Multimode-polariton superradiance via Floquet engineering

1 - As the authors point out, this work adopts a similar idea to ref. [21] for creating Floquet polaritons, and features a multimode superradiant phase which has been already measured in other systems [28-31]. What is the novelty of the platform being put forward? Does it present a technical advantage over the Floquet system in ref. [21] or does it open a new pathway in exploring different physical models? Can this platform generate stronger mode-mode interactions or offer more flexibility in the number of modes you can couple?

The two platforms differ fundamentally in the mechanism underlying polariton formation and their interactions. In ref.[21], cavity photons couple to an internal electronic excitation, while in our platform they dispersively couple to the motion of the center of mass of the atom. The polarizability (or susceptibility) of the medium, which defines the properties of the polaritons, is thus very different in the two cases. In our manuscript, we explicitly discuss the linear susceptibility $\chi$ in eq.(7), which is proportional to the density response function of the ultracold atomic cloud. The latter has in general a frequency and momentum dependence which is different from the one shown by the polarizability of an internal electronic excitation. This difference is not restricted to the linear susceptibility, but also to the nonlinear ones. In particular, in the case of ref.[21], the nonlinear susceptibility is dominated by the strong repulsive Rydberg interactions between excited atoms, which have no counterpart in our case. As a net effect, we can for instance expect the stationary and non-stationary superradiant regimes we discuss in section IV to behave very differently, if present at all, in the case of ref.[21].

Furthermore, the discussed driving scheme does give a lot of tuneability. For a specific linear spacing of modes, the coupling of each mode to the atom cloud is determined by the amplitude of the sidebands, which for the single EOM case is determined by the modulation depth. Using a single EOM comes with a trade-off between frequency and modulation depth. Thus, If the modes are not too far apart, one can use a large modulation depth and thereby couple many modes. However, as mentioned in the paper, it is experimentally more efficient to use a multi-frequency drive like in reference [39]. Here they generate a flat frequency comb with a spacing of 250MHz ,which spans the interval between -9GHz up to 9GHz thereby creating 72 sidebands. If a carrier frequency is chosen such that there are supported modes both above and below it, one can make all 73 sidebands couple to a cavity mode that is close to each specific sideband. The result is an equal superposition of 73 modes. The spacing of 250MHz is comparable to the spacing between modes in the Hamburg experiment [23], which is 600MHz. These numbers are meant to give a flavour of the tuneability of the propsal, as the ideal numbers would depend a lot on the specific nature of the cavity modes.

We have added a more thorough comparison with ref. [21] to the introduction.

2 - In Fig.5 and 6, the authors show that the modes are coupled from the observed avoided crossings. It would be informative to quantify how strong these atom-mediated multimode interactions are with respect to the polariton lifetimes. This would make the overall story clearer, validating if the system is in the mean-field or many-body regime.

The total coupling strengths between the modes is given by the pump strength. How this is distributed between the modes is then determined by the modulation depth. Furthermore, each mode can have different detunings which changes the effective coupling between the modes. As a net effect, one can dramatically change the mode coupling by adjusting any one of these parameters.

To take a concrete example consider the few mode case (Fig. 4). To extract effective coupling between two modes one finds the distance between the two peaks in the spectral function, at an avoided crossing, and divides it by two.
For the first avoided crossing at $(\Lambda/\Lambda_c)^2=0.24$, in Fig.4 (b), the effective coupling strength between the $LG_{00}$ and the $LG_{01}$ mode is $g_{LG_{00},LG_{01}}\approx 3.4\kappa$.

Throughout the manuscript we assume that all modes are equally lossy which means that the linear combinations of these also have loss rate $\kappa$ making it a sensible scale. The second avoided crossing involves the avoided crossing between the composite mode, consisting of $LG_{00}$ and $LG_{01}$, and the $LG_{02}$ mode.
As the energy of the composite mode without coupling to the $LG_{02}$ is unknown, the exact position of the avoided crossing is also unknown. This problem can be overcome by investigating the spectral function for the $LG_{02}$ mode ($A_{22}$). In this spectral function only two peaks appear (one for the composite mode and the one for $LG_{02}$) and the effective coupling is proportional to the distance between the two peaks when they have equal height.
This avoided crossing is found at $(\Lambda/\Lambda_c)^2=0.55$ and the effective coupling strength is $g_{LG_{00,01},LG_{02}}=1.27\kappa$. The magnitude of the effective couplings has a strong dependence on the sidebands weight but the exact dependence is non-trivial to extract, as it has to be disentangled from the effects arising from the continuously changing composite nature of the modes. These effects are essential to include as can be seen from the overlaps at p3 in Fig.5

The effective couplings is a quadratic interaction between two different cavity fields and is therefore exactly treatable on a mean-field level. Higher-order processes can lead to effective cavity-cavity interactions which are beyond mean-field. The presented calculations are in a regime where the cavity mediated atom-atom interactions are long range. Therefore, beyond mean-field interaction processes are suppressed, as shown in ref. [45]. In the presented calculation the different coupled cavity modes have an almost identical overlap with the atom cloud and the scaling of the interaction processes simplifies further to scale with the inverse atom number. For experimental realization the number of atoms are usually around $10^4$ to $10^6$ which justifies neglecting these interaction processes.

The discussion of the effective coupling strength has been included in the manuscript in the discussion of Fig.4~(b) in section 3.B.

3 - The authors justify the multimode superradiant phase from the renormalization of the critical atom-photon coupling and the appearance of instabilities at finite frequencies. Have the authors explored synchronization effects where the collective coupling leads to coherent emission from an ensemble of transverse modes? 4 - Have the authors considered studying this ordered phase by probing photon correlations g2 between different polariton modes?

The ordered phase is indeed very interesting but studying it with the same formalism requires a significant extension due to the broken symmetry of the phase. We are therefore currently investigating the ordered phase through a saddle-point approximation of the action (leading to a mean-field equation). Our ongoing investigations actually hint towards the system having stable ordered phases where the composite cavity field oscillates at different frequencies. An essential part of our investigations is to include the fluctuations induced by the cavity loss, and here g2 might be an insightful probe. As these investigations have a very different focus compared to the current manuscript, we will dedicate a separate manuscript to them. We added this point more clearly to the outlook.

5 - Can the authors explain what they mean by ‘deep multimode regime’ (last paragraph of section I), and how the numbers they get for the coupling strengths and interactions apply to this criteria? Are they referring to the fact that the atom-photon coupling is larger than the damping rate for multiple modes? or is the coupling larger than the free spectral range?

We see now that the wording can be misleading and thank the referee for bringing it to out attention. The phrasing 'deep multimode regime' is meant to refer to the fact that a number of modes much larger than 2 is important in the system. After the referees comment we have come to the conclusion that in the specific paragraph the use of 'deep multimode regime' does not add additional clarity and therefore changed it to 'multimode regime'.

6 - The red crosses in Fig.4 are hardly visible due to the colormap, the authors should consider changing the cross colors

We have changed the cross colors as suggested.