Bilateral photon emission from a vibrating mirror and multiphoton entanglement generation

1-Rich physics described in the manuscript
2-Analytical results well documented
3-Numerical analysis supports results well
4-References are appropriate and comprehensive

1-Suggestion of innovative "undriven" entanglement scheme unclear or incomplete
2-Abstract highlights entanglement but article does not quantify entanglement at all and only briefly discusses "quantum correlations"
3-Discussion of Figure 2 imposes "master-equation-like behaviour" without a concise explanation or comparison
4-Discussion does not cover how accurately the resonance conditions need to be met or said differently how much disorder with regards to the system parameters is permitted without losing the effects.
5-Suggestion to experimental realisation very speculative

The manuscript "Bilateral photon emission from a vibrating mirror and multiphoton entanglement generation" presents a theoretical investigation of the physics of two radiation modes that are coupled in a nonlinear fashion by a moving mirror with perfect reflectivity.

The derivation of the Hamiltonian is covered elsewhere (Ref [38]) in a thorough and plausible manner. The present manuscript covers the resulting physics from the effective Hamiltonian description depending on the system parameters. There exist multiple distinct resonance conditions which enable multiphoton hopping and supposedly lead to 2n-photon entanglement.

While the dynamics of "lowest energy closed dynamics" are considered it is nowhere mentioned how the initial states are supposedly initialised.
It is also said that "path entangled microwave radiation was observed from strongly driven microwave resonators" while "[h]ere, instead, [the authors] propose a scheme to generate 2n-photon entanglement [...] already in an undriven setup". How exactly would the required states be initialised without a drive?

Setting aside this issue and assuming that the considered states can be initialised, it is nowhere quantified which parts of the systems are entangled and which parts are not. More concretely, the discussion of Figure 2 contrasts the behaviour of a quantum jump "whenever one photon is detected in one of the cavities, or when a phonon is detected in the mirror". It is said for the latter case, "when the cavity c jumps, the number of photons in the cavity a immediately goes to zero, as a clear signature of the quantum correlations exhibited in the entangled state $|{\psi^{(2e)}_{i}}$>". On the other hand it is said when "the first jump occurs with a phonon loss, [the system is locked] to the state $|0,1,0>$ " without mentioning any relation to quantum correlations even though the photons in both cavities immediately go to zero. I think the phrasing should be clear when entanglement is present, between which parties it exists, and if it is bipartite or multipartite entanglement, quantified with an appropriate measure (negativity, log negativity, ...)

Moreover, the discussion of Figure 2 says that "Fig. 2(c) shows the master-equation-like behaviour that arises taking the average over 1000 trajectories". It is known that the Monte Carlo wave function (MCWF) method is equivalent to a Markovian master equation. Such a statement creates the urge for me to know the concrete equivalent master equation that matches the results in Figure 2c. Moreover, I would love to know a bit about the convergence behaviour of the MCWF method. I suggest to add an Appendix on the numerical analysis which covers the equivalent master equation and compares the averages over different reasonable numbers of trajectories (maybe 10, 100, 1000 or something more appropriate) and the master equation as the limit the averages should approach.

Another concern that I have with the analysis is that it does not indicate any tolerance for the resonance conditions to be met. Is it possible to see the effects if there is some disorder in the frequencies of the oscillators? If so, what determines the maximal disorder that is allowable?

The final concern is that the manuscript claims "[i]n principle, the effects predicted in this work could be experimentally observed using circuit optomechanical systems" while clarifying that there exist "current limitations of the experimental feasibility". The indicated sources to my understanding have not even shown the realisation of the Hamiltonian under investigation, as usually quadratic optomechanical coupling couples to the square of the position and not to the square of the field amplitude, let alone the required frequency conditions. I agree with the fellow referee and would like to ask the authors to clarify throughout the manuscript, especially in the introduction, that experiments with optomechanical systems have not yet achieved the required regime and that it is more likely to reproduce the physics presented with electronic analogues.

1-At least some convincing comments regarding the state preparation without a drive or a clarification of what is meant by the "undriven setup" or ideally an appendix that outlines the state preparation procedure.
2-At least a clarification of which parts in the system are entangled and carefully phrasing the corresponding statements or ideally adding an evaluation of appropriate entanglement measures of the states
3(a)-A concrete mention of the equivalent Markovian master equation
3(b)-An appendix with an analysis of the convergence of the MCWF method (optional but highly appreciated)
4-A discussion of the precision required for the resonance conditions to observe the presented physics
5-A more thorough discussion of the experimental feasibility of the presented physics, ideally in a dedicated section.

Ask for major revision

1- The paper is presented very clearly.
2- The physics presented is rich and intriguing
3- The numerical corroboration of the analytical results is solid and convincing.

1- The phenomenon presented cannot be implemented easily in an experiment

The authors present an extension of a previous work (Ref 38), where they make use of the dynamical Casimir effect to effectively couple the radiation modes between two modes that are coupled by a moving mirror with perfect reflectivity.

The derivation of the Hamiltonian in Ref 38 is solid, and so did their first presentation of two photon hopping. In this work, the authors explore further resonance conditions and show how 2n photon entanglement occurs, and how the Janus emission (terminology introduced ad hoc) arises.

I enjoyed very much reading their work, I found the language fluent and precise, the mathematical derivations rigorous, the intuition clear, and the numerical corroborations convincing. Overall, I do not have any technical request or questions, except the following small point: why do you define the effective annihilation operator of the dressed mode as $\chi^+$? This seems counter-intuitive, but maybe it is related to some other consideration that gives the meaning to the + and - parts...

Also, I would like to point out that--to my knowledge--there is no experimental implementation of the dynamical Casimir effect to date. To do so, one would need to observe a mechanical displacement close to the speed of light, and the community is not there yet. Instead, the observed dynamical Casimir effects to date are in circuits whose electronic properties mimic those of the mechanical problem. It is also in this kind of settings that we can hope to observe the theoretical predictions of the current manuscript in the short-term future.

-- I would like to ask for a small inset in Fig. 3(a) where it is shown a bit more clearly how the lines oscillate in the intermediate part. Right now, it is not straightforward to see this without the help from the description in the text.

-- I would also like to ask the authors to make more clear in the introduction that the experiments that are more likely to reproduce the physics presented are electronic analogues rather that an optomechanical system.

Publish (easily meets expectations and criteria for this Journal; among top 50%)