological and geometrical band properties (as the curvature or the metric), the authors should dedicate a part of the introduction to a description of the accomplishment already obtained in this direction in cold atoms and photonics, and they could take the opportunity to call the attention on what is different in their system and methodology compared to other methods or platforms. I provide below a list of some experimental and theoretical papers that the authors should cite, but the authors could also add other relevant ones.
Authors reply: We thank the reviewer for these relevant references. We have extended the discussion, adding all of these references, but indeed also some other ones, which have appeared just recently (such as Ref. 60). The new discussion reads: Quantum geometry is currently a subject of active studies. It is studied both globally, at the level of topological invariants, and locally, as a distribution of the Berry curvature and quantum metric in a certain parameter space. Topological invariants with their discrete integer values and associated global effects, such as the presence of edge states determined via the bulk-edge correspondence, are easier to be measured experimentally [50][51][52][53]. The local distribution of the Berry curvature and, later, the quantum metric, have usually been measured via the related dynamical effects [54-60], such as the anomalous Hall drift. Recently, the quantum geometry has been extracted from the eigenstates of a photonic system, like in the present work, with an additional confirmation of the results by the anomalous Hall measurements [24].
We thank both reviewers once again for their constructive criticism, which has allowed us to improve our work. We hope that the reviewers will accept the revised version of the manuscript for publication in Nature Communications.