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Cavityinduced bifurcation in classical rate theory
by Kalle Sulo Ukko Kansanen, Tero Tapio Heikkilä
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Submission summary
Authors (as registered SciPost users):  Kalle Kansanen 
Submission information  

Preprint Link:  https://arxiv.org/abs/2202.12182v4 (pdf) 
Code repository:  https://gitlab.jyu.fi/jyucmt/prt2022 
Date submitted:  20230919 14:03 
Submitted by:  Kansanen, Kalle 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We show how coupling an ensemble of bistable systems to a common cavity field affects the collective stochastic behavior of this ensemble. In particular, the cavity provides an effective interaction between the systems, and parametrically modifies the transition rates between the metastable states. We predict that the cavity induces a collective phase transition at a critical temperature which depends linearly on the number of systems. It shows up as a spontaneous symmetry breaking where the stationary states of the bistable system bifurcate. We observe that the transition rates slow down independently of the phase transition, but the rate modification vanishes for alternating signs of the systemcavity couplings, corresponding to a disordered ensemble of dipoles. Our results are of particular relevance in polaritonic chemistry where the presence of a cavity has been suggested to affect chemical reactions.
List of changes
 Throughout the manuscript, we have updated the mathematical notation so that the states are described with variables s_i obtaining values +1 or 1, and it is easier to see the quantities that depend on these states.
 Throughout the manuscript, we have removed the tautological word 'collective' in connection with 'phase transition'. Similarly, the word 'polaritonic' is removed in several places in order to avoid confusion, and in general the connections between our work and polaritonics should now be more clear.
 The lightmatter coupling model and its assumptions are further discussed below Eq. (4), including the exclusion of direct dipoledipole interaction.
 The meaning of the transition dipole moment in the context of our manuscript is elaborated in Sec. 3.
 Equation (7) expresses the approximation used in the main text.
 The consistency of our approach to thermodynamics is elaborated at the end of Sec. 3.
 The beginning of Sec. 5 presents more clearly the connection between the rate theory approach and the Dicke model.
 Two references added, Refs. [20] and [43], shifting reference numbers.
Current status:
Reports on this Submission
Report
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.
Requested changes
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 qdependent 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)*(qq0)). This is used implicitly in the paper, but not wellexplained and should be clarified.
Report
The authors provided a satisfactory response to my main technical question regarding the omission of the dipole selfenergy 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 twostate 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.
Anonymous Report 1 on 2023103 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2202.12182v4, delivered 20231003, doi: 10.21468/SciPost.Report.7887
Report
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 Dickelike 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 DebyeWaller 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 (BornOppenheimer 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 modelindependent 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 offdiagonal 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 VanderWaals/CasimirPolder interaction because the latter is determined by the offdiagonal 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 offdiagonal 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 dipoledipole interaction would be weaker than the cavityinduced one. Usually it is the opposite, unless the cavity mode is at resonance with a molecular transition. But the effect considered here is nonresonant. 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 nontrivial 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.
Author: Kalle Kansanen on 20231211 [id 4181]
(in reply to Report 1 on 20231003)
We thank the referee for their critical comments which have helped us to clarify the presentation of the manuscript. We recognize the issue of setting up the discussion in a concise and understandable way, choosing the strategy of explaining a much simpler textbook case in Sec. 2 and expanding from that. We have further clarified the manuscript on this starting point which we had taken too much for granted. The physics of the manuscript remains the same, with the model limitations we have discussed also with the other referees. We then respond to the comments in detail.
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. [...]
Our results are valid when thermal activation over the potential barrier dominates over the quantum tunneling rate, which is the proper limit either in the limit of a large mass or a high internal dissipation rate (large linewidth)  as the referee also correctly guessed. One of us has also studied the direct polaritonic effect on the quantum tunneling rate (Phys. Rev. B 107 (3), 035405 (2023)). The conclusion there is that the polaritonic effects on the overall tunneling rate vanish in the large$N$ limit.
The referee's confusion seems to be related with our claim that the cavity effect can still be a quantum effect. Namely, the interaction between the dipoles mediated by the cavity can be obtained also from a quantum treatment.
In other words, we assume that the transition rate (times $\hbar$) is a negligible energy compared to the intrawell energy scales. We have clarified this further below Eq. (2) in the manuscript. On the other hand, the cavity induced coupling between the molecules is nonnegligible compared to the intrawell energy scales in such a way that the parameter alpha defined below Eq. (10) can be larger than unity. In this case the presence of the cavity does not break the assumptions behind the thermal activation model.
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 modelindependent terms.
We believe that the refereen's confusion between our stochastic model and the phase transition in the Dicke model originates from the same misunderstanding of the hierarchy of energy scales. Nevertheless, there is certainly a connection to the Dicke model and its phase transition after some reinterpretation of our starting point. This is our reasoning for writing Sec. 5.
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. [...]
In the molecular systems which we have in mind, the function $d(q)$ is indeed a diagonal matrix element in terms of the dipole operator and the electronic ground state which in turn depends parametrically on the classical $q$ (e.g., distance of two nuclei). As mentioned in Report 3, it is the variation of $d(q)$ with $q$ that gives the vibrational transition moment as nondiagonal matrix elements of the vibrational states. That is, in fully quantum mechanical description, $q$ is promoted to an operator. One certainly can derive this in any gauge they wish but it is the most straightforward to grasp in the dipole gauge.
I see no reason why the Coulomb dipoledipole interaction would be weaker than the cavityinduced one. Usually it is the opposite, unless the cavity mode is at resonance with a molecular transition. But the effect considered here is nonresonant. 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.
We see the situation as follows. In our simple model, we provide a prediction for the bifurcation parameter $\alpha$ below Eq. (10) using Eq. (7) for the parameter $P$. Had we done the complete microscopic calculation that includes the Coulomb dipoledipole interaction, the relation between $\alpha$ and the microscopic parameters would change in a way that reflects the assumptions done in this model (say, the molecule positions within the cavity matter, and also their dipole moment directions not only with respect to each other, but also with respect to the main cavity axis). Nevertheless, the presence of the bifurcation transition would not change. The main point in this work is to illustrate this bifurcation transition and not the details of the dipoledipole interaction.
[...] 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.
The intrawell relaxation (i.e., the internal quality factor) in superconducting circuits can be controlled by the presence of extra shunts. This is the standard way to remove hysteresis from Josephson junction circuits. In effective circuit models such as the RCSJ model (see "The Physics of Nanoelectronics" by T.T. Heikkilä, Oxford University Press 2013, Ch. 9), this thus means adding an extra resistor element to the circuit. Hence getting to "our" limit of low quality factor is straightforward; much easier than getting to the "usual" flux qubit limit mentioned by the referee.
To give one concrete example, Gang Li et al. in Chin. Phys. B 27, 068501 (2018) describe measurements on a much more complicated 'flux qubit' (operated out of the typical flux qubit regime) than discussed here. Their Eq. (1) is equivalent with our Eq. (46) (note that our definitions of $\phi$ differ by a factor of $2\pi$) but they may tune the constants with external fields. They report reaching $b \approx 0.3$ and $f = 0$, putting our works in the same regime. Furthermore, they explicitly measure the thermal activation rates and detailed balance relation.
I admit that Eq.(14) is a valid nontrivial 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.
Does the referee claim that the curves corresponding to different distributions of the dipolar coupling $g$ in Fig. 3 are all the same? To us they look different because they are not all overlapping. This difference comes from the different variances of $g$, thereby illustrating it exactly.
It is true that the figures of our manuscript do not effectively visualize the change in the thermalization time $t_{th}$, defined for instance from $[N_L(t_{th})  N_L(0)]/[N_L(\infty)  N_L(0)] = e^{1}$. Instead, we focus on the initial rates and the steady state values, both of which are obviously experimentally relevant. As an approximation, $t_{th} \approx 1/\Gamma_{LR}$, showing that since the rate modifications are of the order of unity so are the modifications of the thermalization times. As far as our reading of the literature goes, all the claimed effects in polaritonic chemistry are also of the order of unity.
Author: Kalle Kansanen on 20231211 [id 4182]
(in reply to Report 3 on 20231113)We thank for the further clarification on the dipole moment function. We have also added the mentioned linear approximation to the main text, as it was previously only given in the Appendix A in a parametrized form [Eq. (20)].