Bilateral photon emission from a vibrating mirror and multiphoton entanglement generation

Referee's Comment 1

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.

Authors Reply

We thank the Referee for the detailed and correct description of the manuscript.

Referee's Comment 2

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?

Authors Reply

It is worth distinguishing between the system initialization and the state-conversion process (i.e., 2n-photon entangled state generation). Obviously, one needs a drive to initialize the system, otherwise we simply stay in the ground state. On the other hand, after the system is initialized, we can switch off the drive (or a pulse sequence), and leave the spontaneous generation of the entangled state due to the Casimir-Rabi oscillations[^6] derived in our manuscript.

As an example of the system initialization, in the case of Fig. (2) our the current manuscript, the $|0, 2, 0 \rangle$ state can be obtained by using several procedures (see for example Fig 3 of Ref. [^1], or other references [^2][^3][^4][^5]). Nonetheless, we thank the Referee for raising this question and have added a comment in the updated manuscript.

Referee's Comment 3

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 |ψ(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, ...)

Authors Reply

We take the opportunity to address this second concern raised by the Referee.

We have to distinguish between the 2n-photon entangled state $|\psi_+^{(2e)}\rangle = \frac{1}{\sqrt{2}} \left( |2, 0, 0\rangle + |0, 0, 2\rangle \right)$ (which is deterministically generated after half-period of the Rabi oscillation: $t = \frac{\pi}{2 g_\mathrm{eff}}$, where $g_\mathrm{eff} = 2\sqrt{2} g^2 / \omega_b$), and the entangled state $|\psi (t)\rangle = c_1 (t) |0, 2, 0\rangle + c_2(t) |\psi_+^{(2e)} \rangle$ which is governed by the Rabi oscillation itself. Although in our manuscript we mainly focus on the $|\psi_+^{(2e)}\rangle$ entangled state, we have a multipartite entangled state between the mirror and the two photonic fields. Nonetheless, we see the photonic correlations even during the Rabi oscillation.

Since the state $|\psi_+^{(2e)}\rangle$ has a straightforward analytical form, we decided not to explicitly quantify the entanglement entropy of this state. Indeed, it is straightforward to see that it is a non-factorizable pure state. This state belongs to the class of the NOON states (which are very important in quantum sensing and metrology), and their entanglement is very well quantified (see, e.g., Ref. [^7]).

Additionally, in response to Referee’s Comment 5 (see below), we have added an Appendix that presents the logarithmic negativity entanglement as a function of the subsystems’ detuning.

Referee's Comment 4

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.

Authors Reply

The mathematical equivalence between the Lindblad master equation and quantum trajectory methods, such as the Monte Carlo wave function (MCWF) method, is well established in the literature (see, e.g., Refs. [^8][^9][^10][^11]). Nonetheless, we have added an Appendix in the revised manuscript showing the convergence behavior of quantum trajectories toward the master equation as a function of the number of trajectories.

Referee's Comment 5

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?

Authors Reply

The effects studied in this manuscript rely on specific resonance conditions between the three subsystems. For example, the generation of the 2-photon entangled state can be achieved under the condition $\omega_a \simeq \omega_b \simeq \omega_c$. The interaction term enabling this dynamic is given by

in Eq. (3) of the current manuscript. The efficiency of the conversion process between the states $|0, 2, 0\rangle$ and $|\psi_+^{(2e)}\rangle$ depends on the ratio $| g_\mathrm{eff} | / | \omega_{a,c} - \omega_b |$, where $g_\mathrm{eff} = 2\sqrt{2} g^2 / \omega_b$ (see the off-diagonal elements in the matrix in Eq.(4) of the current manuscript). Thus, it depends not only on the effective coupling but also on the detuning. Notably, the efficiency varies continuously with detuning, allowing for some relaxation of the resonance conditions.

We have added a new section in the Appendix that examines the behavior of this process as a function of detuning.

Referee's Comment 6

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.

Authors Reply

As already mentioned in the manuscript, we are aware of the current experimental challenges in constructing such setups and agree with the Referee’s concerns. Nonetheless, we would like to point out that:

1. The quadratic interaction term presented in the manuscript is intrinsic in the optomechanical interaction, and it is already derived in several works (see, e.g., [^12] and its circuit analog [^13]).
2. The state-of-art technology in optomechanics is advancing rapidly, and we recognize the potential for achieving this regime in the future.

On point 2, we would like to discuss the current state of optomechanics technology.

Over the past two decades, circuit optomechanics has achieved remarkable results. Cryogenic cooling can bring a microwave-frequency mechanical mode (4–6 GHz) to its quantum ground state [^14]. Additionally, experiments have demonstrated resonant quantum interactions between a superconducting phase qubit and mechanical modes, modeled by the quantum Rabi Hamiltonian [^15]. The addition of a quantum two-level system has increased coupling strength and non-linearities, bringing the radiation-pressure interaction near the strong-coupling regime. [^16]. This is achieved with a Josephson junction qubit setup operating in the microwave regime. By analyzing the coupling between a mechanical resonator and a flux qubit, and using the experimentally achieved qubit-oscillator coupling strength parameter in Ref.[^14], the Supplemental Material of Ref.[^17] shows that the radiation-pressure interaction strength between the high-frequency mechanical resonator and an electromagnetic one can be achieved with this technology.

Given advances in circuit-QED, it is likely that our proposed results could be observed on this platform.

Despite these advancements, circuit-QED analogs of optomechanics may provide an additional suitable platform [^18][^19]. For instance, the mechanical membrane can be simulated by a Superconducting Quantum Interference Device (SQUID). As an example, it's worth to mention the demonstration of photons-pair extraction through vacuum perturbation (dynamical Casimir effect[^20][^21]).

Finally, as stated in the manuscript, we hope that this theoretical proposal can stimulate future experimental realizations.

Bibliography

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[^2]: Tan, H. (2014). Deterministic quantum superpositions and fock states of mechanical oscillators via quantum interference in single-photon cavity optomechanics. Physical Review A, 89(5), 053829.

[^3]: Garziano, L., Stassi, R., Macrí, V., Savasta, S., & Di Stefano, O. (2015). Single-step arbitrary control of mechanical quantum states in ultrastrong optomechanics. Physical Review A, 91(2), 023809.

[^4]: Bergholm, V., Wieczorek, W., Schulte-Herbrüggen, T., & Keyl, M. (2019). Optimal control of hybrid optomechanical systems for generating non-classical states of mechanical motion. Quantum Science and Technology, 4(3), 034001.

[^5]: de Moraes Neto, G. D., Montenegro, V., Teizen, V. F., & Vernek, E. (2019). Dissipative phonon-Fock-state production in strong nonlinear optomechanics. Physical Review A, 99(4), 043836.

[^6]: Macrì, V., Ridolfo, A., Di Stefano, O., Kockum, A. F., Nori, F., & Savasta, S. (2018). Nonperturbative dynamical Casimir effect in optomechanical systems: Vacuum Casimir-Rabi splittings. Physical Review X, 8(1), 011031.

[^7]: Bohmann, M., Sperling, J., & Vogel, W. (2017). Entanglement verification of noisy NOON states. Physical Review A, 96(1), 012321.

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We thank the Referee for the detailed report of our manuscript, and for considering the physics presented in this work as "rich and intriguing". In the following, we provide responses to the raised comments.

Referee's Comment 1

why do you define the effective annihilation operator of the dressed mode as $\hat{\mathcal{X}}^+$?

Authors Reply

We thank the Referee for having raised this important question.

The description of an open quantum system in terms of dressed operators is more general than using the "bare" operators in the Lindblad form. This approach is thoroughly discussed in the book The Theory of Open Quantum Systems by Breuer and Petruccione, and then applied to ultrastrongly coupled systems in, e.g., Refs [30,57] of the current Manuscript. In the limit of linear systems and weak coupling, the standard Lindblad form is restored. Here, we used the dressed operators to have a more general description of our system, but we could also use the standard Lindblad form, with similar results. The physical meaning of $\hat{\mathcal{X}}^+$ is to get an operator that only oscillates with positive frequencies. It is the analogous of the annihilation operator $\hat{a}$, which oscillates as $e^{-i \omega t}$ for a linear system. More generally, the $\hat{\mathcal{X}}^+$ operator would involve multiple positive frequencies.

Referee's Comment 2

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.

Authors Reply

We completely agree with the Refeeree. We acknowledge that observing this effect in optomechanical platforms remains challenging. However, as the Referee points out, it may be observable in circuit QED simulators.

Hence, we have revised the manuscript to include a more detailed explanation of this point.

Referee's Comment 3

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.

Authors Reply

We modified the layout of Fig. 3(a) to include an inset that highlights the intermediate oscillations more clearly.

Referee's Comment 4

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.

Authors Reply

As noted in our response to Referee’s Comment 2, we have revised the introduction to clarify that electronic analogues are more likely than optomechanical systems to reproduce the physics presented. The manuscript has been updated accordingly.