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REVIEW 3 major objections 5 minor 57 references

Generation and Enhancement of Bipartite and Tripartite Entanglement in an Electro-Optomechanical Ring Cavity

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

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Pith's one-line read This paper claims that Coulomb coupling generates entanglement and an optical parametric amplifier amplifies it in an electro-optomechanical ring cavity.

desk verdict Interesting new optomechanical architecture, but the numerical results use a steady-state amplitude that does not solve the stated equations; the quantitative entanglement claims are unreliable as written. read the letter →

arxiv 2608.03578 v1 pith:2CB37VCI submitted 2026-08-04 quant-ph physics.optics

Fouad Essaadi , Yassine Oussarhan , Mohamed Ouhammou , Said Mouslih , Mohamed Jakha , Bouzid Manaut , Souad Taj This is my paper · ORCID
classification quant-phphysics.optics PACS 03.67.-a03.67.Bg42.50.Lc42.50.Wk42.65.Yj
keywords optomechanicsringcavitybipartiteentanglementtripartiteparametricnonlinearitynonlinearopticsCoulombcouplingopticalamplifier
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 claims that in a triangular electro-optomechanical ring cavity—two charged mechanical resonators coupled by the Coulomb force, a third uncharged mechanical oscillator, and an optical parametric amplifier (OPA)—entanglement is created only by the Coulomb interaction, and the OPA then acts as a tunable amplifier for that entanglement. Working with the linearized quantum Langevin equations and Gaussian states, the authors compute steady-state covariance matrices and quantify bipartite entanglement by logarithmic negativity $E_N$ and tripartite entanglement by the minimum residual contangle $R_{\min}^{\tau}$. They report that increasing the OPA gain $G$ and choosing its phase $\theta$ can push the mechanical–mechanical entanglement from roughly $0.29$ to about $0.52$, while tripartite measures grow from about $0.018$ to $0.04$, with the caveat that high gain or strong Coulomb coupling shrinks the stable operating window. If correct, the result identifies two practical control knobs—charge voltage and parametric drive—for distributing stronger quantum correlations across mechanical and optical modes in a single hybrid device.

What carries the argument

The load-bearing object is the $8\times 8$ drift matrix $A$ of the linearized quantum Langevin equations, written in the fluctuation basis $(\delta q_1,\delta p_1,\delta q_2,\delta p_2,\delta q_3,\delta p_3,\delta x,\delta y)$. Its entries contain the Coulomb coupling $\lambda$, the linearized optomechanical couplings $G_1=\sqrt{2}g_1\cos^2\phi\,\alpha$ and $G_3$, and the OPA terms $\pm 2G\cos\theta$, $\pm 2G\sin\theta$; stability requires all its eigenvalues to have negative real parts. For each stable parameter set, the steady-state covariance matrix $V$ solves the Lyapunov equation $AV+VA^T=-D$, and bipartite entanglement is read from the smallest symplectic eigenvalue of the partially transposed reduced state, while tripartite entanglement comes from the minimum residual contangle obeying the Coffman–Kundu–Wootters monogamy inequality.

What would settle it

Set the Coulomb coupling to zero ($\lambda=0$) and run the same covariance-matrix calculation: the paper predicts every entanglement measure is exactly zero for all detunings and gains; observing a nonzero logarithmic negativity in that case would refute the claim that the Coulomb interaction is indispensable.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a division of labour between two nonlinear couplings. The Coulomb interaction $\lambda q_1 q_2$ between the two charged membranes is the indispensable generator: at $\lambda=0$ every computed entanglement measure, including entanglement between the optical mode and the third mechanical oscillator, vanishes. The intracavity OPA, with gain $G$ and phase $\theta$, does not create correlations from nothing but dramatically amplifies the ones the Coulomb coupling seeds; by scanning detuning $\Delta$, gain, phase, Coulomb strength, laser power, and temperature, the paper maps out stable parameter regions and reports maxima such as $E_{M_1M_2}\approx 0.52$ at $\lambda=0.95\,\omega_m$ and $\Delta=0.65\,\omega_m$, and $R_{M_1M_2C}^{\min}\approx 0.04$ at $\lambda=0.94\,\omega_m$, $\Delta=0.31\,\omega_m$. The dependence on $\lambda$ is non-monotonic for the optical–mechanical bipartition, and all correlations degrade with temperature, so the paper's conclusion is that low-temperature operation and careful parameter selection are required to exploit the OPA enhancement.

Load-bearing premise

The paper's numbers depend on assuming the cavity field settles into a stable steady state whose phase can be rotated away; if that rotation changes the meaning of the detuning and amplifier phase, the reported maxima may not be physically reachable.

Editorial extensions

If this is right

  • At $\lambda=0$ none of the predicted bipartite or tripartite entanglement survives, so any experiment aiming to realize this scheme must first establish a nonzero charge-mediated coupling between the two mechanical resonators.
  • The OPA converts a modest baseline ($E_{M_1M_2}\simeq 0.29$ at $G=0$) into a substantially stronger correlation ($\simeq 0.52$ near $G=0.5\kappa$, $\theta=\pi$, $\lambda=0.95\omega_m$), giving experimentalists a single optical knob to tune entanglement strength without changing the mechanical setup.
  • Detuning $\Delta$ controls where the maximum sits: increasing OPA gain shifts the optimal detuning downward, so the operating point must be re-optimized whenever $G$ or $\theta$ is changed.
  • Thermal noise degrades every correlation measure, and the OPA only partially compensates; the paper's numbers therefore presuppose cryogenic operation (order 1 mK) for the reported magnitudes.
  • Strong coupling and high gain trade against stability: near $\lambda\simeq\omega_m$ or $G/\kappa>0.5$ the stable detuning window narrows, so maximizing entanglement requires operating close to an instability boundary.

Reading between the lines

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

  • A natural extension the paper does not pursue is to compute Gaussian quantum steering or Bell-nonlocality measures from the same covariance matrix; if the OPA amplifies logarithmic negativity, it likely also amplifies steering, which would make the device more directly useful for one-sided quantum communication.
  • Because the third oscillator M3 is not directly charged, the nonzero M3–cavity entanglement at intermediate $\lambda$ suggests the Coulomb pair acts as a nonlinear mediator; this could be tested by replacing the Coulomb coupling with a different quadratic interaction and checking whether the same 'seeding plus OPA amplification' pattern survives.
  • The reported stability threshold at $G/\kappa\simeq 0.5$ is a sharp, testable prediction: an experiment should see the output field transition from a stationary state to an unstable or oscillatory regime as the parametric pump crosses that gain, at fixed $\theta=\pi$.
  • The paper fixes the geometric angle $\phi$ in the equations but never specifies its value in the numerics; since the effective couplings scale as $\cos^2\phi$, a quantitative comparison with experiment requires $\phi$ to be reported or measured. This is an editorial caution, not a claim in the paper.
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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. This manuscript proposes a hybrid electro-optomechanical ring-cavity system containing two Coulomb-coupled charged mechanical resonators, a third uncharged mechanical resonator, and an intracavity optical parametric amplifier. The authors derive the rotating-frame Hamiltonian, write nonlinear quantum Langevin equations, linearize around steady-state solutions, and analyze Gaussian fluctuations through the covariance matrix and the Lyapunov equation. Bipartite entanglement is quantified by logarithmic negativity and tripartite entanglement by the minimum residual contangle, computed as functions of detuning, OPA gain and phase, Coulomb coupling strength, laser power, and temperature. The central claims are that the Coulomb interaction is indispensable for entanglement generation and that the OPA dramatically enhances all bipartite and tripartite correlations, subject to a stability trade-off.

Significance. If the quantitative results were correct, the paper would provide a useful parameter map for entanglement engineering in a hybrid optomechanical platform and a concrete example of OPA-based control of Gaussian multipartite entanglement. The work has genuine strengths: the optomechanical coupling is derived with oblique incidence, the covariance-matrix formalism is applied in a standard way, stability is checked eigenvalue-by-eigenvalue, and the parameter values are physically motivated. However, the numerical foundation of the paper is compromised by an algebraic error in the steady-state optical amplitude, and because every numerical curve in Section 5 is generated from that amplitude, the claimed enhancement factors and quantitative maxima are not currently supported.

major comments (3)
  1. [§2.3.2, Eq. (19)] The steady-state optical amplitude in Eq. (19) is not the solution of the steady-state condition obtained from Eq. (16). Setting ⟨ȧ⟩ = 0 in Eq. (16) gives (κ + iΔ)α − 2G e^{iθ} α* = E_L, whereas Eq. (19) solves (κ + iΔ − 2G e^{iθ})α = E_L, i.e., it replaces α* by α. The correct solution is α = E_L[(κ + 2G cosθ) − i(Δ − 2G sinθ)]/(κ² + Δ² − 4G²), which differs from Eq. (19) except in the special cases G = 0 or Δ = 2G sinθ. Because Section 4 constructs G1,3 from this α and every subsequent stability check, Lyapunov solution, and entanglement curve depends on it, all numerical results in Section 5 are computed for a point that is not a steady state of the model. The authors must correct Eq. (19), rerun the numerical pipeline, and verify whether the claimed maxima (e.g., EM1M2 ≈ 0.52 and RM1M2C ≈ 0.04) survive.
  2. [§2.3.3 and §4] The linearization in Section 2.3.3 assumes α can be taken real 'without loss of generality' and defines G1,3 = √2 g1,3 cos²φ α, while Section 4 states G1,3 = √2 g1,3 |α|. For a genuinely complex α, the optomechanical interaction linearizes into both δx and δy quadrature couplings, and the drift matrix entries involving G1,3 change; the δy couplings are absent from Eq. (20) and Eq. (22). A phase rotation of the cavity mode can make α real, but only at the cost of also rotating the effective detuning and the OPA phase. Since Section 5 scans Δ and θ independently as if no such rotation had been performed, the drift matrix used in the numerics is not the linearization of Eq. (16) at the actual steady state. The authors should either carry the complex α through the linearization or define and apply the full phase rotation consistently before scanning parameters.
  3. [§5] Because of the steady-state error, the quantitative claims in Section 5 are not reliable in their present form. This includes the reported maxima EM1M2 ≈ 0.52 at λ = 0.95ωm and Δ = 0.65ωm, RM1M2C ≈ 0.04 at λ = 0.94ωm and Δ = 0.31ωm, and the OPA-enhancement statements for EM3C. In addition, the text in Fig. 5(a) reports a peak logarithmic negativity of 0.81 for G/κ = 0.3, which is inconsistent with the nearby values in Figs. 5 and 6 for the same quantity and is likely a typographical error. After correcting the steady state, all figures, peak values, and the conclusions drawn from them must be regenerated and re-evaluated.
minor comments (5)
  1. [§3.2, Eq. (30)] The formula for ϱ_{r|st} is written as a minimum eigenvalue, but the residual contangle requires the minimum symplectic eigenvalue of the partially transposed covariance matrix; please correct Eq. (30) and specify the quadrature ordering for the partial-transposition matrices P_{r|st}, P_{s|rt}, and P_{t|sr}.
  2. [§5, Fig. 5(a)] The reported peak EM3C ≈ 0.81 at G/κ = 0.3 should be checked against Fig. 6, where the same quantity with G/κ = 0.3 is approximately 0.08; either the value, the axis label, or the figure caption contains an error.
  3. [§5] The geometric angle φ is never given a numerical value, although every optomechanical coupling strength G1,3 contains cos²φ; please state the value used or explain why the results are insensitive to it.
  4. [§2.3.2–§5] The symbol Δ denotes the effective detuning in Eq. (19) but is used as the scan variable in Section 5; please clarify whether the plotted Δ/ωm is the effective detuning including the mechanical steady-state shifts or the bare detuning, and if necessary redefine the variable to avoid ambiguity.
  5. [General] Some cross-references are imprecise, e.g., the text refers to 'Fig. 6(a)' although the caption of Fig. 6 does not label panels; please correct these references.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: entanglement results are consequences of the stated model, with OPA and coupling parameters scanned rather than fitted.

full rationale

The paper's derivation is self-contained in the relevant sense: it starts from an explicitly stated rotating-frame Hamiltonian (Eq. 9), derives nonlinear quantum Langevin equations (Eq. 16), linearizes them around a steady state to obtain the drift matrix (Eq. 22), and then computes bipartite and tripartite entanglement from the covariance-matrix solution of the Lyapunov equation (Eq. 23) using standard measures (logarithmic negativity and minimum residual contangle). The OPA gain G, phase θ, detuning Δ, Coulomb coupling λ, and input power P_L enter as control parameters that are scanned in Section 5; none of the quoted entanglement maxima (e.g., E_M1M2 ≈ 0.52 at λ = 0.95ωm and Δ = 0.65ωm, or R_M1M2C ≈ 0.04 at λ = 0.94ωm and Δ = 0.31ωm) is obtained by fitting a parameter to the entanglement value itself. The claim that λ = 0 yields no entanglement is a direct consequence of the model, since M2 is coupled only through λq1q2 and is otherwise isolated from the optical and M3 subsystems. The OPA-enhancement statements are likewise consequences of the linearized optomechanical couplings and modified optical damping; they are not produced by defining the OPA parameters in terms of the entanglement output. No load-bearing self-citation chain appears: cited prior work supplies standard formalisms (Lyapunov equation, PPT criterion, contangle) and comparison models, not the paper's central conclusion. The algebraic concern raised by a skeptical reading, namely that Eq. (19) treats α* as α when solving the steady-state amplitude and that taking α real omits δy-quadrature couplings in Eq. (20), is a correctness or internal-consistency issue rather than circularity: it concerns whether the drift matrix is the correct linearization of the stated equations, not whether the reported prediction is equivalent to the model input by construction. Therefore no circular step meeting the quoted-evidence standard is present.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The paper's central results depend on the standard assumptions of linearized optomechanics and on a set of experimentally motivated parameter values. The only quantities not pinned down numerically are the geometric angle ϕ (which enters as cos²ϕ in the optomechanical coupling) and the exact phase convention used to make the steady-state amplitude α real; these are the principal reproducibility risks.

free parameters (6)
  • OPA gain ratio G/κ = 0.5
    Control parameter scanned in figures; optimal near 0.5 but high gain compresses the stable detuning window.
  • OPA phase θ = π
    Scanned; maximum enhancement for the M1-M2 subsystem occurs at θ≈π.
  • Coulomb coupling strength λ/ωm = 0.9
    Chosen from experimental parameters (charges, distance) and scanned; 0.9ωm is near the stability limit.
  • Laser detuning Δ/ωm = 0.8
    Scanned to optimize entanglement; best values generally near the mechanical frequency.
  • Input laser power P_L = 60 mW
    Scanned; higher power increases effective coupling but adds noise.
  • Temperature T = 1 mK
    Chosen to reduce thermal noise; entanglement degrades as T rises.
assumptions (4)
  • domain assumption The system operates in the linearized regime where quantum fluctuations are Gaussian and fully described by the covariance matrix.
    Used throughout Section 2.3 and required for the Lyapunov equation solution in Eq. (23); valid for strong coherent drive |α|≫1, which is assumed but not demonstrated quantitatively.
  • domain assumption Mechanical reservoirs are Markovian with the correlation function in Eq. (17).
    Standard for high-Q mechanical oscillators; this assumption underlies the diagonal noise matrix D in Eq. (25).
  • domain assumption The Coulomb interaction can be truncated at the quadratic term λ q1 q2, neglecting higher-order nonlinearities.
    Justified by the small zero-point motion relative to the electrode separation (parameters in Section 2.2), so the expansion in Eq. (7) is extremely well controlled.
  • domain assumption The cavity supports a single optical mode, and the OPA is described by the degenerate parametric Hamiltonian in Eq. (5).
    Standard model assumption; the paper does not discuss multi-mode effects or OPA bandwidth.

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Cite this review

Pith. "Pith review of Generation and Enhancement of Bipartite and Tripartite Entanglement in an Electro-Optomechanical Ring Cavity." pith.science (2026). https://pith.science/paper/2CB37VCI

@misc{pith2026260803578,
  author       = {Pith},
  title        = {Pith review of: Generation and Enhancement of Bipartite and Tripartite Entanglement in an Electro-Optomechanical Ring Cavity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2CB37VCI}},
  note         = {Machine review of arXiv:2608.03578}
}
read the original abstract

This study investigates the generation and enhancement of quantum entanglement in an electro-optomechanical ring cavity system. The setup integrates two Coulomb-coupled mechanical resonators, which serve as the fundamental mechanism for the generation of bipartite and tripartite entanglement via charge mediated coupling. We then demonstrate the significant enhancement of this entanglement via a nonlinear parametric drive (an optical parametric amplifier, OPA), which injects a controllable nonlinearity into the cavity. We derive the system's Hamiltonian and the corresponding quantum Langevin equations, which are linearized around steady-state solutions to analyze Gaussian quantum fluctuations. Employing the covariance matrix formalism, we quantify bipartite entanglement via logarithmic negativity and tripartite entanglement via the minimum residual contangle. Our results unequivocally show that while the Coulomb interaction is indispensable for creating entanglement, the OPA acts as a powerful control tool, dramatically amplifying the degree of quantum correlations for all subsystems. We find that the strength of entanglement is highly sensitive to several parameters and can be optimized through the strategic selection of the OPA's gain and phase, the laser detuning, and the input power. A key finding is the existence of a trade-off, where parameters that maximize entanglement also constrain the stable operating regime of the system. Furthermore, thermal noise is shown to progressively degrade all quantum correlations, underscoring the necessity for low-temperature operation. These findings provide comprehensive guidance for parameter optimization, outlining a clear path from generation to enhancement, and highlight the potential of such hybrid systems as versatile platforms for controlling multipartite entanglement in quantum technologies.

Figures

Figures reproduced from arXiv: 2608.03578 by the authors.

Figure 1
Figure 1. Schematic representation of the optomechanical system where H0 = ωca † a + X 3 i=1 ωmi 2 (q 2 i + p 2 i ), (2) Hint = HOM + HC, (3) Hdrive = iEL(a † e −iωLt − aeiωLt ), (4) HOPA = iG  e iθ (a † ) 2 e −2iωLt − e −iθ a 2 e 2iωLt  . (5) The Hamiltonian is composed of several key compo￾nents. H0 describes the free energy of the cavity optical field and the three mechanical oscillators. The interaction Hamiltonian, Hin… view at source ↗

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Works this paper leans on

57 extracted references · 57 canonical work pages

  1. [1]

    Jammer,The Conceptual Development of Quantum Me- chanics.(McGraw-Hill, New York, 1966)

    M. Jammer,The Conceptual Development of Quantum Me- chanics.(McGraw-Hill, New York, 1966)

    work page 1966

  2. [2]

    Pais,Subtle is the Lord: The Science and the Life of Albert Einstein(Oxford University Press, Oxford, 1982)

    A. Pais,Subtle is the Lord: The Science and the Life of Albert Einstein(Oxford University Press, Oxford, 1982)

    work page 1982

  3. [3]

    Aspect,Bell’s Theorem: The Naive View of an Experimen- talist, inQuantum [Un]speakables: From Bell to Quantum Information(Springer, Berlin, 2002)

    A. Aspect,Bell’s Theorem: The Naive View of an Experimen- talist, inQuantum [Un]speakables: From Bell to Quantum Information(Springer, Berlin, 2002)

    work page 2002

  4. [4]

    Einstein, B

    A. Einstein, B. Podolsky and N. Rosen,Phys. Rev.47, 777 (1935)

    work page 1935

  5. [5]

    J. S. Bell,Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, 1987)

    work page 1987

  6. [6]

    J. S. Bell,Physics Physique Fizika1, 195 (1964)

    work page 1964

  7. [7]

    Aspect, P

    A. Aspect, P. Grangier and G. Roger,Phys. Rev. Lett.47, 460 (1981)

    work page 1981

  8. [8]

    Aspect, P

    A. Aspect, P. Grangier and G. Roger,Phys. Rev. Lett.49, 91 (1982)

    work page 1982

Show all 57 references
  1. [9]

    Aspect, J

    A. Aspect, J. Dalibard and G. Roger,Phys. Rev. Lett.49, 1804 (1982)

    1982

  2. [10]

    C. H. Bennett and G. Brassard,Quantum Cryptography: Public Key Distribution and Coin Tossing(Bangalore, 1984)

    1984

  3. [11]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information(Cambridge University Press, Camb- ridge, 2010)

    2010

  4. [12]

    H. J. Kimble,Nature453, 1023 (2008)

    2008

  5. [13]

    A. K. Ekert,Phys. Rev. Lett.67, 661 (1991)

    1991

  6. [14]

    Pathak,Elements of Quantum Computation and Quantum Communication(CRC Press, Boca Raton, 2013)

    A. Pathak,Elements of Quantum Computation and Quantum Communication(CRC Press, Boca Raton, 2013)

    2013

  7. [15]

    Aspelmeyer, T

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

    2014

  8. [17]

    W. P. Bowen and G. J. Milburn,Quantum Optomechanics (CRC Press, Boca Raton, 2015)

    2015

  9. [18]

    C. M. Caves,Phys. Rev. Lett.45, 75 (1980)

    1980

  10. [19]

    et al,Nat

    Xia, Y ., Agrawal, A.R., Pluchar, C.M. et al,Nat. Photon.17, 470–477 (2023)

    2023

  11. [20]

    Ghobadi, A

    R. Ghobadi, A. R. Bahrampour and C. Simon,Phys. Rev. A84, 033846 (2011)

    2011

  12. [21]

    et al.Nat

    Youssefi, A., Kono, S., Chegnizadeh, M. et al.Nat. Phys.19, 1697–1702 (2023)

    2023

  13. [22]

    Chan et al.,Nature478, 89 (2011)

    J. Chan et al.,Nature478, 89 (2011)

    2011

  14. [23]

    Vitali et al.,Phys

    D. Vitali et al.,Phys. Rev. Lett.98, 030405 (2007)

    2007

  15. [24]

    Amazioug, M

    M. Amazioug, M. Nassik and N. Habiballah,Int. J. Quantum Inf.16, 1850043 (2018)

    2018

  16. [25]

    H. Xu, Y . Ma and X. Zhang,Nat. Phys.18, 172 (2022)

    2022

  17. [26]

    J. Li, S. Gr ¨oblacher and M. Poot,Adv. Quantum Technol.4, 2100030 (2021)

    2021

  18. [27]

    G. A. Brawley, M. R. Vanner, P. E. Larsen, S. Schmid, A. Boisen and W. P. Bowen,Nat. Commun.7, 10988 (2016)

    2016

  19. [28]

    T. J. Kippenberg and K. J. Vahala,Science321, 1172 (2008)

    2008

  20. [29]

    Marquardt and S

    F. Marquardt and S. M. Girvin,Physics2, 40 (2009)

    2009

  21. [30]

    Huang and G

    S. Huang and G. S. Agarwal,New J. Phys.11, 103044 (2009)

    2009

  22. [31]

    R. J. Schulze and C. Genes,Phys. Rev. A81, 063810 (2010)

    2010

  23. [32]

    Huang and G

    S. Huang and G. S. Agarwal,Phys. Rev. A79, 013821 (2009)

    2009

  24. [33]

    G. S. Agarwal and S. Huang,Phys. Rev. A93, 043844 (2016)

    2016

  25. [34]

    X. Mi, J. Bai and S. Ke-hui,Eur. Phys. J. D67, 109 (2013)

    2013

  26. [35]

    C. S. Hu, X. R. Huang, L. T. Shen, Z. B. Yang and H. Z. Wu, Eur. Phys. J. D71, 109 (2017)

    2017

  27. [36]

    Xuereb, M

    A. Xuereb, M. Barbieri and M. Paternostro,Phys. Rev. A86, 013809 (2012)

    2012

  28. [37]

    Gao, J.-G

    X.-Y . Gao, J.-G. Liu, G.-W. Wang,Appl. Math. Lett.171, 109615 (2025)

    2025

  29. [38]

    Gao,Appl

    X.-Y . Gao,Appl. Math. Lett.159, 109262 (2025)

    2025

  30. [39]

    Gao,China Ocean Eng.39, 541–547 (2025)

    X.-Y . Gao,China Ocean Eng.39, 541–547 (2025)

    2025

  31. [40]

    Gao, X.-Q

    X.-Y . Gao, X.-Q. Chen, Y .-J. Guo, W.-R. Shan,Qual. Theory Dyn. Syst.24, 73 (2025)

    2025

  32. [41]

    Gao,Chin

    X.-Y . Gao,Chin. J. Phys.92, 1233–1239 (2024)

    2024

  33. [42]

    H.-W. Shan, B. Tian, C.-D. Cheng, X.-T. Gao, Y .-Q. Chen, H.-D. Liu,Qual. Theory Dyn. Syst.23(Suppl. 1), 267 (2024)

    2024

  34. [43]

    C.-H. Feng, B. Tian, X.-T. Gao,Qual. Theory Dyn. Syst.24, 100 (2025)

    2025

  35. [44]

    Wang, Z.-X

    G.-W. Wang, Z.-X. Tan, X.-Y . Gao, J.-G. Liu,Appl. Math. Lett. 172, 109720 (2026) Pramana–J. Phys.(2026)123: #### Page 13 of 12 ####

    2026

  36. [45]

    Gao, W.-G

    X.-Y . Gao, W.-G. Zhao, D.-W. Zuo,Qual. Theory Dyn. Syst. 24, 234 (2025)

    2025

  37. [46]

    J. Q. Zhang, Y . Li, M. Feng and Y . Xu,Phys. Rev. A86, 053806 (2012)

    2012

  38. [47]

    C. H. Bai, D. Y . Wang, H. F. Wang, A. D. Zhu and S. Zhang, Sci. Rep.7, 2545 (2017)

    2017

  39. [48]

    et al,Sci

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

    2023

  40. [49]

    Benguria and M

    R. Benguria and M. Kac,Phys. Rev. Lett.46, 1 (1981)

    1981

  41. [50]

    D. F. Walls and J. D. Harvey,Quantum Optics VI: Proceed- ings of the Sixth International Symposium on Quantum Optics (Springer, 2012)

    2012

  42. [51]

    E. X. DeJesus and C. Kaufman,Phys. Rev. A35, 5288 (1987)

    1987

  43. [52]

    Vitali, S

    D. Vitali, S. Mancini and P. Tombesi,J. Phys. A: Math. Theor. 40, 8055 (2007)

    2007

  44. [53]

    Vidal and R

    G. Vidal and R. F. Werner,Phys. Rev. A65, 032314 (2002)

    2002

  45. [54]

    M. B. Plenio,Phys. Rev. Lett.95, 090503 (2005)

    2005

  46. [55]

    Simon,Phys

    R. Simon,Phys. Rev. Lett.84, 2726 (2000)

    2000

  47. [56]

    Adesso and F

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

    2006

  48. [57]

    H.-D. Liu, B. Tian, Y .-Q. Chen, T.-Y . Zhou, X.-T. Gao,Nonlin- ear Dyn.113, 3655–3672 (2025)

    2025

  49. [58]

    Gao,Appl

    X.-Y . Gao,Appl. Math. Lett.152, 109018 (2024)

    2024

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