Cavity-induced bifurcation in classical rate theory

In our view, the broader physics interest from our work is the rate theory construction of interacting systems (mediated by a cavity) and how collective effects arise in that framework. We hope that our response to the limitations of this approach could affect your consideration.

Only a single cavity mode is considered. This can capture a significant part of the physics of excited states resonantly coupling to that mode, and thus is widely used to describe polariton properties and dynamics. However, the effects discussed in the manuscript do not depend on resonance behavior as far as I understood, and all modes are expected to contribute. Restricting the model to just a single cavity mode is thus a severe approximation.

In principle, taking into account many modes phenomenologically does not really change the picture. This is probably best exemplified by the section 5 where we explicitly take into account infinitely many modes of the cavity. While we do not discuss this, it is clear from Eq. (16) that we change the single-mode result to a sum over all the modes. The same would happen in the rate theory approach. But, based on the second point,

The molecular dipoles are assumed to be aligned to the electric field of the cavity. In typical experiments, they are disordered and point in arbitrary directions. While for a single-mode description, non-perfect orientation can be included by adjusting the coupling strengths, when all modes are taken into account, the orientation of the dipoles becomes much more important.

it seems to us that one would hope for a full calculation starting from Maxwell's equations to determine all the constants that are currently merely phenomenological in terms of physical parameters of the cavity. Doing this and cutting the infinite sums is something we have thought of, but it is a difficult task we could not finish for this work. While this result would be fantastic, our intuition is that it doesn't change the rate theory approach or the qualitative results (e.g., disorder in the dipole moments removes the cavity effect on the initial rates). We note that the issue of self-interaction terms in the QED description we have discussed with another referee is relevant for such a calculation.

In response to this comment, we have added the text "Moreover, the model containing only a single cavity mode is chosen for simplicity. Generalizing the results to the case of several cavity modes is relatively straightforward and does not lead to qualitative changes in the results." in the paragraph below Eq. (4).

We appreciate the thorough reading of our manuscript.

The question of equilibrium phase transitions in QED is something we stumbled upon, not knowing its fifty year history beforehand. And we agree, it is an important and interesting question. Moreover, as discussed in our response to referee 1, the superradiant transition in the Dicke model is not the same transition as what we found within the transition state theory.

While the statement

By now, it seems more or less well established that the phase transition does not exists in the context of that model (as discussed, e.g., in Ref. 54 of the manuscript). Although the authors attempt to explain away this contradiction with their results, the arguments used (see Sec. 5 of the manuscript) clearly do not make sense, because the ''strong assumptions'' underlying the proofs of the absence of the phase transition are the same assumptions used by the authors in this work.

makes it sound as if the issue is fully resolved, the more we looked into it, the more nuanced and difficult it appears. Indeed, Ref. 54 claims that there can be no phase transitions, yet Ref. 55 from the same authors published only a year after Ref. 54 says that there is a phase transition. (Note that these reference numbers point to the manuscript before resubmission, v3 in arXiv.) The relaxed assumption in that case was that the EM field varies over the molecules. This means that, in Ref. 54, the molecules are effectively in the same location within a cavity - clearly not a valid assumption in most experiments. Therefore, our reading of the literature so far is that QED predicts equilibrium phase transitions.

An added difficulty is that the dipole gauge is explicitly derived from the Coulomb gauge assuming this "long wavelength" -limit where the position of molecules is fully neglected. This means that the EM vector field is also position independent. When one relaxes this assumption, the EM vector field depends on position, and moreover, does not commute with itself at different positions. This makes the unitary transformation to the dipole gauge very complicated, so much so that we have not seen it in the literature. As a consequence, comparing results in different gauges is not straightforward.

There is also a minor problem with the text of the manuscript that should be corrected before possible publication. The term ''collective phase transition'' used in the manuscript is very confusing, because it implies that there are also phase transitions that are not ''collective''. There are none. All phase transitions are collective by their very nature, and the term should be changed to simply ''phase transition''.

The use of the word collective is indeed tautological. Our main hope was to differentiate our work from those, e.g., in the ultrastrong coupling regime. Judging from the comment, it was not a successful choice and has been removed everywhere. We renamed Sec. 4 as "Cavity-induced phase transition and collective dynamics" to better reflect the content of Sec. 4.3.

We thank for the numerous good suggestions regarding the presentation of the manuscript. The overview touches upon all the important parts of the manuscript except the description of how the initial relaxation or reaction rate is modified. While there are some points to be clarified, we agree upon this overview until the discussion around the classicality of the cavity mode. That part of the report mostly concerns the way we have presented the theory, not so much the actual results. On the other hand, the interpretations of the manuscript that follow we do not share.

Our initial goal was to understand what kind of polaritonic chemistry there can be when the typical chemistry tools such as transition-state theory suggest that no effects should be observed in large ensembles of weakly coupled molecules. We admit that we have probably misused the term "polaritonic" in this regard, as the word polariton refers to the hybrid mode between the cavity field and the molecules, and our result, while starting from the same setting as the usual polaritonic chemistry papers, does not rely on the avoided crossing between the cavity and molecular resonances. The main idea was that perhaps one has to be careful in constructing the rate equations themselves, which is why our manuscript is framed in that language. This seemed to us like a general physics question of wider interest. We found interesting, for instance, that there can be a phase transition which does not affect the reaction rates. After establishing the overall picture in the rate equations, we noticed a similarity to the phase transition in the Dicke model discussed in Sec. 5. However, unlike many of the reviewers seem to assume, these phase transitions are not the same. In particular, the phase transition in the reaction rate theory has an additional parameter compared to the Dicke model, namely the (effective) height of the reaction barrier.

We have now modified the text to clarify the precise connections to the polaritonics and to the superradiant transition in the Dicke model.

Next, we comment on specific parts of the report.

Each minimum is labeled by a set of N discrete Ising-like variables which the authors denote by sign(q_i), instead of introducing a more compact notation s_i = +1 or -1. The energies of the minima are given by Eq.(6) in which V_tot = V_tot(s_1,...,s_N), a function of N discrete variables. More careful notations could save some effort for the reader here.

We agree on the suggestion, and the sentiment of the following paragraph that some notations were left too implicit. The functional dependencies are now stated more clearly throughout the manuscript, especially sections 2 and 3 and appendices A, B and F.

Usually, the term "transition dipole moment" refers to an off-diagonal matrix element of the dipole moment between two stationary states.

This is to our understanding standard nomenclature in the dipole gauge, see for instance Eqs. (1) and (4) in this JCPL by Yang and Cao. Here, the relevant states are the vibrational states in the right and left well which are approximately like harmonic oscillators. Another way to think of the terminology is that if the dipole function $d_j(q_j)$ of Eq. (4) is a constant function, there is no light-matter coupling at all in the model.

[...] the self-consistent equation can be found from the thermal Gibbs distribution with the energy V_tot(s_1,..,s_N) from Eq.(6).

This is true, and an important point we left implicit. To clarify this, we have added a sentence "We note that our approach is consistent with thermodynamics by construction, and hence the stationary properties could also be found by treating the light-matter system as a canonical ensemble and solving its thermal distribution" before Sec. 4. However, as we not only want to discuss the stationary states but also the transients, we prefer to resort to the rate equation model.

I wonder, is it possible to proceed analogosly to Appendix C and derive a closed kinetic equation for the distribution function p(g), the probability for a TLS whose coupling g_i is in the interval g < g_i < g + dg to have s_i = -1? Such a kinetic equation might be tractable.

This is definitely a good idea. One can indeed derive a kinetic equation with the same approach. However, the problem is that one would need the time evolution of the order parameter $\Delta$, and in turn, the kinetic equation for $\Delta$ depends on the integral over the distribution function. The coupled equations do not close in any straightforward way. It is mentioned in the paragraph containing Eqs. (11) and (12).

In the authors' model the cavity mode is treated classically. Strictly speaking, this is valid only when the reorganization energy is much larger than the mode frequency (that is, the shifted ground state contains many photons). I did not see this condition checked anywhere in the paper. At the same time, in Sec. 5 the authors obtain a similar Ising energy functional V_tot from the quantum Dicke-like model. However, (i) this only works in the absence of the energy difference between the two states of each TLS, and (ii) transition rates will be dressed by Debye-Waller factors (overlaps between the shifted cavity ground states). So I still doubt that the presented calculation really applies to cavity polaritons.

We fail to see how exactly it is important that the cavity mode would be treated classically. Since the cavity's contribution to the energy is linear and we find the effective interaction it mediates, it would seem that we would end up to the same result with a quantum mechanical cavity mode. We do not understand what is meant here by ''reorganization energy'' (which is a concept used at least in charge transfer and open quantum systems) and hence the argued validity condition.

To clarify: the most important effect of the cavity is to mediate the interaction between the dipoles. To some extent, this can be viewed as the cavity field providing an effective quantum vacuum to the molecules. It is hence fully compliant with quantum mechanics. Whether the effect is quantum mechanical or classical is a semantic problem. One can compare this to the effective interaction between electrons mediated by phonons, forming the basis of the theory of conventional superconductivity.

We do not fully understand the second part of this comment, related with the Ising energy functional, energy differences between the states, and the Debye-Waller factors. First, the Ising functional is presented only as a reference point, as it does not represent the same physics as we found in the transition state theory. We also fail to see the relevance of Debye-Waller factors in our model, and do not know which shifts of the cavity ground states the referee talks about. In other words, it is unclear if the referee here is questioning the starting model (e.g., Eq. (4)), or the approach used to describe the system behavior. We note that we assume an ensemble of same molecules (i.e., the same microscopic parameters), but allow for randomness in their positions and orientations, along with the stochasticity of the individual transition states of each molecule. Is this setting not well-defined enough?

1. I do not see any polaritonic chemistry here. I only see the good old ferroelectric transition, the same as found in the equilibrium Dicke model. The case of TLSs not being identical has been treated in the literature [Phys. Rev. B 64, 235101 (2001); Nucl. Phys. B 919, 218 (2017)]. That case was not exactly the same as here, since here the authors include randomness in the dipole moments instead of randomness in TLS energies, but I do not see any qualitative difference.

Physically, the dipole moments play a fully different role than the eigenfrequency of the molecular or atomic mode. In the picture of cavity-mediated dipole-dipole interaction, the dipole moments determine if the effective interaction is attractive or repulsive in nature. Mere changes in eigenfrequency will not achieve this. We also point out that although seemingly analogous, the Dicke model transition is different than the transition in the rate equation model, because the latter has an extra parameter: the height $E_b$ of the potential barrier in the potential energy surface. As we point out in Sec. 5, the transition temperature in the rate equation model is $2 E_b/\omega_b$ times larger than in the (simple) Dicke model.

We admit that the connection between the two models (and the fact that they are not the same) was quite unclear in the manuscript, so we have clarified it in the beginning of Sec. 5.

2. I have strong doubts about the applicability of the authors' model to polaritonic systems. Besides the classical treatment of the cavity mode that I already mentioned, I see one more problem. If the TLSs couple to the cavity mode's electric field via their dipoles, they should also couple to each other directly via the Coulomb dipole-dipole interaction. Why is it not included?

As we already mention above, we disagree with the claim of disregarding some quantum effects related with the presence of the cavity.

The direct coupling via Coulomb dipole-dipole interaction is definitely possible. However, we assume that such coupling will be too weak to induce the bifurcation transition, whereas the cavity mediated coupling can be much stronger. Taking into account everything amounts to calculating the Casimir-Polder-van der Waals interaction between the molecules inside a cavity. The detailed form of this interaction will then depend on the precise locations of the molecules inside the cavity and with respect to each other, and their dipole orientation not only with respect to each other, but also to the cavity electric field, and on the type of the electromagnetic cavity. After specifying the system completely this calculation is in principle possible, but it is quite tedious and quite definitely outside the scope of the present work. We fear that its technical details would also obfuscate the big picture related with the bifurcation transition. Our model should therefore be considered as the relevant toy model indicating the possibility of the bifurcation dynamics, and the relevance of the orientation of the molecular dipoles.

We have now clarified these issues below Eq. (4).

3. The circuit QED implementation discussed in Appendix G.2. resembles very much the one proposed in Phys. Rev. Lett. 104, 023601 (2010). Also, superconducting circuits typically have low dissipation, so quantum Hamiltonian treatment should be more relevant for such systems than the classical dissipative model studied in the present paper. So we are back to the model applicability problem.

In fact, our motivation of writing Appendix G.2 stemmed from the need of understanding the physics of the sign of the dipole moment. In the case discussed in Appendix G.2 this sign is directly connected with the relative windings of the inductors mediating the coupling.

The model in the quoted reference indeed looks at first quite similar to the one we have in Appendix G.2, and is an interesting manifestation of a model quite analogous to the Dicke model. However, we are in a different regime of parameters: in that PRL, similar to the usual flux qubit variants (as in Ref. [63]), the relevant parameter of interest is the coupling energy between the two wells of the "flux potential" (this parameter is $\omega_F$ in that PRL). The resonance condition directly compares to this energy. In our case, we assume this energy to be quite small (in other words, the potential barrier between the wells is high), and the cavity frequency is compared only to the resonance frequencies of the two harmonic potentials inside the wells.

We disagree on the necessity of using a fully quantum mechanical treatment for any superconducting circuit. The (classical) transition rate theory is valid in the case where after each transition the system thermalizes to the local equilibrium state before a new transition takes place. In other words, when the transition rate is much smaller than the internal dissipation rate, the classical description is accurate. In the circuit model, the transition rates are effectively controlled by the Josephson energy, so we see no fundamental reason why one could not reach such a parameter regime. As is painfully known in the superconducting qubit community, even superconducting circuits exhibit finite dissipation (and thereby dephasing/relaxation).

4. The only part of the paper that looks to me somewhat novel with respect to the earlier literature is the study of the relaxation dynamics presented in Appendices E, F. But its results could be expected without any calculations. Yes, the system relaxes towards equilibrium. The relaxation time scale naturally matches the rates that were plugged into the model. So I do not see what one can learn from these results.

For us, the initial rate modification, Eq. (14), is not obvious but a result of some effort, so we would be delighted to know how it can be derived without any calculations. In polaritonics, at least, we often encounter a view where only the Rabi splitting should matter, as it determines the energy spectrum, so having the rate modification to be proportional to the average of the light-matter coupling constants is not something we understood before starting.