pith. sign in

arxiv: 2605.07371 · v1 · submitted 2026-05-08 · 🪐 quant-ph

Breaking mechanical dark mode via the Coulomb interaction

Jian-Song Zhang , Yuan Chen , Guang-Ling Cheng , Ai-Xi Chen This is my paper

Pith reviewed 2026-05-11 02:08 UTC · model grok-4.3

classification 🪐 quant-ph
keywords optomechanicsmechanical dark modeCoulomb interactionmechanical squeezingquantum entanglementground-state coolingparametric amplification
0
0 comments X

The pith

Coulomb interaction between two degenerate mechanical resonators breaks their dark mode, enabling simultaneous ground-state cooling and squeezing beyond 3 dB.

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

The paper establishes that a tunable Coulomb interaction can lift the dark-mode decoupling that normally prevents one of two identical mechanical resonators from interacting with the optical field. With an added optical parametric amplifier, both resonators reach the quantum ground state simultaneously even outside the resolved-sideband regime. The same Coulomb coupling generates mechanical parametric amplification that, combined with the optical amplifier, produces mechanical squeezing stronger than the 3 dB limit. The setup further yields robust bipartite entanglement between the resonators and the cavity field together with genuine tripartite entanglement among all three modes. A reader would care because these effects appear in a simple, frequency-degenerate system without requiring auxiliary tuning of resonance frequencies.

Core claim

By introducing the Coulomb interaction between two degenerate mechanical resonators in an optomechanical cavity, the dark mode is broken so that both resonators couple to the optical field. Combined with an optical parametric amplifier, this permits simultaneous ground-state cooling of the two resonators beyond the resolved-sideband limit. The Coulomb force additionally supplies mechanical parametric amplification; together with the optical amplifier it generates mechanical squeezing exceeding 3 dB and supports robust bipartite as well as genuine tripartite entanglement in the otherwise degenerate system.

What carries the argument

The Coulomb interaction, which supplies tunable mechanical parametric amplification that breaks the symmetry of the dark mode and mixes the two resonator modes.

If this is right

  • Both mechanical resonators can be cooled to the ground state at the same time even outside the resolved-sideband regime.
  • Mechanical squeezing stronger than 3 dB is produced by the combined action of optical and mechanical parametric amplification.
  • Robust bipartite entanglement appears between the two resonators and the cavity field.
  • Genuine tripartite entanglement is generated among the two mechanical modes and the optical mode.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same symmetry-breaking principle could be tested in other degenerate mechanical arrays by replacing the Coulomb force with capacitive or piezoelectric coupling.
  • Varying the Coulomb strength offers a continuous experimental knob for tuning the amount of squeezing and the degree of tripartite entanglement.
  • The approach may simplify the fabrication of multi-mode quantum sensors that require simultaneous cooling and entanglement without individual frequency control.

Load-bearing premise

The Coulomb interaction can be introduced and tuned in strength without adding enough extra noise or decoherence to block ground-state cooling or destroy the generated squeezing and entanglement.

What would settle it

If experiments show that the mechanical resonators fail to reach the ground state or that squeezing remains at or below 3 dB once the Coulomb coupling is activated and the optical parametric amplifier is turned on, the central claim would be falsified.

Figures

Figures reproduced from arXiv: 2605.07371 by Ai-Xi Chen, Guang-Ling Cheng, Jian-Song Zhang, Yuan Chen.

Figure 2
Figure 2. Figure 2: FIG. 2: Steady-state mechanical squeezing versus the bias [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Steady-state bipartite entanglement [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
read the original abstract

We propose a method to break the dark mode of two degenerate mechanical resonators (MRs) in optomechanical systems via the Coulomb interaction. Two degenerate MRs can be cooled to their ground-state simultaneously beyond the resolved sideband regime using the Coulomb interaction and an optical parametric amplifier (OPA). We show that strong and robust mechanical squeezing beyond 3 dB can be generated using the OPA and mechanical parametric amplification (MPA) introduced by the Coulomb interaction. Our results manifests that robust bipartite and genuine tripartite entanglement can be produced in a degenerate optomechanical system.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes using the Coulomb interaction between two degenerate mechanical resonators (MRs) in an optomechanical system to break the mechanical dark mode. Combined with an optical parametric amplifier (OPA), this enables simultaneous ground-state cooling of both resonators beyond the resolved sideband regime. The work further claims that the OPA together with mechanical parametric amplification (MPA) induced by the Coulomb interaction generates strong and robust mechanical squeezing beyond 3 dB, and produces robust bipartite as well as genuine tripartite entanglement.

Significance. If the central claims hold under realistic conditions, the approach would provide a practical route to overcome dark-mode limitations in degenerate optomechanical systems, enabling ground-state cooling, squeezing, and multipartite entanglement in parameter regimes where standard radiation-pressure cooling is inefficient. This could have implications for quantum sensing and continuous-variable quantum information processing with mechanical modes.

major comments (1)
  1. The central claims of ground-state cooling, squeezing >3 dB, and robust entanglement all rest on the assumption that the Coulomb interaction contributes only a coherent MPA term in the effective master equation with no additional Lindblad operators for charge fluctuations or position-dependent damping. The abstract provides no derivation, parameter regime, or numerical evidence showing that realistic charge noise (common in charged MR experiments) keeps the steady-state phonon occupancy below 1 or preserves the reported logarithmic negativity values, particularly in the bad-cavity limit. This assumption is load-bearing for every quantitative claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for raising this important point about the modeling assumptions underlying our central claims. We address the concern directly below.

read point-by-point responses

  1. Referee: The central claims of ground-state cooling, squeezing >3 dB, and robust entanglement all rest on the assumption that the Coulomb interaction contributes only a coherent MPA term in the effective master equation with no additional Lindblad operators for charge fluctuations or position-dependent damping. The abstract provides no derivation, parameter regime, or numerical evidence showing that realistic charge noise (common in charged MR experiments) keeps the steady-state phonon occupancy below 1 or preserves the reported logarithmic negativity values, particularly in the bad-cavity limit. This assumption is load-bearing for every quantitative claim.

    Authors: We appreciate the referee highlighting this modeling assumption. In the manuscript, the Coulomb interaction is introduced strictly as a coherent bilinear term in the system Hamiltonian, H_C = ħ g_c x_1 x_2 (with g_c determined by the product of the fixed charges on the two resonators). This term is derived from the electrostatic energy and, after transformation to the interaction picture and adiabatic elimination of the cavity mode (in the presence of the OPA), produces the mechanical parametric amplification (MPA) that breaks the dark mode. The effective master equation for the mechanical modes is obtained under the standard Markovian approximation for the optical bath, without additional dissipators arising from the Coulomb coupling itself. This treatment follows the conventional approach used in prior theoretical works on Coulomb-coupled mechanical resonators, where charges are taken as stable control parameters. We acknowledge that realistic charge fluctuations (e.g., random telegraph noise or position-dependent damping) would introduce extra Lindblad operators not present in our ideal model. Our quantitative results (phonon numbers, squeezing spectra, and logarithmic negativities) are therefore obtained in the absence of such noise. We agree that this is a load-bearing assumption, especially in the bad-cavity regime. To address the concern, we will revise the manuscript by (i) explicitly stating the constant-charge assumption in the model section, (ii) adding a discussion of typical experimental charge-noise levels and the parameter window in which the coherent coupling dominates, and (iii) including numerical simulations that incorporate a phenomenological charge-noise term to verify that ground-state cooling, >3 dB squeezing, and the reported entanglement persist revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper proposes introducing Coulomb interaction to break the mechanical dark mode in a degenerate optomechanical system, then derives cooling to ground state, mechanical squeezing beyond 3 dB via OPA and MPA, and bipartite/tripartite entanglement as consequences of the resulting effective dynamics. No self-definitional steps appear where an output is defined in terms of itself, no fitted parameters are relabeled as predictions, and no load-bearing self-citations or uniqueness theorems imported from prior author work are invoked in the provided abstract or claims. The central results follow from the standard optomechanical Hamiltonian augmented by the Coulomb term, with the physical assumptions (tunable interaction without excess decoherence) stated separately from the derived quantities. This is the normal case of an independent model-to-result chain.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so free parameters, axioms, and invented entities cannot be exhaustively identified. The proposal implicitly relies on standard quantum optics assumptions and tunable Coulomb coupling whose strength is not specified.

free parameters (1)
  • Coulomb coupling strength
    Must be tuned to produce the claimed MPA effect; value not given in abstract.
axioms (1)
  • domain assumption Standard Markovian master equation and rotating-wave approximation hold for the combined optomechanical-Coulomb system.
    Typical for such proposals but not stated explicitly.

pith-pipeline@v0.9.0 · 5384 in / 1248 out tokens · 31008 ms · 2026-05-11T02:08:17.631045+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

75 extracted references · 75 canonical work pages

  1. [1]

    The above Hamiltonian can be linearized using the fol- lowing displacement transformations a → α + δa and bj → βj + δbj

    of this equation. The above Hamiltonian can be linearized using the fol- lowing displacement transformations a → α + δa and bj → βj + δbj. Using the Hamiltonian of Eq. (7) and the displacement transformations, we obtain the follow- ing quantum Langevin equations ˙α = −(i∆a + κ 2 )α + ig1α(β∗ 1 + β1) +ig2α(β∗ 2 + β2) + 2χeiθα∗ + E, ˙β1 = −(iω′ m1 + γ1 2 )β...

    work page

  2. [2]

    The noise quadrature operators are defined as follows: X in O=a,b1,b2 = (Oin† + Oin)/ √ 2 and Y in O=a,b1,b2 = i(Oin† − Oin)/ √

    work page

  3. [3]

    Note that the dynamics of the present system can be described by a 6 × 6 covariance matrix V with Vjk = (⟨fjfk + fkfj⟩)/2

    From the above quantum Langevin equations, we get ˙⃗f = A ⃗f + ⃗ n, (9) with ⃗f = ( Xa, Ya, Xb1 , Yb1 , Xb2 , Yb2 )T , ⃗ n = (√κX in a , √κY in a , √γ1X in b1 , √γ1Y in b1 , √γ2X in b2 , √γ2Y in b2 )T and A = 0 BBBBB@ κ+ ∆a,+ 0 0 0 0 ∆a,− κ− 2eg1 0 2 eg2 0 0 0 − γ1 2 ωm1 0 0 2eg1 0 −ωm1 − 4λ − γ1 2 0 0 0 0 0 0 − γ2 2 ωm2 2eg2 0 0 0 −ωm2 − γ2 2 1 CCCCCA ,(...

    work page

  4. [5]

    Aspelmeyer, T

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

    work page 2014

  5. [7]

    Genes, D

    C. Genes, D. Vitali, et al. , New J. Phys. 10, 095009 (2008)

    work page 2008

  6. [8]

    C. Dong, V. Fiore, M. C. Kuzyk, et al. , Science 338, 1609 (2012)

    work page 2012

  7. [9]

    Y. D. Wang and A. A. Clerk, Phys. Rev. Lett. 108, 153603 (2012)

    work page 2012

  8. [10]

    A. B. Shkarin, et al., Phys. Rev. Lett. 112, 013602 (2014)

    work page 2014

  9. [11]

    M. C. Kuzyk and H. L. Wang, Phys. Rev. A 96, 023860 (2017)

    work page 2017

  10. [12]

    C. F. Ockeloen-Korppi, et al. , Phys. Rev. A 99, 023826 (2019)

    work page 2019

  11. [13]

    D. G. Lai, F. Zou, et al., Phys. Rev. A 98, 023860 (2018)

    work page 2018

  12. [14]

    D. G. Lai, J. F. Huang, et al. , Phys. Rev. A 102, 011502(R) (2020)

    work page 2020

  13. [15]

    D. G. Lai, J. Huang, et al. , Phys. Rev. A 103, 063509 (2021)

    work page 2021

  14. [16]

    M. T. Naseem and O. E. Mustecaplioglu, Commun. Phys. 4, 95 (2021)

    work page 2021

  15. [17]

    Sommer and C

    C. Sommer and C. Genes, Phys. Rev. Lett. 123, 203605 (2019)

    work page 2019

  16. [18]

    Huang, D

    J. Huang, D. G. Lai, et al. , Phys. Rev. A 106, 013526 (2022)

    work page 2022

  17. [19]

    Huang, D

    J. Huang, D. G. Lai, et al. , Phys. Rev. A 106, 063506 (2022)

    work page 2022

  18. [20]

    S. S. Chu, H. Q. Zhang, et al., Phys. Rev. A 109, 033509 (2024)

    work page 2024

  19. [21]

    Y. Cao, C. Yang, et al. , Phys. Rev. Lett. 134, 043601 (2025)

    work page 2025

  20. [22]

    P. C. Ma, J. Q. Zhang, et al. , Phys. Rev. A 90, 043825 (2014)

    work page 2014

  21. [23]

    C. Kong, H. Xiong, et al., Phys. Rev. A 95, 033820 (2017)

    work page 2017

  22. [24]

    C. H. Bai, D. Y. Wang, et al. , Sci. Rep. 7, 2545 (2017)

    work page 2017

  23. [25]

    A. A. Nejad, H. R. Askari, et al., J. Opt. Soc. Am. B 35, 2237 (2018)

    work page 2018

  24. [26]

    H. D. Mekonnen, T. G. Tesfahannes, et al. , Sci. Rep. 13, 13800 (2023)

    work page 2023

  25. [27]

    J. Q. Zhang, Y. Li, et al. , J. Phys.: Condens. Matter 25, 142201 (2013)

    work page 2013

  26. [28]

    Chen and A

    Y. Chen and A. X. Chen, Opt. Express 33, 21566 (2025)

    work page 2025

  27. [29]

    E. X. DeJesus and C. Kaufman, Phys. Rev. A 35, 5288 (1987)

    work page 1987

  28. [30]

    X. Y. Lv, J. Q. Liao, et al. , Phys. Rev. A 91, 013834 (2015)

    work page 2015

  29. [31]

    Zhang, Y

    R. Zhang, Y. N. Fang, et al. , Phys. Rev. A 99, 043805 (2019)

    work page 2019

  30. [32]

    J. S. Zhang and A. X. Chen, Opt. Express 28, 36620 (2020)

    work page 2020

  31. [33]

    Vidal and R

    G. Vidal and R. F. Werner, Phys. Rev. A 65, 032314 (2002)

    work page 2002

  32. [34]

    Adesso and F

    G. Adesso and F. Illuminati, New J. Phys. 8, 15 (2006)

    work page 2006

  33. [35]

    Adesso and F

    G. Adesso and F. Illuminati, J. Phys. A 40, 7821 (2007)

    work page 2007

  34. [36]

    W. Y. Qiu, X. H. Cheng, et al., Phys. Rev. A 105, 063718 (2022)

    work page 2022

  35. [37]

    Asjad, N

    M. Asjad, N. E. Abari, et al. , Opt. Express 27, 32427 (2019)

    work page 2019

  36. [38]

    M. D. LaHaye, O. Buu, et al. , Science 304, 74 (2004)

    work page 2004

  37. [39]

    J. D. Thompson, B. M. Zwickl, et al. , Nature 452, 72 (2008)

    work page 2008

  38. [40]

    W. K. Hensinger, et al. , Phys. Rev. A 72, 041405(R) (2005). 6

    work page 2005

  39. [41]

    G. S. Agarwal, Quantum Optics (Cambridge University Press, Cambridge, 2013)

    work page 2013

  40. [42]

    Aspelmeyer, T

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

    work page 2014

  41. [43]

    W. P. Bowen and G. J. Milburn, Quantum Optomechan- ics (CRC Press, 2015)

    work page 2015

  42. [44]

    Genes, D

    C. Genes, D. Vitali, and P. Tombesi, ”Simultaneous cool- ing and entanglement of mechanical modes of a micromir- ror in an optical cavity”, New J. Phys. 10, 095009 (2008)

    work page 2008

  43. [45]

    C. Dong, V. Fiore, M. C. Kuzyk, and H. L. Wang, ”Op- tomechanical dark mode”, Science 338, 1609 (2012)

    work page 2012

  44. [46]

    Y. D. Wang and A. A. Clerk, ”Using interference for high fidelity quantum state transfer in optomechanics”, Phys. Rev. Lett. 108, 153603 (2012)

    work page 2012

  45. [47]

    A. B. Shkarin, N. E. Flowers-Jacobs, S. W. Hoch, A. D. Kashkanova, C. Deutsch, J. Reichel, and J. G. E. Harris, ”Optically mediated hybridization between two mechanical modes”, Phys. Rev. Lett. 112, 013602 (2014)

    work page 2014

  46. [48]

    M. C. Kuzyk and H. L. Wang, ”Controlling multimode optomechanical interactions via interference”, Phys. Rev. A 96, 023860 (2017)

    work page 2017

  47. [49]

    C. F. Ockeloen-Korppi, M. F. Gely, E. Damskagg, M. Jenkins, G. A. Steele, and M. A. Sillanpaa, ”Sideband cooling of nearly degenerate micromechanical oscillators in a multimode optomechanical system”, Phys. Rev. A 99, 023826 (2019)

    work page 2019

  48. [50]

    D. G. Lai, F. Zou, B. P. Hou, Y. F. Xiao, and J. Q. Liao, ”Simultaneous cooling of coupled mechanical resonators in cavity optomechanics”, Phys. Rev. A 98, 023860 (2018)

    work page 2018

  49. [51]

    D. G. Lai, J. F. Huang, X. L. Yin, B. P. Hou, W. Li, D. Vitali, F. Nori, and J. Q. Liao, ”Nonreciprocal ground- state cooling of multiple mechanical resonators”, Phys. Rev. A 102, 011502(R) (2020)

    work page 2020

  50. [52]

    D. G. Lai, J. Huang, B. P. Hou, F. Nori, and J. Q. Liao, ”Domino cooling of a coupled mechanical-resonator chain via colddamping feedback”, Phys. Rev. A 103, 063509 (2021)

    work page 2021

  51. [53]

    M. T. Naseem and O. E. Mustecaplioglu, ”Ground-state cooling of mechanical resonators by quantum reservoir engineering”, Commun. Phys. 4, 95 (2021)

    work page 2021

  52. [54]

    Sommer and C

    C. Sommer and C. Genes, ”Partial optomechanical re- frigeration via multimode cold-damping feedback”, Phys. Rev. Lett. 123, 203605 (2019)

    work page 2019

  53. [55]

    Huang, D

    J. Huang, D. G. Lai, C. Liu, J. F. Huang, F. Nori, and J. Q. Liao, ”Multimode optomechanical cooling via general dark-mode control”, Phys. Rev. A 106, 013526 (2022)

    work page 2022

  54. [56]

    Huang, D

    J. Huang, D. G. Lai, and J. Q. Liao, ”Thermal-noise- resistant optomechanical entanglement via general dark- mode control”, Phys. Rev. A 106, 063506 (2022)

    work page 2022

  55. [57]

    S. S. Chu, H. Q. Zhang, J. S. Zhang, W. X. Zhong, G. L. Cheng, A. X. Chen, ”Simultaneous cooling of degenerate mechanical modes in unresolved sideband regime via op- tical and mechanical nonlinearities”, Phys. Rev. A 109, 033509 (2024)

    work page 2024

  56. [58]

    Y. Cao, C. Yang, J. T. Sheng, H. B. Wu, ”Optomechani- cal Dark-Mode-Breaking Cooling”, Phys. Rev. Lett. 134, 043601 (2025)

    work page 2025

  57. [59]

    P. C. Ma, J. Q. Zhang, Y. Xiao, et al. , ”Tunable double optomechanically induced transparency in an optome- chanical system”, Phys. Rev. A 90, 043825 (2014)

    work page 2014

  58. [60]

    C. Kong, H. Xiong, and Y. Wu, ”Coulomb-interaction- dependent effect of high-order sideband generation in an optomechanical system”, Phys. Rev. A 95, 033820 (2017)

    work page 2017

  59. [61]

    C. H. Bai, D. Y. Wang, H. F. Wang, et al. , ”Classical- to-quantum transition behavior between two oscillators separated in space under the action of optomechanical interaction”, Sci. Rep. 7, 2545 (2017)

    work page 2017

  60. [62]

    A. A. Nejad, H. R. Askari, and H. R. Baghshahi, ”Normal mode splitting in an optomechanical system: effects of coulomb and parametric interactions”, J. Opt. Soc. Am. B 35, 2237 (2018)

    work page 2018

  61. [63]

    H. D. Mekonnen, T. G. Tesfahannes, T. Y. Darge, et al. , ”Quantum correlation in a nano-electro-optomechanical system enhanced by an optical parametric amplifier and coulomb-type interaction”, Sci. Rep. 13, 13800 (2023)

    work page 2023

  62. [64]

    J. Q. Zhang, Y. Li, and M. Feng, ”Cooling a charged mechanical resonator with time-dependent bias gate volt- ages”, J. Phys.: Condens. Matter 25, 142201 (2013)

    work page 2013

  63. [65]

    Chen and A

    Y. Chen and A. X. Chen, ”Mechanical squeezing in a cavity optomechanical system with the Coulomb interac- tion and an optical parametric amplifier”, Opt. Express 33, 21566 (2025)

    work page 2025

  64. [66]

    E. X. DeJesus and C. Kaufman, ”Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations”, Phys. Rev. A 35, 5288 (1987)

    work page 1987

  65. [67]

    X. Y. Lv, J. Q. Liao, L. Tian, et al. , ”Steady-state me- chanical squeezing in an optomechanical system via duff- ing nonlinearity”, Phys. Rev. A 91, 013834 (2015)

    work page 2015

  66. [68]

    Zhang, Y

    R. Zhang, Y. N. Fang, Y. Y. Wang, S. Chesi, and Y. D. Wang, ”Strong mechanical squeezing in an unresolved- sideband optomechanical system”, Phys. Rev. A 99, 043805 (2019)

    work page 2019

  67. [69]

    J. S. Zhang and A. X. Chen, ”Large and robust mechan- ical squeezing of optomechanical systems in a highly un- resolved sideband regime via Duffing nonlinearity and in- tracavity squeezed light”, Opt. Express 28, 36620 (2020)

    work page 2020

  68. [70]

    Vidal and R

    G. Vidal and R. F. Werner, ”Computable measure of entanglement”, Phys. Rev. A 65, 032314 (2002)

    work page 2002

  69. [71]

    Adesso and F

    G. Adesso and F. Illuminati, ”Continuous variable tan- gle, monogamy inequality, and entanglement sharing in Gaussian states of continuous variable systems”, New J. Phys. 8, 15 (2006)

    work page 2006

  70. [72]

    Adesso and F

    G. Adesso and F. Illuminati, ”Entanglement in continuous-variable systems: recent advances and cur- rent perspectives”, J. Phys. A 40, 7821 (2007)

    work page 2007

  71. [73]

    W. Y. Qiu, X. H. Cheng, A. X. Chen, Y. H. Lan and W. J. Nie, ”Controlling quantum coherence and entanglement in cavity magnomechanical system”, Phys. Rev. A 105, 063718 (2022)

    work page 2022

  72. [74]

    Asjad, N

    M. Asjad, N. E. Abari, S. Zippilli, et al. , ¡°Optomechan- ical cooling with intracavity squeezed light,¡± Opt. Ex- press 27, 32427 (2019)

    work page 2019

  73. [75]

    M. D. LaHaye, O. Buu, B. Camarota and K. Schwab, ”Approaching the quantum limit of a nanomechanical resonator”, Science 304, 74 (2004)

    work page 2004

  74. [76]

    J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Mar- quardt, S. M. Girvin and J. G. E. Harris, ”Strong dis- persive coupling of a hing-finesse to a micromechanical membrane”, Nature 452, 72 (2008). 7

    work page 2008

  75. [77]

    W. K. Hensinger, D. W. Utami, H. S. Goan, K. Schwab, C. Monroe and G. J. Milburn, ”Ion trap transducers for quantum electromechanical oscillators”, Phys. Rev. A 72, 041405(R) (2005)

    work page 2005