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REVIEW 3 major objections 5 minor 98 references

Routing thermal noise flow and ground-state cooling in an optomechanical plaquette

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read By setting the overall loop phase to $\pi$ and matching the coupling-to-decay ratios of the two optical paths, the paper shows that thermal phonons stop flowing between two mechanical resonators, so each can be cooled to its dual-cavity…

desk verdict A genuinely new impedance-matching condition for routing thermal noise in a four-mode optomechanical plaquette; the equal-cavity-decay caveat should be stated. read the letter →

arxiv 2506.16705 v1 pith:SAR3ZEF6 submitted 2025-06-20 quant-ph

Guang-Zheng Ye , Tian-Le Yang , Wan-Jun Su , Yong Li , Huaizhi Wu This is my paper · ORCID
classification quant-ph
keywords optomechanicsmechanicalground-statecoolingthermalnoiseroutingplaquettephaseimpedancematchingdissipationengineeringnonreciprocalphonontransportmultimode
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes that in a four-mode optomechanical plaquette—two optical cavities and two mechanical resonators—thermal noise flow between the mechanical modes can be switched off completely by setting the overall laser-driven loop phase to $\Phi=\pi$ and matching the coupling strengths and cavity decays of the two paths. With that destructive interference, each resonator is left alone with its cavity cooling, so both can be sideband-cooled to the dual-cavity cooling limit; simulations give $\bar n_1=\bar n_2\approx 0.135$ phonons when the thermal baths start at $\bar m_1=\bar m_2=10^3$. The suppression works even when the two baths have very different temperatures, and it extends to nonreciprocal, one-way routing of phonon noise. A careful reader would care because it offers a dissipation-based alternative to coherent dark-mode control for multimode cooling and for thermal management in optomechanical networks.

What carries the argument

The load-bearing object is the overall plaquette phase $\Phi=\phi_{11}+\phi_{21}-\phi_{12}-\phi_{22}$ inherited from the phases of the four driving lasers, combined with the impedance-matching condition Eq. (8), which balances products of effective optomechanical couplings against the complex cavity susceptibilities $\chi_{a_j}=(\kappa_j/2-i\omega)^{-1}$. For $\Phi=\pi$ the two paths connecting $b_1$ and $b_2$ (through $a_1$ and through $a_2$) acquire opposite signs, and the impedance condition makes their amplitudes equal, so the cross-noise terms cancel. Because the cancellation involves the cavity decay rates $\kappa_j$, not just the coherent couplings, this is dissipation engineering at the optomechanical interfaces rather than purely coherent control. The same structure also produces the dark/supermode decomposition in which each mechanical mode couples to its own optical supermode and no phonons are exchanged.

What would settle it

Measure the steady-state phonon number $\bar n_2$ in a four-mode plaquette at $\Phi=\pi$ with the impedance condition $G_{11}G_{12}/(\kappa_1/2)=G_{21}G_{22}/(\kappa_2/2)$ while the hot-bath occupation $\bar m_1$ is raised from $10^3$ to $10^5$. The paper predicts $\bar n_2$ stays pinned at its dual-cavity value, independent of $\bar m_1$; any observable rise of $\bar n_2$ with $\bar m_1$ would show the cross thermal-noise path is not fully suppressed.

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Extended reading notes

Core claim

The central claim is that thermal noise flow between two non-degenerate mechanical resonators in the four-mode plaquette is governed by a gauge-invariant loop phase $\Phi$, and that the two cross-noise channels $T_{R_2\to b_1}$ and $T_{R_1\to b_2}$ interfere destructively when $\Phi=\pi$ and the optomechanical couplings and cavity dampings satisfy $G_{11}G_{12}/(\kappa_1/2-i\omega)=G_{21}G_{22}/(\kappa_2/2-i\omega)$ (Eq. 8). Under this condition the off-diagonal terms $H_{12}$ and $H_{21}$ in the mechanical response vanish, meaning no thermal phonons pass from one resonator to the other regardless of the bath temperatures. The Hamiltonian then splits into two independent beam-splitter couplings $G_1 b_1 \alpha_{1,-}^\dagger + G_2 b_2 \alpha_{2,+}^\dagger$ (plus Hermitian conjugate) to orthogonal optical supermodes, and each resonator cools toward the dual-cavity cooling limit. For equal baths with $\bar m_1=\bar m_2=10^3$ the simulations give $\bar n_1=\bar n_2\approx 0.135$; for a hot bath $\bar m_1=10^5$ and a cold bath $\bar m_2=10^3$, resonator 2 still reaches its dual-cavity limit while resonator 1 is cooled but not to its ground state. The paper also shows the same phase control can make the phonon flow unidirectional at $\Phi=\pi/2$ or $3\pi/2$ when cavity detunings are imbalanced.

Load-bearing premise

The linearized model uses a rotating-wave approximation in which the two mechanical frequencies and their difference must be far larger than the optical linewidths and optomechanical couplings; at the strong-coupling parameters and in the quoted experiments this separation is only moderate, so counter-rotating corrections could shift the exact decoupling point and the predicted occupancies.

Editorial extensions

If this is right

  • With equal thermal baths, both non-degenerate resonators cool to about $\bar n\approx 0.135$ phonons, which is near the ground state and equal to the dual-cavity cooling limit.
  • When one resonator sits in a much hotter bath, the colder resonator's final phonon number is independent of the hot bath's occupation as long as the impedance condition holds, so its ground-state cooling is protected.
  • The noise suppression survives from weak coupling up to strong coupling $G/\kappa\sim 1$, so it is not limited to the perturbative sideband regime.
  • By imbalancing cavity detunings, the same loop phase can make the thermal transfer unidirectional, with one transmission direction vanishing at $\Phi=\pi/2$ or $3\pi/2$.
  • The scheme is robust to laser detuning variations and does not require the resonators to be degenerate, unlike standard dark-mode-based cooling approaches.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The plaquette is a natural building block for larger optomechanical networks: the same loop-phase and impedance conditions could be applied edge-by-edge to route heat away from selected nodes, a multi-cell extension the paper does not itself work out.
  • A practical by-product that the authors leave implicit is a diagnostic: sweeping $\Phi$ and watching the colder resonator's temperature would locate the impedance-matched operating point with no prior knowledge of the coupling phases.
  • One testable extension is to deliberately break the impedance condition and use the phase to steer phonon noise directionally, effectively programming a thermal router in a phonon network; this would go beyond the single nonreciprocal pair demonstrated in the appendix.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes a four-mode optomechanical plaquette consisting of two non-degenerate mechanical resonators coupled through two cavity modes, and studies the routing of thermal noise flow among the mechanical modes. The authors linearize the optomechanical interaction around red-sideband drives, eliminate the optical modes, and derive scattering amplitudes that describe how thermal noise from each mechanical bath reaches the two resonators. The central result is that when the overall loop phase is Phi = pi and the coupling strengths and cavity decay rates satisfy an impedance-matching condition, the cross-coupling coefficients H12 and H21 vanish, so that thermal noise flow between the two mechanical resonators is suppressed. In that regime each resonator cools toward the dual-cavity cooling limit, and the numerical spectra give phonon occupations of about 0.135 for both resonators when each bath has 10^3 thermal quanta. The paper also studies intermediate and strong coupling, unequal cavity decay rates, laser detuning effects, and a nonreciprocal extension with unbalanced detunings.

Significance. If the central claim holds, the paper offers a conceptually distinct route to multimode mechanical ground-state cooling that works through dissipative interference of thermal noise channels rather than through coherent dark-mode breaking. The derivation is parameter-free in the sense that the destructive-interference condition is derived from the linearized quantum Langevin equations, not fitted, and the comparison against the independently defined dual-cavity cooling limit provides a clear quantitative benchmark. The numerical spectra in Figs. 2-6 confirm the predicted suppression at Phi = pi for equal cavity decays, and the proposed scheme is tied to experimentally demonstrated microwave optomechanical platforms. The idea of routing thermal noise flow with a loop phase is likely to be of interest to the optomechanics and phonon-transport communities. However, as detailed in the major comments, the exactness of the suppression condition for unequal cavity decays and the validity regime of the rotating-wave approximation need to be stated more carefully before the strongest claims are accepted.

major comments (3)
  1. [Section 'Gauge-invariant phase', Eq. (8)] The destructive-interference condition (8), G11G12/(kappa1/2 - i omega) = G21G22/(kappa2/2 - i omega), cannot hold exactly for all omega unless kappa1 = kappa2. Cross-multiplying gives G11G12(kappa2/2 - i omega) = G21G22(kappa1/2 - i omega); equating the constant and linear coefficients forces both G11G12 = G21G22 and kappa1 = kappa2. For kappa1 != kappa2 with positive real couplings, the equality is only satisfied at omega = 0, which is the content of the approximate condition (10). The text should explicitly state that exact, frequency-independent suppression of the cross-noise terms requires equal cavity decay rates, and that for unequal rates the suppression is approximate, becoming exact only in the zero-bandwidth limit. The abstract and the conclusion say the thermal noise flow "can be fully suppressed" without this qualification; they should be revised, and the inset of Fig. 6 should be translated into a quantitative bound on the residual phonon number, for example an upper bound on nbar2 - nbar2^dual for kappa1/kappa2 = 1/2.
  2. [Section 'Model', Eq. (1); Figs. 2(d), 2(h), and 4] The linearized Hamiltonian (1) is obtained under the rotating-wave approximation, which requires the mechanical frequencies and their difference to be much larger than all optical and coupling rates, i.e., omega_b,k, |omega_b,1 - omega_b,2| >> G_jk, kappa_j, gamma_k. In the strong-coupling figures the authors set G/kappa = 1, and in the experimental parameters quoted in the discussion omega_b,1/kappa = 5 with omega_b,2/kappa = 25. While the frequency difference is large, the ratio G/omega_b,1 is about 0.2, which is not 'much smaller than unity'. Counter-rotating terms may therefore contribute at the few-percent level and could shift the predicted phonon occupations and the exactness of the Phi = pi decoupling in the strong-coupling panels. The strong-coupling results should either be restricted to the regime where RWA is quantitatively justified, or checked against the full model retaining counter-rotating terms.
  3. [Abstract and conclusion] The statement that thermal noise flow is suppressed 'irrespective of their thermal temperatures' is correct only for the cross-flow itself; the achievable ground-state cooling of each resonator still depends on the temperature of its own bath. In Fig. 3(c), when one bath has mbar1 = 10^5 and the other has mbar2 = 10^3, the hot resonator (MR1) is not cooled to the ground state, as the paper itself states. The abstract's phrase 'even down to the ground state, can be realized in this regime' should therefore be accompanied by the explicit condition that the relevant bath temperatures (or thermal occupations) are low enough for the dual-cavity cooling limit to lie below one phonon.
minor comments (5)
  1. [Page 3, paragraph after Eq. (4)] The sentence 'the thermal flow into MR1 comes no only from its own heat bath R1' contains a typo: 'no only' should be 'not only'.
  2. [Page 3, paragraph 'Gauge-invariant phase'] The phrase 'which is exactly the same to TR2->b2(omega) and TR1->b2(omega)' should read 'exactly the same as'.
  3. [Conclusion] The claim that the method is 'robust to variations in laser detunings' appears too strong in view of Appendix C, where Fig. 7(a) shows that the cooling of MR1 degrades substantially as Delta1/kappa increases at Phi = pi. The robustness statement should be qualified to small detunings or to MR2 only.
  4. [Further discussion and conclusion] When the authors write that their parameters 'both ensure that the system is well in the resolved sideband regime', the ratio omega_b,1/kappa = 5 is only moderately resolved. Using 'in the resolved-sideband regime' without 'well' would be more accurate.
  5. [Appendix D, Eq. (20)] The two equations in (20) are written with absolute-value bars around expressions that are already set to zero; this is confusing. It should be clarified that the conditions are that the two complex combinations vanish, i.e., that both real and imaginary parts are zero.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the cooling result is derived from the linearized Langevin equations and checked against a separately defined dual-cavity limit, with no fitted parameters driving the prediction.

full rationale

The central claim—that at Φ=π with G11G12/(κ1/2−iω)=G21G22/(κ2/2−iω) the inter-resonator thermal-noise terms H12 and H21 vanish—is obtained algebraically from the frequency-domain coupled equations (Eq. 6 and Appendix B), not by fitting. The cooling benchmark called the dual-cavity cooling limit is computed from a distinct three-mode setup (a2−b1−a1) in the same paper, so reaching that limit at the destructive-interference point is an independent structural consequence rather than an input. The later approximate impedance condition for κ1≠κ2 (Eq. 10) is explicitly labeled approximate and its finite-bandwidth residual coupling is acknowledged in Fig. 6; this is a matter of scope and correctness, not circularity. Self-citations (Refs. 26, 64, 79, 84, 94) support experimental parameters or model realizability but do not supply the derivation of the suppression condition or the cooling numbers. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' prior work, and no known result is repackaged under new coordinates. The derivation chain is self-contained against an external, separately defined benchmark.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central derivation rests on standard input-output quantum Langevin equations, the RWA linearization, and independent Markovian baths; these are explicitly stated in the paper. No new physical entities are introduced; the loop phase is a derived combination of laser phases, and the thermal-noise-flow quantity is a defined observable, not a new force or particle.

free parameters (2)
  • Overall plaquette phase Phi = pi
    Destructive interference between the two noise-flow pathways requires Phi = pi (Eqs. 7 and 8). It is a controllable laser-phase combination, not a fit to data.
  • Impedance-matching coupling ratio G11G12 / G21G22 = kappa1 / kappa2 (equal to 1 when kappa1 = kappa2)
    The condition H12 = H21 = 0 (Eqs. 7-10) requires this equality; it is set by choosing laser powers and cavity parameters, not fitted.
assumptions (5)
  • domain assumption Rotating-wave approximation and linearization of the optomechanical interaction under resolved-sideband and weak-coupling conditions.
    Used to obtain Eq. (1) from Eq. (11); requires omega_b,k and |omega_b,1 - omega_b,2| >> G_jk, kappa_j, gamma_k.
  • domain assumption Cavity modes are coupled to zero-temperature vacuum baths; mechanical modes to independent Markovian thermal baths.
    The input-noise correlations in Appendix B; the whole thermal-noise-flow formalism assumes independent baths with occupation mbar_k.
  • standard math Gaussian white noise statistics for input operators and standard input-output quantum Langevin equations.
    Used in Appendix B to compute spectra and phonon numbers.
  • standard math Gauge transformation and the extraction of the single gauge-invariant plaquette phase Phi (Eq. 5).
    The loop phase is the only physical combination of the four drive phases; used to define the interference condition.
  • domain assumption Narrow-band approximation around omega = 0 when kappa1 != kappa2.
    Eq. (10) is derived for |delta omega| << kappa1, kappa2; violations cause deviations from the dual-cavity limit (Fig. 6).

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Pith. "Pith review of Routing thermal noise flow and ground-state cooling in an optomechanical plaquette." pith.science (2026). https://pith.science/paper/SAR3ZEF6

@misc{pith2026250616705,
  author       = {Pith},
  title        = {Pith review of: Routing thermal noise flow and ground-state cooling in an optomechanical plaquette},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SAR3ZEF6}},
  note         = {Machine review of arXiv:2506.16705}
}
read the original abstract

We propose an effective method for cooling two non-degenerate mechanical resonators by routing thermal noise flow in a four-mode optomechanical plaquette. The thermal noise flow between the mechanical resonators can be fully suppressed by addressing the overall loop phase in the plaquette, irrespective of their thermal temperatures. We find that optimal mechanical cooling, even down to the ground state, can be realized in this regime. The thermal noise routing, achieved by dissipation engineering at optomechanical interfaces, provides a valuable and complementary approach to conventional coherent dark-mode control theory. It can be generalized to nonreciprocal control of phonon transport and mechanical cooling, and may find applications in optomechanical networks with complex thermal environments.

Figures

Figures reproduced from arXiv: 2506.16705 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Schematic of a four-mode optomechanical [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. shows the scattering coefficients TR1→b1 (ω) [Figs. 2(a)-(d)] and TR2→b1 (ω) [Figs. 2(e)-(h)] from the own heat bath R1 and from the bath R2, respectively, which is exactly the same to TR2→b2 (ω) and TR1→b2 (ω) for MR2. Under the weak coupling condition G/κ = 0.1 and setting ∆k = 0 for simplicity (we have discussed the influence of laser detunings in Appendix C), both TR1→b1 (ω) and TR2→b1 (ω) [as shown by [PITH_FU… view at source ↗
Figure 3
Figure 3. FIG. 3. (a) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (6 more)
Figure 5
Figure 5. Figure 5: , by considering the thermal baths with ¯m1 = m¯ 2 = 103 [Figs. 5(a) and 5(b)]. The destructive interference regime is indicated by the white solid lines. For comparison, we also indicate the regime G22/G11 = G21/G12 with the black solid curves, which is referred to as…
Figure 4
Figure 4. Figure 4: (a). However, since the thermal phonon exchange around Φ = π/2, 3π/2 can not be fully suppressed, the ground-state cooling of MR2 could not be possible if far more than one thermal phonon from R1 flows into MR2 [i.e. N in 2 ≫ 1, see [PITH_FULL_IMAGE:figures/full_fig_p…
Figure 6
Figure 6. Figure 6: FIG. 6. Phonon occupation number ¯n [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Phonon occupation numbers ¯n [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Phonon occupation number ¯n [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. (a) The transmission between the two MRs, ( [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

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Works this paper leans on

98 extracted references · 59 canonical work pages

  1. [1]

    3(a) and 3(b)

    When the phonon number of the thermal bathsR1 and 5 R2 are both equal to ¯m1 = ¯m2 = 103, the thermal noise flow between the two MRs is reciprocal, i.e.N out 2 =N out 1 and N in 2 = N in 1 as denoted by the solid and dotted curves in Figs. 3(a) and 3(b). We find that the phonon occupation numbers ¯n1 = ¯n2≈ 0.135 are achieved, see the gray curve in Fig. 3...

  2. [2]

    Aspelmeyer, T

    M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014)

    2014

  3. [3]

    Barzanjeh, A

    S. Barzanjeh, A. Xuereb, S. Gr¨ oblacher, M. Paternostro, C. A. Regal, and E. M. Weig, Nat. Phys. 18, 15 (2022)

    2022

  4. [4]

    Kurizki, P

    G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, Proc. Natl. Acad. Sci.USA 112, 3866 (2015)

    2015

  5. [5]

    Blais, A

    A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Rev. Mod. Phys. 93, 025005 (2021)

    2021

  6. [6]

    J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, Nature (London) 471, 204 (2011)

    2011

  7. [7]

    Liao and L

    J.-Q. Liao and L. Tian, Phys. Rev. Lett. 116, 163602 (2016)

    2016

  8. [8]

    Riedinger, A

    R. Riedinger, A. Wallucks, I. Marinkovi´ c, C. L¨ oschnauer, M. Aspelmeyer, S. Hong, and S. Gr¨ oblacher, Nature (London) 556, 473 (2018)

    2018

Show all 98 references
  1. [9]

    Hu, L.-T

    C.-S. Hu, L.-T. Shen, Z.-B. Yang, H. Wu, Y. Li, and S.-B. Zheng, Phys. Rev. A 100, 043824 (2019)

    2019

  2. [10]

    Kotler, G

    S. Kotler, G. A. Peterson, E. Shojaee, F. Lecocq, K. Cicak, A. Kwiatkowski, S. Geller, S. Glancy, E. Knill, R. W. Simmonds, J. Aumentado, and J. D. Teufel, Science 372, 622 (2021)

    2021

  3. [11]

    R. A. Thomas, M. Parniak, C. Østfeldt, C. B. Møller, C. Bærentsen, Y. Tsaturyan, A. Schliesser, J. Appel, E. Zeuthen, and E. S. Polzik, Nat. Phys. 17, 228 (2021)

    2021

  4. [12]

    Bassi, K

    A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, Rev. Mod. Phys. 85, 471 (2013)

    2013

  5. [13]

    V. C. Vivoli, T. Barnea, C. Galland, and N. Sangouard, Phys. Rev. Lett. 116, 070405 (2016)

    2016

  6. [14]

    Marinkovi´ c, A

    I. Marinkovi´ c, A. Wallucks, R. Riedinger, S. Hong, M. Aspelmeyer, and S. Gr¨ oblacher, Phys. Rev. Lett. 121, 220404 (2018)

    2018

  7. [15]

    S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77, 513 (2005)

    2005

  8. [16]

    Rosenberg, Q

    J. Rosenberg, Q. Lin, and O. Painter, Nat. Photon. 3, 478 (2009)

    2009

  9. [17]

    A. P. Reed, K. H. Mayer, J. D. Teufel, L. D. Burkhart, W. Pfaff, M. Reagor, L. Sletten, X. Ma, R. J. Schoelkopf, E. Knill, and K. W. Lehnert, Nat. Phys. 13, 1163 (2017)

    2017

  10. [18]

    Pfaff, C

    W. Pfaff, C. J. Axline, L. D. Burkhart, U. Vool, P. Reinhold, L. Frunzio, L. Jiang, M. H. Devoret, and R. J. Schoelkopf, Nat. Phys. 13, 882 (2017)

    2017

  11. [19]

    Fiaschi, B

    N. Fiaschi, B. Hensen, A. Wallucks, R. Benevides, J. Li, T. P. M. Alegre, and S. Gr¨ oblacher, Nat. Photon. 15, 817 (2021)

    2021

  12. [20]

    Giovannetti and D

    V. Giovannetti and D. Vitali, Phys. Rev. A 63, 023812 (2001)

    2001

  13. [21]

    M. D. LaHaye, O. Buu, B. Camarota, and K. C. Schwab, Science 304, 74 (2004)

    2004

  14. [22]

    Giovannetti, S

    V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 1330 (2004)

    2004

  15. [23]

    Zhang, Y

    J.-Q. Zhang, Y. Li, M. Feng, and Y. Xu, Phys. Rev. A 86, 053806 (2012)

    2012

  16. [24]

    Peano, H

    V. Peano, H. G. L. Schwefel, C. Marquardt, and F. Marquardt, Phys. Rev. Lett. 115, 243603 (2015)

    2015

  17. [25]

    Motazedifard, A

    A. Motazedifard, A. Dalafi, F. Bemani, and M. H. Naderi, Phys. Rev. A 100, 023815 (2019)

    2019

  18. [26]

    Clarke, P

    J. Clarke, P. Neveu, K. E. Khosla, E. Verhagen, and M. R. Vanner, Phys. Rev. Lett. 131, 053601 (2023)

    2023

  19. [27]

    X.-W. Xu, Y. Li, A.-X. Chen, and Y.-x. Liu, Phys. Rev. A 93, 023827 (2016)

    2016

  20. [28]

    D. Malz, L. D. T´ oth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, Phys. Rev. Lett. 120, 023601 (2018)

    2018

  21. [29]

    N. Lauk, N. Sinclair, S. Barzanjeh, J. P. Covey, M. Saffman, M. Spiropulu, and C. Simon, Quantum Sci. Technol. 5, 020501 (2020)

    2020

  22. [30]

    N. J. Lambert, A. Rueda, F. Sedlmeir, and H. G. L. Schwefel, Adv. Quantum Technol. 3, 1900077 (2020)

    2020

  23. [31]

    J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gr¨ oblacher, M. Aspelmeyer, and O. Painter, Nature (London) 478, 89 (2011)

    2011

  24. [32]

    Verhagen, S

    E. Verhagen, S. Del´ eglise, S. Weis, A. Schliesser, and T. J. Kippenberg, Nature (London) 482, 63 (2012)

    2012

  25. [33]

    E. E. Wollman, C. U. Lei, A. J. Weinstein, J. Suh, A. Kronwald, F. Marquardt, A. A. Clerk, and K. C. Schwab, Science 349, 952 (2015)

    2015

  26. [34]

    Pirkkalainen, E

    J.-M. Pirkkalainen, E. Damsk¨ agg, M. Brandt, F. Massel, and M. A. Sillanp¨ a¨ a, Phys. Rev. Lett. 115, 243601 (2015)

    2015

  27. [35]

    Barzanjeh, E

    S. Barzanjeh, E. S. Redchenko, M. Peruzzo, M. Wulf, D. P. Lewis, G. Arnold, and J. M. Fink, Nature (London) 570, 480 (2019)

    2019

  28. [36]

    Heinrich, M

    G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, Phys. Rev. Lett. 107, 043603 (2011)

    2011

  29. [37]

    Xuereb, C

    A. Xuereb, C. Genes, and A. Dantan, Phys. Rev. Lett. 109, 223601 (2012)

    2012

  30. [38]

    Ludwig and F

    M. Ludwig and F. Marquardt, Phys. Rev. Lett. 111, 073603 (2013)

    2013

  31. [39]

    Xuereb, C

    A. Xuereb, C. Genes, G. Pupillo, M. Paternostro, and A. Dantan, Phys. Rev. Lett. 112, 133604 (2014)

    2014

  32. [40]

    Xuereb, A

    A. Xuereb, A. Imparato, and A. Dantan, New J. Phys. 17, 055013 (2015)

    2015

  33. [41]

    O. c. ˇCernot´ ık, S. Mahmoodian, and K. Hammerer, Phys. Rev. Lett. 121, 110506 (2018)

    2018

  34. [42]

    Carollo, K

    F. Carollo, K. Brandner, and I. Lesanovsky, Phys. Rev. Lett. 125, 240602 (2020)

    2020

  35. [43]

    J. P. Dowling and G. J. Milburn, Philos. Trans. R. Soc. London, A 361, 1655 (2003). 8

    2003

  36. [44]

    Mancini, D

    S. Mancini, D. Vitali, and P. Tombesi, Phys. Rev. Lett. 90, 137901 (2003)

    2003

  37. [45]

    Vitali, S

    D. Vitali, S. Gigan, A. Ferreira, H. R. B¨ ohm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, Phys. Rev. Lett. 98, 030405 (2007)

    2007

  38. [46]

    Lai, J.-Q

    D.-G. Lai, J.-Q. Liao, A. Miranowicz, and F. Nori, Phys. Rev. Lett. 129, 063602 (2022)

    2022

  39. [47]

    Lai, Y.-H

    D.-G. Lai, Y.-H. Chen, W. Qin, A. Miranowicz, and F. Nori, Phys. Rev. Res. 4, 033112 (2022)

    2022

  40. [48]

    Wilson-Rae, N

    I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, Phys. Rev. Lett. 99, 093901 (2007)

    2007

  41. [49]

    Marquardt, J

    F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, Phys. Rev. Lett. 99, 093902 (2007)

    2007

  42. [50]

    Deli´ e, M

    U. Deli´ e, M. Reisenbauer, K. Dare, D. Grass, V. Vuleti´ c, N. Kiesel, and M. Aspelmeyer, Science 367, 892 (2020)

    2020

  43. [51]

    Toroˇ s, U

    M. Toroˇ s, U. Deli´ c, F. Hales, and T. S. Monteiro, Phys. Rev. Res. 3, 023071 (2021)

    2021

  44. [52]

    Genes, D

    C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, Phys. Rev. A 77, 033804 (2008)

    2008

  45. [53]

    Y. Dong, F. Bariani, and P. Meystre, Phys. Rev. Lett. 115, 223602 (2015)

    2015

  46. [54]

    D.-G. Lai, W. Qin, B.-P. Hou, A. Miranowicz, and F. Nori, Phys. Rev. A 104, 043521 (2021)

    2021

  47. [55]

    Mancini, D

    S. Mancini, D. Vitali, and P. Tombesi, Phys. Rev. Lett. 80, 688 (1998)

    1998

  48. [56]

    P. F. Cohadon, A. Heidmann, and M. Pinard, Phys. Rev. Lett. 83, 3174 (1999)

    1999

  49. [57]

    Kleckner and D

    D. Kleckner and D. Bouwmeester, Nature (London) 444, 75 (2006)

    2006

  50. [58]

    Corbitt, C

    T. Corbitt, C. Wipf, T. Bodiya, D. Ottaway, D. Sigg, N. Smith, S. Whitcomb, and N. Mavalvala, Phys. Rev. Lett. 99, 160801 (2007)

    2007

  51. [59]

    Poggio, C

    M. Poggio, C. L. Degen, H. J. Mamin, and D. Rugar, Phys. Rev. Lett. 99, 017201 (2007)

    2007

  52. [60]

    Y. Guo, K. Li, W. Nie, and Y. Li, Phys. Rev. A 90, 053841 (2014)

    2014

  53. [61]

    Gu and G.-x

    W.-j. Gu and G.-x. Li, Phys. Rev. A 87, 025804 (2013)

    2013

  54. [62]

    Liu, Y.-F

    Y.-C. Liu, Y.-F. Xiao, X. Luan, and C. W. Wong, Phys. Rev. Lett. 110, 153606 (2013)

    2013

  55. [63]

    Liu, Y.-F

    Y.-C. Liu, Y.-F. Xiao, X. Luan, Q. Gong, and C. W. Wong, Phys. Rev. A 91, 033818 (2015)

    2015

  56. [64]

    M. T. Naseem and ¨O. E. M¨ ustecaplio˘ glu, Commun. Phys. 4, 95 (2021)

    2021

  57. [65]

    Liu, G.-Z

    Y. Liu, G.-Z. Ye, C.-S. Hu, Y. Li, and H. Wu, Phys. Rev. A 111, 023504 (2025)

    2025

  58. [66]

    Schliesser, R

    A. Schliesser, R. Rivi` ere, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, Nat. Phys. 4, 415 (2008)

    2008

  59. [67]

    de los R´ ıos Sommer, N

    A. de los R´ ıos Sommer, N. Meyer, and R. Quidant, Nat. Commun. 12, 276 (2021)

    2021

  60. [68]

    J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, Nature (London) 475, 359 (2011)

    2011

  61. [69]

    Massel, S

    F. Massel, S. U. Cho, J. M. Pirkkalainen, P. J. Hakonen, T. T. Heikkil¨ a, and M. A. Sillanp¨ a¨ a, Nat. Commun.3, 987 (2012)

    2012

  62. [70]

    Mercier de L´ epinay, E

    L. Mercier de L´ epinay, E. Damsk¨ agg, C. F. Ockeloen- Korppi, and M. A. Sillanp¨ a¨ a, Phys. Rev. Appl. 11, 034027 (2019)

    2019

  63. [71]

    J. Guo, R. Norte, and S. Gr¨ oblacher, Phys. Rev. Lett. 123, 223602 (2019)

    2019

  64. [72]

    D.-G. Lai, F. Zou, B.-P. Hou, Y.-F. Xiao, and J.-Q. Liao, Phys. Rev. A 98, 023860 (2018)

    2018

  65. [73]

    C. F. Ockeloen-Korppi, M. F. Gely, E. Damsk¨ agg, M. Jenkins, G. A. Steele, and M. A. Sillanp¨ a¨ a, Phys. Rev. A 99, 023826 (2019)

    2019

  66. [74]

    Lai, J.-F

    D.-G. Lai, J.-F. Huang, X.-L. Yin, B.-P. Hou, W. Li, D. Vitali, F. Nori, and J.-Q. Liao, Phys. Rev. A 102, 011502 (2020)

    2020

  67. [75]

    Huang, D.-G

    J. Huang, D.-G. Lai, C. Liu, J.-F. Huang, F. Nori, and J.-Q. Liao, Phys. Rev. A 106, 013526 (2022)

    2022

  68. [76]

    Xu, D.-G

    R. Xu, D.-G. Lai, B.-P. Hou, A. Miranowicz, and F. Nori, Phys. Rev. A 106, 033509 (2022)

    2022

  69. [77]

    D.-G. Lai, W. Qin, A. Miranowicz, and F. Nori, Phys. Rev. Res. 4, 033102 (2022)

    2022

  70. [78]

    P. Wen, X. Mao, M. Wang, C. Wang, G.-Q. Li, and G.-L. Long, Opt. Lett. 47, 5529 (2022)

    2022

  71. [79]

    J.-Y. Liu, W. Liu, D. Xu, J.-C. Shi, H. Xu, Q. Gong, and Y.-F. Xiao, Phys. Rev. A 105, 053518 (2022)

    2022

  72. [80]

    Y. Cao, C. Yang, J. Sheng, and H. Wu, Phys. Rev. Lett. 134, 043601 (2025)

    2025

  73. [81]

    S. Kim, X. Xu, J. M. Taylor, and G. Bahl, Nat. Commun. 8, 205 (2017)

    2017

  74. [82]

    H. Xu, L. Jiang, A. A. Clerk, and J. G. Harris, Nature (London) 568, 65 (2019)

    2019

  75. [83]

    Barzanjeh, M

    S. Barzanjeh, M. Aquilina, and A. Xuereb, Phys. Rev. Lett. 120, 060601 (2018)

    2018

  76. [84]

    G. A. Peterson, F. Lecocq, K. Cicak, R. W. Simmonds, J. Aumentado, and J. D. Teufel, Phys. Rev. X 7, 031001 (2017)

    2017

  77. [85]

    C. Yang, X. Wei, J. Sheng, and H. Wu, Nat. Commun. 11, 4656 (2020)

    2020

  78. [86]

    M. J. Weaver, F. Buters, F. Luna, H. Eerkens, K. Heeck, S. De Man, and D. Bouwmeester, Nat. Commun. 8, 824 (2017)

    2017

  79. [87]

    D. P. Lake, M. Mitchell, B. C. Sanders, and P. E. Barclay, Nat. Commun. 11, 2208 (2020)

    2020

  80. [88]

    Liang, X

    B. Liang, X. S. Guo, J. Tu, D. Zhang, and J. C. Cheng, Nat. Mater. 9, 989 (2010)

    2010

  81. [89]

    Boechler, G

    N. Boechler, G. Theocharis, and C. Daraio, Nat. Mater. 10, 665 (2011)

    2011

  82. [90]

    Popa and S

    B.-I. Popa and S. A. Cummer, Nat. Commun. 5, 3398 (2014)

    2014

  83. [91]

    Fleury, D

    R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and A. Al` u, Science343, 516 (2014)

    2014

  84. [92]

    H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, Nature (London) 537, 80 (2016)

    2016

  85. [93]

    Sasaki, Y

    R. Sasaki, Y. Nii, Y. Iguchi, and Y. Onose, Phys. Rev. B 95, 020407 (2017)

    2017

  86. [94]

    Chen, Y.-L

    Y. Chen, Y.-L. Zhang, Z. Shen, C.-L. Zou, G.-C. Guo, and C.-H. Dong, Phys. Rev. Lett. 126, 123603 (2021)

    2021

  87. [95]

    Lan, W.-J

    Y.-T. Lan, W.-J. Su, H. Wu, Y. Li, and S.-B. Zheng, Opt. Lett. 47, 1182 (2022)

    2022

  88. [96]

    Shen, Y.-L

    Z. Shen, Y.-L. Zhang, Y. Chen, Y.-F. Xiao, C.-L. Zou, G.-C. Guo, and C.-H. Dong, Phys. Rev. Lett. 130, 013601 (2023)

    2023

  89. [97]

    Tang and X.-W

    Z.-X. Tang and X.-W. Xu, Phys. Rev. Appl. 19, 034093 (2023). 9 Appendix A Linearized Hamiltonian The Hamiltonian of the four-mode optomechanical system is ( ℏ = 1) H = X j=1,2 ωa,ja† jaj + X k=1,2 ωb,kb† kbk + X j,k gj,ka† jaj(b† k +bk), (11) where (aj,a† j)j=1,2 are the annih...

    2023

  90. [98]

    1 ( A12B11 +e−iΦA22B21)χ(12,22) F (A11B12 +eiΦA21B22)χ(11,21) F 1 # Γ1b1,in Γ2b2,in +

    (17) For the four-mode setup, the MRs under optomechanical interactions response to the input noises via − →χ−1 B (Φ) b1 b2 = " 1 ( A12B11 +e−iΦA22B21)χ(12,22) F (A11B12 +eiΦA21B22)χ(11,21) F 1 # Γ1b1,in Γ2b2,in + " B21e−iΦA22B12χ(12,22) F +B11χ(12,22) FF B11A12B22χ(22,12) F +...

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.