Cavity-induced bifurcation in classical rate theory

The authors have responded to all the points raised by the referees. While I agree that some of the issues that one of the other referees pointed out could be clarified, I believe that the paper is still more complete than many other works in the field and that it is worth publishing and appropriate for SciPost Physics. I therefore recommend publication.

1- The issue of the (transition) dipole moments mentioned by one of the other referees seems to be a confusion between the adiabatic "electronic" molecular dipole d(q), which is indeed the q-dependent diagonal expectation value, and the vibrational transition dipole between the (quantised) vibrational states within the adiabatic potential (which to lowest order is proportional to d'(q0), i.e., the spatial derivative of the adiabatic dipole, since d(q) = d(q0) + d'(q0)*(q-q0)). This is used implicitly in the paper, but not well-explained and should be clarified.

The authors provided a satisfactory response to my main technical question regarding the omission of the dipole self-energy terms in the starting equation (4) for potential energy.
In my view, the manuscript in its present form is suitable for publication. The fact that the ``Dicke model’’ has a phase transition in the regime when the dynamics of the two-state systems in the model is classical is new, and at least one of the suggested experimental realizations of this proposal based on the bistable dynamics of flux ``qubits’’ also looks technically sound and interesting.

I thank the authors for their clarifications, but unfortunately I am still not satisfied. I guess, my main problem is that Eqs.(3) and (4) are not sufficient for me to define the model and to decide on its applicability to various physical systems. I also need the kinetic energy and the associated hierarchy of energy scales. Since the kinetic energy is not mentioned at all in the model, my first guess was that all degrees of freedom are in the overdamped limit, the frequency is a small energy scale, the masses are heavy, and the wave function is well localized in the coordinates x,q_i at all times; this is what I meant by the classical treatment. Apparently, my guess was wrong, and the authors have in mind something else. If we adopt the fully quantum picture for all degrees of freedom, like in the Dicke-like model in Sec. 5, then transitions flipping sigma^i_z must be due to some perturbation proportional to sigma^i_x or sigma^i_y. In this case, since the unitary transformation shifts the a_alpha modes to different ground states for different sigma^i_z, the transition rate must involve the overlap between these shifted ground states (the Debye-Waller factor), which the authors do not consider. So my current guess about the authors' model is that the cavity mode has high frequency, so it remains in the instantaneous ground state which adiabatically follows the slow overdamped degrees of freedom q_i (Born-Oppenheimer approximation). In this case, the frequencies must be strongly different and one cannot speak about polaritons.

I still think that the transition found by the authors is physically indistinguishable from the Dicke phase transition. The authors seem to have agreed that it has nothing to do with the kinetics of the system, but is an equilibrium property. Essentially, it is an instability of the system "dipoles + cavity mode". One can use different models for dipoles and their coupling, they may be richer than the origianl Dicke model and have more parameters, but essentially the phenomenon is the same, if one tries to define the transition in model-independent terms.

Originally, I thought the issue of "transition dipole moment" to be a purely terminological one, but now I start to suspect that it is a manifestation of some deeper confusion. The term "transition matrix element" is a standard and general one in quantum mechanics, and it denotes an off-diagonal matrix element of some operator between two stationary states of the system, when this operator acts as a perturbation. It has nothing to do with the dipole gauge. In the authors' construction, d(q) is the diagonal matrix element, as far as I can understand. For this reason, to me the authors' construction seems different from Van-der-Waals/Casimir-Polder interaction because the latter is determined by the off-diagonal dipole matrix elements, rather than by the static (diagonal) dipoles. For the same reason, I do not see any relation between the dipoles introduced by the authors and the Rabi frequency, because the latter is determined by the off-diagonal dipole matrix elements. Also, if the frequency of the mode is strongly different from that of the q_i variables, one cannot speak about splitting.

I see no reason why the Coulomb dipole-dipole interaction would be weaker than the cavity-induced one. Usually it is the opposite, unless the cavity mode is at resonance with a molecular transition. But the effect considered here is non-resonant. If the main reason to neglect Coulomb is not to obfuscate the big picture, then let the authors study the big picture, but not claim its relevance to molecules in a cavity.

Concerning the potential realization in superconducting circuits, I agree with the general statement that the authors' construction is valid when the transition rate is much smaller than the internal dissipation rate, but my point is that in superconducting circuits it is usually the opposite, especially in qubits. So to claim applicability to superconducting circuits, the authors must check if realistic circuits with suitable hierarchy of scales really exist.

I admit that Eq.(14) is a valid non-trivial result. But I do not see much of its manifestation in the relaxation curves shown in Figs. 3,4, and 6. All these curves relax on the time scale ~ 1/Gamma_0.

To conclude, the paper still apears to be very vague to me, applicability of the presented results to real physical systems looks doubtful, so I cannot recommend it for publication.