Multimode-polariton superradiance via Floquet engineering

The authors have addressed my concerns and have made the appropriate changes in the manuscript. I would also like to echo the second reviewer's criticism regarding the mathematical formulation. It is important to make the manuscript more accessible for the readers interested in following these calculations. I am recommending publication in Scipost, but I highly encourage the authors to present their theoretical formulation with more rigor before submitting the final version of the manuscript.

- Interacting multimode polaritons are highly topical subject
- The paper is well organized and very descriptive
- Creative approach how to realize interacting multimode polaritons using Floquet engineering
- Gives experimental estimates

- The mathematical formulation should be more stringent

In this paper the authors consider a cloud of ultracold atoms within a near-planar Fabry-Perot resonator where interactions among the photons are mediated by the atomic excitations. The special point of this proposal is an additional implementation of a pump laser field with a phase modulation which is also periodic in time. The driving frequency of the pump laser is far detuned with respect to the atomic transition. The phase modulation frequency is comparable to the energy difference between the transverse electromagnetic modes of the cavity which gives room to transfer energy from the cavity modes to the Floquet modes. The resulting scattering processes lead to a polarization of the medium and in particular to non-diagonal matrix elements between initial and final cavity modes. These are the interacting polaritons. In the revised version the authors describe how their proposal diverges from pervious ones.
The general subject of interacting multimode polaritons is very topical. I find the paper well organized and very descriptive. The experimentally relevant quantities and scales are discussed thoroughly. However, in my opinion the mathematical formulation of the calculation does not reach the same standard. For readers who orientate themselves along the equations there are hurdles from typos and undefined symbols or symbols which are only introduced much later in the text. I also strongly recommend to give more details on the derivation of the most important equations or to quote appropriate references. I will list some examples below but I believe the strong focus on descriptive text with much less rigor in the correct theoretical formulation is present throughout the text. Therefore, before recommending this manuscript for publication in Scipost I would ask to improve on that point. In the following, I will make my criticism more precise in form of explicit examples:

1) In Fig. 1) there are misleading notations. What is \omega_p? Is the \lambda correct in the inset? The figure is discussed in the text in chapter II A while the depth B_m is only introduced in chapter II C. This makes the text and figure confusing.
2) Below Eq. (3) the argument of the spatial mode function should be a vector
3) Eq. 4: alpha is not clarified.
4) How is Q defined in the recoil energy?
5) Give more details on the derivation of Eqs. (7) and (8) from Eq. (6)
6) Below Eq. (8) it should read \bar{c}_\alpha c_\beta
7) Below Eq. (11) there should be a semicolon separating the indices in the Nambu matrix (Minor typo)
8) The indices in Eq. (12) do not seem to be correct (and should be comma separated as done before).
9) Above Eq. (14): are the indices of the delta functions correct? They do not agree with Eq. (11).
10) Fig 4): The labeling LG_00 on the left top of the figures is confusing as other modes appear in the discussion. Distinguish more clearly between cavity modes and atomic wave modes. The spectral function with only two indices has not been defined before.
11) Eq. (14): the same symbol has been used before with a different meaning.
12) Fig. 5 inset: I is not defined.
13) How are the effective coupling strengths defined? This discussion appears handwaving with no precise theoretical background.
14) It should also be stated in the main text that the laser form factor is taken to be constant.
15) Eq. (A2) the index n^\prime does not seem to be correct.

In conclusion, I am missing a stringent mathematical formulation but if the authors make their concrete calculations more accessible to the interested readers by improving in that direction I believe the paper would be quite substantial and I would recommend publication in Scipost.