Coherent coupling completes an unambiguous optomechanical classification framework

Mikhail Korobko; Roman Schnabel; Xiang Li; Yanbei Chen; Yiqiu Ma

REVIEW 1 major objections 4 minor 87 references

Coherent coupling completes an unambiguous optomechanical classification framework

T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Optomechanical coupling has a third, mutually exclusive type: coherent coupling.

desk verdict A useful taxonomy and a checkable ring-cavity derivation, but the 'unambiguous' claim breaks down in degenerate mode subspaces where the canonical basis is not unique. read the letter →

arxiv 1908.03372 v1 pith:JP6NO4AE submitted 2019-08-09 quant-ph

Xiang Li , Mikhail Korobko , Yiqiu Ma , Roman Schnabel , Yanbei Chen This is my paper · ORCID

classification quant-ph MSC 81V80

keywords optomechanicscoherentcouplingdispersivedissipativeringcavitysidebandcoolingHamiltonianclassificationmodes

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

The paper argues that every optomechanical system can be classified into exactly one of three mutually exclusive coupling types once its Hamiltonian is written in a canonical form that separates optical eigenfrequencies, coupling to the environment, and the eigenmodes themselves. Mechanical oscillation can shift the eigenfrequencies (dispersive coupling), modify the coupling to the outside world (dissipative coupling), or mix the eigenmodes without shifting frequencies (coherent coupling), which is the new category the paper names and defines. If correct, this gives experimenters an unambiguous checklist for identifying what a setup actually does, and it turns coherent coupling into a deliberate design option. The paper then derives a ring-cavity realization from first principles and shows that this coupling can cool a mechanical oscillator faster than an equivalent dispersive cavity under the same input power.

What carries the argument

The load-bearing object is the canonical Hamiltonian of Eq. (6), a multi-mode optomechanical Hamiltonian written in the eigenmode basis with diagonal frequency matrix $\Omega(x)$ and diagonal environment-coupling matrix $\Gamma(x)$, so that all $x$-dependence must appear in and only in $\Omega(x)$, $\Gamma(x)$, or the mode operators $\hat{a}(x)$. The classification step is to inspect this Hamiltonian and read which of the three objects carries the mechanical displacement. For the ring cavity, the machinery is the closed transfer matrix $T_c(k)$ for the circulating fields; its determinant condition yields two resonant modes split by $\omega_s=c\arcsin(r)/L$, whose phase reference is the membrane position, and the $x$-dependent superposition of Eq. (36) converts a displacement into the coherent interaction of Eq. (38).

What would settle it

Find one optomechanical Hamiltonian that yields two different canonical forms assigning the same physical system to different coupling types, for example a system with degenerate optical modes where one natural basis gives $x$-dependent frequencies (dispersive) and another gives $x$-dependent mode mixing (coherent); equivalently, measure in the ring cavity whether the two mode frequencies remain exactly constant while their spatial profiles shift with membrane position, since any frequency shift linear in $x$ at fixed membrane position would contradict the pure-coherent derivation.

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Extended reading notes

Core claim

The central discovery is that optomechanical coupling admits a third irreducible form, coherent coupling, defined by $x$-dependent eigenmodes in the canonical Hamiltonian $\hat{H}(x)=\hbar\hat{a}^\dagger(x)\Omega(x)\hat{a}(x)+i\hbar(\hat{a}^\dagger(x)\Gamma(x)\hat{b}-\mathrm{h.c.})$. The paper claims that any optomechanical Hamiltonian can be brought uniquely into this form, so that all dependence on the mechanical displacement $x$ falls in exactly one of three places: the diagonal frequencies $\Omega(x)$ (dispersive), the environment-coupling rates $\Gamma(x)$ (dissipative), or the eigenmode basis $\hat{a}(x)$ (coherent). For a ring cavity with a movable low-reflectivity membrane, it derives from the cavity transfer matrix that the mechanical displacement couples the two non-degenerate symmetric and antisymmetric modes through $\hat{H}_{\mathrm{int}}(x)=2i\omega_s\hbar k_p x(\hat{c}_-^\dagger\hat{c}_+-\mathrm{h.c.})$, with no frequency shift, which is pure coherent coupling. It then shows that when the mechanical frequency equals the mode splitting, pumping the lower mode makes both the pump and the upper sideband resonant, giving an optical damping rate $\gamma_{\mathrm{opt}}=\Omega_m k_p^2\hbar|A_{\mathrm{in}}|^2/(m\gamma^2)$ and a cooling advantage over a single dispersive cavity whenever the mechanical frequency exceeds the geometric mean of the free spectral range and the linewidth.

Load-bearing premise

The whole classification rests on the assertion, stated without proof, that every optomechanical Hamiltonian can be rewritten uniquely in the diagonal canonical form of Eq. (6); if two equally valid canonical forms placed $x$-dependence in different slots, or if some systems admit no diagonal form, the claim that the classification is unambiguous and exhaustive would fail.

Editorial extensions

If this is right

Every existing optomechanical setup can be assigned a unique coupling type, or a defined coexistence of types, by the same canonical-form procedure, removing the basis-choice ambiguity illustrated in the introduction.
Coherent coupling becomes a design option: any cavity with two non-degenerate modes close enough to be coupled by a mechanical element can realize mode mixing without shifting optical frequencies.
Ring-cavity coherent coupling cools more efficiently than dispersive coupling when the mechanical frequency satisfies $\Omega_m\gg\sqrt{\Delta\omega_{\mathrm{FSR}}\gamma}$, with an explicit example of a 2.5 MHz membrane giving a 2.4 times higher cooling rate at equal length and input power.
The framework explains parametric-instability-type heating in three-mode optoacoustic systems: pumping the upper of two coupled non-degenerate modes makes the lower mechanical sideband resonant and enhances heating, the counterpart of coherent cooling.
Because the classification is Hamiltonian-based, it resolves the seeming contradiction where one physical system looks dispersive in one basis and mode-coupling in another; the canonical-form prescription fixes the answer.

Reading between the lines

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

The uniqueness assumption could be tested by constructing two explicit Hamiltonians that are unitarily equivalent but whose canonical forms put $x$-dependence in different slots; if such a pair exists, the scheme would need an added convention for degenerate subspaces.
The same canonical-form test can be applied to quadratic or nonlinear optomechanical couplings, which the paper explicitly excludes, potentially revealing further coupling species at second order in $x$.
The predicted cooling advantage suggests a direct experiment: hold input power, cavity length, and linewidth fixed, swap a dispersive cavity for the coherent ring cavity, and compare the mechanical temperature; the predicted damping-rate ratio is $R_{\mathrm{rc/sc}}=\Omega_m^4 L_{\mathrm{sc}}^2/(8c^2\gamma^2)$.
For setups with several mechanical degrees of freedom coupled to the same optical pair, the framework would classify each mechanical mode separately, which could expose hybrid dispersive-coherent behavior not visible in single-mode analyses.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know up front. The paper builds something real: a three-channel taxonomy of optomechanical coupling—dispersive, dissipative, coherent—defined by where x-dependence appears in a canonical Hamiltonian, plus a first-principles derivation of a purely coherent interaction in a ring cavity (Eqs. 23–38, Appendix C). I re-checked the resonance condition, the splitting omega_s = c arcsin(r)/L, and the structure of Eq. (38): they are internally consistent, and the interaction H_int = 2i omega_s hbar k_p x (c_-^dagger c_+ - h.c.) has exactly the claimed coherent form with x-independent eigenfrequencies. The cooling comparison is parameter-free and dimensionally consistent, and the physics—coherent coupling lets the pump and upper sideband be resonant simultaneously—is sound. The authors also disclose prior work honestly: Ref. [62] saw eigenmode x-dependence without naming a category, and [64–66] lack the mode definitions this paper supplies.

The soft spot is the one the stress-test flagged, and it lands. Section II asserts the canonical form is 'always unique' without proof, and for exactly degenerate mode subspaces the assertion is false. Take two degenerate modes with no interaction: any x-dependent unitary rotation inside the subspace preserves the diagonal Hamiltonian but makes the basis x-dependent, so Eq. (10) classifies a non-interacting system as coherent. The f_ii(x) ≡ 1 condition doesn't rescue the claim, because at the linear order the framework itself uses, a smooth rotation satisfies it. The paper's motivating example (Eqs. 1–3) is itself a degenerate case; the symmetric/antisymmetric basis choice is a convention, not a derivation. The ring-cavity example uses non-degenerate modes, so the concrete result survives—but the 'unambiguous, basis-independent' framing needs an additional gauge rule for degenerate subspaces, after which the classification claim needs restating.

Second, the stated 2.4x cooling advantage is inconsistent with the paper's own Eq. (52) at the stated parameters (2.5 MHz, 0.01% transmission, 40 cm): the formula gives roughly 9.6x. The qualitative conclusion holds; the number in the text looks like a bookkeeping slip. The linear-regime caveat is disclosed in Section V, so the 'collectively exhaustive' claim is at least honestly scoped.

Verdict: worth a serious referee. I would send it to review, with the central request being a fix to the uniqueness claim and a corrected numerical comparison. Reading group maybe—the flaw makes it a good case for discussing what 'basis-independent' should mean.

Referee Report

1 major / 4 minor

Summary. The paper proposes a classification scheme for optomechanical couplings based on a canonical Hamiltonian form whose x-dependence is supposed to be unique. Within that framework, it identifies a third coupling type, 'coherent coupling', where the mechanical displacement x appears in the optical eigenmodes rather than in the eigenfrequencies or in the coupling to the environment. The paper then derives, from first principles, a ring-cavity system with a movable membrane that exhibits purely coherent coupling, and shows that this system can provide enhanced sideband cooling relative to a standard dispersive single-cavity setup. The ring-cavity derivation is detailed and internally consistent, and the cooling comparison is parameter-free in the sense that all quantities are physical inputs rather than fitted parameters.

Significance. If the classification framework is sound, it provides a useful methodological tool for optomechanical design and introduces a previously unnamed coupling type with a concrete physical realization. The ring-cavity derivation is a genuine contribution: the mode splitting, the interaction Hamiltonian, and the optical damping rate are derived from transfer-matrix boundary conditions without ad hoc assumptions. The paper also explicitly demonstrates a cooling advantage with a dimensionless ratio that depends only on physical parameters, which is a falsifiable prediction. However, the central claim of uniqueness of the canonical form is not proven and, as discussed below, is not correct as stated for degenerate mode subspaces. This gap undermines the 'unambiguous' and 'collectively exhaustive' status of the proposed classification, so the framework needs revision before it can serve as a reliable classification tool.

major comments (1)

[Sec. II, Eqs. (6)-(10)] The assertion that 'the canonical form in Eq. (6) is always unique' is not established and is false without an additional gauge convention for degenerate subspaces. Consider a system of two degenerate cavity modes with H = ħω0(a1†a1 + a2†a2) and no mechanical coupling. The constant basis a_i(x)=a_i(0) gives no coupling. But an x-dependent unitary rotation within the degenerate subspace, U(x)=exp[iθ(x)(a1†a2 + h.c.)] with θ(x)=gx, leaves Ω(x)=ω0 I and Γ(x) diagonal, while ∂a_i'/∂x at x=0 involves the other mode. Eq. (10) then classifies the system as coherent coupling, even though there is no physical optomechanical interaction. The side condition f_ii(x)=1 in Eq. (7) does not remove this ambiguity at linear order, since the diagonal coefficient is 1+O(x^2). Section II provides no rule for choosing the eigenbasis inside degenerate subspaces, and the motivating example (Eqs. 1-3) makes one such choice without a general prescription. Consequently, the claimed mutually exclusive and collectively exhaustive classification is not well-defined for systems with degenerate optical modes, including cases where the degeneracy persists to all orders in x. This is a structural gap in the central claim; it is independent of the ring-cavity derivation, which is detailed and self-consistent, but it directly affects the paper's main promise of an unambiguous classification. The authors should either prove uniqueness under a precise gauge-fixing condition, or restrict the classification to non-degenerate subspaces, or formulate the classification in terms of invariants that do not depend on the choice of basis inside a degenerate subspace.

minor comments (4)

[Sec. IV.B, after Eq. (53)] The numerical example stating that a ring cavity and a single cavity with equal length ~40 cm, front-mirror transmission 0.01%, and mechanical frequency 2.5 MHz give a cooling-rate ratio of 2.4 appears inconsistent with Eq. (52). Using L=L_sc=0.4 m, Ω_m=2π×2.5 MHz, and amplitude transmittance t0=0.01 (power transmittance 10^-4), Eq. (52) yields R≈9.6; if t0 is instead taken as 10^-4, the ratio is on the order of 10^8. Please check the parameter definitions (amplitude versus power transmittance, angular versus cyclic frequency) and the numerical evaluation.
[Sec. II, Eq. (7)] The condition f_ii(x)≡1 in Eq. (7) is ambiguous and does not appear to hold for the paper's own example: the symmetric/antisymmetric basis in Eq. (3) has diagonal coefficients 1/√2 when expanded in the original modes. The authors should clarify whether Eq. (7) is meant as a gauge-fixing condition at all x, only at x=0, or only to linear order in x, and how it is compatible with the example in Eqs. (1)-(3).
[Sec. II, paragraph after Eq. (6)] There is a typo in the sentence 'we consider the last the last possible x-dependence', which should read 'the last possible x-dependence'.
[Fig. 4 caption] The caption of Fig. 4 mentions a 'power and signal-recycled interferometer' and labels BS, SRM, and PRM, but these abbreviations are not defined in the caption or in the main text. Please add definitions or refer to the original source more explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the coherent-coupling classification is definitional, and the ring-cavity derivation is self-contained with no fitted inputs or load-bearing self-citations.

full rationale

The central derivation chain is not circular. The paper defines coherent coupling as x-dependence in the eigenmodes a(x) in Eq. (10) and then derives, from a transfer-matrix model, the ring-cavity interaction Hamiltonian H_int(x) = 2i omega_s hbar k_p x (c_-^dag(0)c_+(0) - h.c.) in Eq. (38). That derivation is parameter-free from the stated physical setup (L, r, t0, etc.); no quantity entering the cooling ratio (Eqs. 51-53) is fitted to the claimed prediction. The taxonomy itself is definitional: given the canonical form of Eq. (6), the three possible x-dependences are a(x), Omega(x), and Gamma(x), and the paper calls the first one coherent coupling. Applying that definition to a system whose eigenmodes are x-dependent is classification, not a circular derivation of a predicted effect. The only structural weakness is the assertion in Section II that 'the canonical form in Eq.(6) is always unique.' This is an unproven well-definedness premise, and the degenerate-subspace example in Eqs. (1)-(3) shows that basis choices inside degeneracies can affect the apparent x-dependence. However, this is a correctness or completeness risk, not a circularity: the paper does not derive uniqueness from the classification, nor does it import it from a self-citation. The cited works, including Refs. [62] and [63] involving the present authors, are used as examples and background, but the classification and the ring-cavity result do not reduce to those citations. Since there is no fitted input called a prediction, no self-citation chain carrying the argument, and no definitional equivalence between the derivation's output and its input, the appropriate finding is no significant circularity.

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

The central claims rest on no fitted parameters: all numerical inputs (L, gamma, Omega_m, t0, r) are physical parameters or design operating points. The axioms above are the structural assumptions the framework and the cooling comparison depend on. No new physical entities are introduced: 'coherent coupling' is a classification label for x-dependence in the eigenmodes, not a new force, particle, or conserved quantity. The resonance condition 2 omega_s = Omega_m used in the cooling comparison is a stated design operating point, counted as an axiom rather than a fitted parameter.

assumptions (6)

domain assumption The canonical Hamiltonian form (Eq. 6) with diagonal Omega(x) and Gamma(x) exists and is unique for every optomechanical system, making the classification mutually exclusive and collectively exhaustive.
Section II asserts 'the canonical form in Eq.(6) is always unique' without proof. At frequency degeneracies (e.g., Eq. 2 with omega1 = omega2) the eigenbasis is non-unique, and the framework gives no explicit rule for tracking the basis through the degeneracy.
domain assumption Restriction to linear x-dependence with x much smaller than the optical wavelength (conventional linear regime).
Section II explicitly limits the framework to the linear regime and excludes quadratic optomechanical coupling and nonlinear optical effects, as acknowledged in Section V.
domain assumption Quasi-static treatment of x: optical relaxation is much faster than the mechanical motion, so x is treated as a parameter when constructing the cavity modes.
Section II states x 'can be treated as quasi-stationary parameters here because the time scale for optical relaxation is much smaller than the mechanical one'; this adiabatic approximation underlies the mode construction in Appendix C.
domain assumption Low-reflectivity membrane (r much less than 1) and low-transmittance front mirror (t0 much less than 1), allowing a two-mode, single-FSR description of the ring cavity.
Section IV A restricts to r much less than 1 so that only the two resonances within one FSR are retained (omega_s much less than Delta_omega_FSR); the linewidth formula gamma = c t0^2/(2L) requires t0 much less than 1.
domain assumption Resolved-sideband condition Omega_m much greater than gamma for the cooling derivation.
Appendix D derives the optical damping rate and the quantum cooling limit 'under the resolved sideband condition Omega_m much greater than gamma'; the cooling advantage comparison inherits this condition.
domain assumption Cooling comparison operates each system at its optimal resolved-sideband operating point (ring: 2 omega_s = Omega_m with pump on the lower mode; single cavity: pump detuned by -Omega_m).
Section IV B states 'both of the two systems are pumped with frequency omega_p' and sets the ring cavity resonance to 2 omega_s = Omega_m; the comparison assumes equal input amplitude, equal gamma, equal Omega_m, and similar round-trip length, which are stated conditions rather than fitted values.

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Pith. "Pith review of Coherent coupling completes an unambiguous optomechanical classification framework." pith.science (2026). https://pith.science/paper/JP6NO4AE

@misc{pith2026190803372,
  author       = {Pith},
  title        = {Pith review of: Coherent coupling completes an unambiguous optomechanical classification framework},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JP6NO4AE}},
  note         = {Machine review of arXiv:1908.03372}
}

read the original abstract

In most optomechanical systems a movable mirror is a part of an optical cavity, and its oscillation modulates either the resonance frequency of the cavity, or its coupling to the environment. There exists the third option -- which we call a "coherent coupling" -- when the mechanical oscillation couples several non-degenerate optical modes supported by the cavity. Identifying the nature of the coupling can be an important step in designing the setup for a specific application. In order to unambiguously distinguish between different optomechanical couplings, we develop a general framework based on the Hamiltonian of the system. Using this framework we give examples of different couplings, and discuss in details one particular case of a purely coherent coupling in a ring cavity with a movable mirror inside. We demonstrate that in certain cases coherent coupling can be beneficial for cooling the motion of the mechanical oscillator. Our general framework allows to approach the design of optomechanical experiments in a methodological way, for precise exploitation of the strengths of particular optomechanical couplings.

Figures

Figures reproduced from arXiv: 1908.03372 by the authors.

**Figure 2.** FIG. 2. Single cavity with a movable end mirror. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. On-chip Optomechanical coupling between the curved [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: FIG. 5. The coupled cavity configuration. [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 6.** Figure 6: FIG. 6. Ring cavity configuration and field labeling. Here [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 8.** Figure 8: FIG. 8. The illustrating plot of [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]

**Figure 9.** Figure 9: FIG. 9. Pumping regime of sideband cooling. Coherent cou [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

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Reference graph

Works this paper leans on

87 extracted references · 72 canonical work pages

[62]

Khorasani, Applied Sciences7, 656 (2017)

S. Khorasani, Applied Sciences7, 656 (2017)

work page 2017
[1]

Unlessclaimedotherwise, we will viewx as a parameter in the following contents as the two counterpropagating ﬁeldsˆe1,2(k) are deﬁned at the instantaneous position of the membrane

Input-output relation We start the rigorous derivation by writing down the input-output relations[23] for the coupling of incoming electromagnetic ﬁeldsˆa1,2(k) of wavenumberk to all in- tracavityﬁeldsshowninFig.6. Unlessclaimedotherwise, we will viewx as a parameter in the following contents as the two counterpropagating ﬁeldsˆe1,2(k) are deﬁned at the i...
[2]

In this case no outside ﬁeld can couple in,i.e

Resonance structure To derive the Hamiltonian, we ﬁrst consider resonant cavity modes assuming a perfectly reﬂective front mir- ror M0. In this case no outside ﬁeld can couple in,i.e. Tin(k;x)≡ 0, and therefore Eq.(C4) becomes: Tc(k)ˆe(k) = ˆ0. (C7) The resonance condition can be obtained from the non- trivial solutions of Eq.(C7) which requires detTc = 0...
[3]

Theoriginofthis z-coordinate isthefrontmirror M0 anditincreasesclockwisealongthe optical axis of the ring cavity

Electric ﬁeld standing wave distribution To quantitatively describe the electric ﬁeld distribu- tion, we construct a coordinate system inside the ring cavity, asshowninFIG.6. Theoriginofthis z-coordinate isthefrontmirror M0 anditincreasesclockwisealongthe optical axis of the ring cavity. It becomeszx =L/2+x at the instantaneous position of the membrane an...
[4]

(C23) Substituting Eq.(C19) in, we can obtain: ˆHcav = ¯hω−ˆc† −ˆc− + ¯hω+ˆc† +ˆc+

Conservative cavity Hamiltonian The cavity Hamiltonian can be obtained from the total optical energy inside the ring cavity[75]: ˆHcav = 2Aϵ0 ∫ L 0 ˆE−(z;x) ˆE+(z;x)dz. (C23) Substituting Eq.(C19) in, we can obtain: ˆHcav = ¯hω−ˆc† −ˆc− + ¯hω+ˆc† +ˆc+. (C24) The cavity Hamiltonian doesn’t havex-dependence be- cause the ring cavity is a closed quantum syst...
[5]

The cav- ity linewidthγ can be obtained from the imaginary part of the pole of Eq.(C5) and it only depends onL and front mirror transmittancet0: γ = ct2 0 2L

Interaction with the environment To complete the total Hamiltonian derivation and re- veal thex-dependence, we consider the coupling of the cavity modes to the outside continuum by assuming the front mirror to have low transmittancet0≪ 1. The cav- ity linewidthγ can be obtained from the imaginary part of the pole of Eq.(C5) and it only depends onL and fro...
[6]

˙ˆc±≡ ∂ˆc±/∂t; ˆc±(in) are used to representˆc±(in)(0);C±≡⟨ ˆc±⟩ are used to represent the expectation value of optical modes, i.e

Coupled optical and mechanical equations of motion For notational convenience, in the following contents, all expressions without explicit arguments are by default in time domain with temporal argument t; all deriva- tives represented by dot are with respect tot, i.e. ˙ˆc±≡ ∂ˆc±/∂t; ˆc±(in) are used to representˆc±(in)(0);C±≡⟨ ˆc±⟩ are used to represent t...
[7]

Based on the Hamiltonian in Eq.(41), the equations of motion forˆc± modes read: ˙ˆc− = 2ωskpˆxˆc+−γˆc− + √ 2γˆc−in, (D1a) ˙ˆc+ =−2iωsˆc+− 2ωskpˆxˆc−−γˆc+ + √ 2γˆc+in

We work in the ro- tating frame with pumping frequencyω−. Based on the Hamiltonian in Eq.(41), the equations of motion forˆc± modes read: ˙ˆc− = 2ωskpˆxˆc+−γˆc− + √ 2γˆc−in, (D1a) ˙ˆc+ =−2iωsˆc+− 2ωskpˆxˆc−−γˆc+ + √ 2γˆc+in. (D1b) The intracavity amplitudes of the two modes are given by the static solutions of Eq.(D1) withˆx = 0: C− = C−in √2γ γ = A1 √γ, ...

Show all 87 references

[8]

Sideband feature and optical damping To solve the coupled optical and mechanical EOMs in Eqs.(D3)(D4a)(D5), we transfer them into frequency do- main[3]. The ˆx-dependenceinsidebandsofeachmode ˆc± is obtained by scattering from the other mode amplitude C∓ and can be expressed a...
[9]

Quantum limit of mechanical occupation number We then calculate the occupation number limit. In the case where the mechanical object is a high-Q-oscillator, we represent the mechanical oscillation in terms of me- chanical creation and annihilation operators ˜m† and ˜m in the r...
[10]

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

2008
[11]

Favero and K

I. Favero and K. Karrai, Nature Photonics3, 201 (2009)

2009
[12]

Aspelmeyer, T

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Reviews of Modern Physics86, 1391 (2014)

2014
[13]

W.P.BowenandG.J.Milburn, Quantum optomechanics (CRC press, 2015)

2015
[14]

Bhattacharya and P

M. Bhattacharya and P. Meystre, Physical Review Let- ters 99, 073601 (2007)

2007
[15]

L.Yong-Chun, H.Yu-Wen, W.C.Wei, andX.Yun-Feng, Chinese Physics B22, 114213 (2013)

2013
[16]

Sawadsky, H

A. Sawadsky, H. Kaufer, R. M. Nia, S. P. Tarabrin, F. Y. Khalili, K.Hammerer, andR.Schnabel,PhysicalReview Letters 114, 043601 (2015)

2015
[17]

T. P. Purdy, P.-L. Yu, R. Peterson, N. Kampel, and C. Regal, Physical Review X3, 031012 (2013)

2013
[18]

Kronwald, F

A. Kronwald, F. Marquardt, and A. A. Clerk, New Jour- nal of Physics16, 063058 (2014)

2014
[19]

Aggarwal, T

N. Aggarwal, T. Cullen, J. Cripe, G. D. Cole, R. Lanza, A.Libson, D.Follman, P.Heu, T.Corbitt, andN.Maval- vala, arXiv preprint arXiv:1812.09942 (2018)

2018 arXiv
[20]

Schnabel, Physics Reports684, 1 (2017)

R. Schnabel, Physics Reports684, 1 (2017)

2017
[21]

S. Bose, K. Jacobs, and P. Knight, Physical Review A 56, 4175 (1997)

1997
[22]

S. Bose, K. Jacobs, and P. L. Knight, Physical Review A 59, 3204 (1999)

1999
[23]

Vitali, S

D. Vitali, S. Gigan, A. Ferreira, H. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. As- pelmeyer, Physical Review Letters98, 030405 (2007)

2007
[24]

H. Miao, S. Danilishin, and Y. Chen, Physical Review A 81, 052307 (2010)

2010
[25]

Mancini, V

S. Mancini, V. Giovannetti, D. Vitali, and P. Tombesi, Physical Review Letters88, 120401 (2002)

2002
[26]

Bhattacharya, P.-L

M. Bhattacharya, P.-L. Giscard, and P. Meystre, Phys- ical Review A77, 030303 (2008)

2008
[27]

M. J. Hartmann and M. B. Plenio, Physical Review Let- ters 101, 200503 (2008)

2008
[28]

Schnabel, Physical Review A92, 012126 (2015)

R. Schnabel, Physical Review A92, 012126 (2015)

2015
[29]

Nation, Physical Review A88, 053828 (2013)

P. Nation, Physical Review A88, 053828 (2013)

2013
[30]

Brunelli, O

M. Brunelli, O. Houhou, D. W. Moore, A. Nunnenkamp, M. Paternostro, and A. Ferraro, Physical Review A98, 063801 (2018)

2018
[31]

E. J. Davis, Z. Wang, A. H. Safavi-Naeini, and M. H. Schleier-Smith, Physical Review Letters 121, 123602 (2018)

2018
[32]

Chen, Journal of Physics B: Atomic, Molecular and Optical Physics46, 104001 (2013)

Y. Chen, Journal of Physics B: Atomic, Molecular and Optical Physics46, 104001 (2013)

2013
[33]

T. P. Purdy, R. W. Peterson, and C. Regal, Science339, 801 (2013)

2013
[34]

Bawaj, C

M. Bawaj, C. Biancoﬁore, M. Bonaldi, F. Bonﬁgli, A. Borrielli, G. Di Giuseppe, L. Marconi, F. Marino, R. Natali, A. Pontin,et al., Nature Communications6, 7503 (2015)

2015
[35]

J. Li, S. Zippilli, J. Zhang, and D. Vitali, Physical Re- view A93, 050102 (2016)

2016
[36]

Belenchia, D

A. Belenchia, D. M. Benincasa, S. Liberati, F. Marin, F. Marino, and A. Ortolan, Physical Review D 95, 026012 (2017)

2017
[37]

Tian and H

L. Tian and H. Wang, Physical Review A82, 053806 (2010)

2010
[38]

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, Nature Communications3, 1196 (2012)

2012
[39]

Ockeloen-Korppi, E

C. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, T. Heikkilä, F. Massel, and M. Sillanpää, Physical Re- view X6, 041024 (2016)

2016
[40]

Lecocq, J

F. Lecocq, J. Clark, R. Simmonds, J. Aumentado, and J. Teufel, Physical Review Letters116, 043601 (2016)

2016
[41]

Huang, Y

J. Huang, Y. Li, L. K. Chin, H. Cai, Y. Gu, M. F. Karim, J. Wu, T. Chen, Z. Yang, Y. Hao,et al., Applied Physics Letters 112, 051104 (2018)

2018
[42]

M. Wu, A. C. Hryciw, C. Healey, D. P. Lake, H. Jayaku- mar, M. R. Freeman, J. P. Davis, and P. E. Barclay, Physical Review X4, 021052 (2014)

2014
[43]

D. E. McClelland, N. Mavalvala, Y. Chen, and R. Schnabel, Laser & Photonics Reviews5, 677 (2011)

2011
[44]

B. P. Abbott et al. (LIGO Scientiﬁc Collaboration and Virgo Collaboration), Physical Review Letters116, 061102 (2016)

2016
[45]

B. P. Abbottet al. (KAGRA Collaboration, LIGO Scien- tiﬁc Collaboration and Virgo Collaboration), Living Re- views in Relativity21, 3 (2018)

2018
[46]

G.M.HarryandtheLIGOScientiﬁcCollaboration,Clas- sical and Quantum Gravity27, 084006 (2010)

2010
[47]

Aasi et al

J. Aasi et al. (The LIGO scientiﬁc Collaboration), Clas- sical and quantum gravity32, 074001 (2015)

2015
[48]

Acernese et al

F. Acernese et al. (The Virgo Collaboration), Classical and Quantum Gravity32, 024001 (2014)

2014
[49]

Acerneseet al

F. Acerneseet al. (The Virgo Collaboration), inJournal of Physics: Conference Series , Vol. 610 (IOP Publishing,
[50]

Lück and the GEO600 Team, Classical and quantum gravity 14, 1471 (1997)

H. Lück and the GEO600 Team, Classical and quantum gravity 14, 1471 (1997)

1997
[51]

Aﬀeldt, K

C. Aﬀeldt, K. Danzmann, K. Dooley, H. Grote, M. He- witson, S.Hild, J.Hough, J.Leong, H.Lück, M.Prijatelj, et al., Classical and quantum gravity31, 224002 (2014)

2014
[52]

Y. Aso, Y. Michimura, K. Somiya, M. Ando, O. Miyakawa, T. Sekiguchi, D. Tatsumi, and H. Ya- mamoto (The KAGRA Collaboration), Physical Review D 88, 043007 (2013)

2013
[53]

Somiya, Classical and Quantum Gravity29, 124007 (2012)

K. Somiya, Classical and Quantum Gravity29, 124007 (2012)

2012
[54]

Kleckner, W

D. Kleckner, W. Marshall, M. J. de Dood, K. N. Dinyari, 18 B.-J. Pors, W. T. Irvine, and D. Bouwmeester, Physical Review Letters96, 173901 (2006)

2006
[55]

Xuereb, R

A. Xuereb, R. Schnabel, and K. Hammerer, Physical Review Letters107, 213604 (2011)

2011
[56]

S. P. Vyatchanin and A. B. Matsko, Physical Review A 93, 063817 (2016)

2016
[57]

F. Y. Khalili, S. P. Tarabrin, K. Hammerer, and R. Schnabel, Physical Review A94, 013844 (2016)

2016
[58]

Xuereb, P

A. Xuereb, P. Horak, and T. Freegarde, Journal of Modern Optics 58, 1342 (2011), https://doi.org/10.1080/09500340.2011.559316

2011
[59]

S.Chesi, Y.-D.Wang, andJ.Twamley,ScientiﬁcReports 5, 7816 EP (2015)

2015
[61]

Law, Physical Review A51, 2537 (1995)

C. Law, Physical Review A51, 2537 (1995)

1995
[63]

B. H. Pang,Theoretical Foundations for Quantum Mea- surement in a General Relativistic Framework,Ph.D.the- sis, California Institute of Technology (2018)

2018
[64]

Thompson, B

J. Thompson, B. Zwickl, A. Jayich, F. Marquardt, S. Girvin, and J. Harris, Nature452, 72 (2008)

2008
[65]

Karuza, M

M. Karuza, M. Galassi, C. Biancoﬁore, C. Molinelli, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, Journal of Optics15, 025704 (2012)

2012
[66]

Xie, C.-G

H. Xie, C.-G. Liao, X. Shang, M.-Y. Ye, and X.-M. Lin, Physical Review A96, 013861 (2017)

2017
[67]

Machado, R

J. Machado, R. Slooter, and Y. M. Blanter, Physical Review A99, 053801 (2019)

2019
[68]

H. Miao, C. Zhao, L. Ju, and D. G. Blair, Physical Re- view A79, 063801 (2009)

2009
[69]

Braginsky, S

V. Braginsky, S. Strigin, and S. P. Vyatchanin, Physics Letters A287, 331 (2001)

2001
[70]

Evans, S

M. Evans, S. Gras, P. Fritschel, J. Miller, L. Barsotti, D. Martynov, A. Brooks, D. Coyne, R. Abbott, R. X. Adhikari, et al. , Physical Review Letters 114, 161102 (2015)

2015
[71]

H. Miao, S. Danilishin, T. Corbitt, and Y. Chen, Phys- ical Review Letters103, 100402 (2009)

2009
[72]

Y. Ma, S. L. Danilishin, C. Zhao, H. Miao, W. Z. Korth, Y. Chen, R. L. Ward, and D. G. Blair, Physical Review Letters 113, 151102 (2014)

2014
[73]

Xuereb, P

A. Xuereb, P. Horak, and T. Freegarde, Journal of Mod- ern Optics58, 1342 (2011)

2011
[74]

Chesi, Y.-D

S. Chesi, Y.-D. Wang, and J. Twamley, Scientiﬁc reports 5, 7816 (2015)

2015
[75]

Yilmaz, S

A. Yilmaz, S. Schuster, P. Wolf, D. Schmidt, M. Eisele, C. Zimmermann, and S. Slama, New Journal of Physics 19, 013038 (2017)

2017
[76]

Nagorny, T

B. Nagorny, T. Elsässer, and A. Hemmerich, Physical Review Letters91, 153003 (2003)

2003
[77]

Kruse, C

D. Kruse, C. von Cube, C. Zimmermann, and P. W. Courteille, Physical Review Letters91, 183601 (2003)

2003
[78]

Elsässer, B

T. Elsässer, B. Nagorny, and A. Hemmerich, Physical Review A69, 033403 (2004)

2004
[79]

Klinner, M

J. Klinner, M. Lindholdt, B. Nagorny, and A. Hem- merich, Physical Review Letters96, 023002 (2006)

2006
[80]

Slama, S

S. Slama, S. Bux, G. Krenz, C. Zimmermann, and P. W. Courteille, Physical Review Letters98, 053603 (2007)

2007
[81]

Ritsch, P

H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, Reviews of Modern Physics85, 553 (2013)

2013
[82]

Schmidt, H

D. Schmidt, H. Tomczyk, S. Slama, and C. Zimmer- mann, Physical Review Letters112, 115302 (2014)

2014
[83]

Mivehvar, S

F. Mivehvar, S. Ostermann, F. Piazza, and H. Ritsch, Physical Review Letters120, 123601 (2018)

2018
[84]

Cheung and C

H. Cheung and C. Law, Physical Review A84, 023812 (2011)

2011
[85]

M. O. Scully and M. S. Zubairy,Quantum Optics (Cam- bridge University Press, 1997)

1997
[86]

W. H. P. Nielsen, Y. Tsaturyan, C. B. Møller, E. S. Polzik, and A. Schliesser, Proceedings of the National Academy of Sciences114, 62 (2017)

2017
[87]

H. Miao, Y. Ma, C. Zhao, and Y. Chen, Physical Review Letters 115, 1 (2015), arXiv:1506.00117v1

2015 arXiv
[88]

J. J. Sakurai and J. Napolitano,Modern quantum me- chanics; 2nd ed. (Addison-Wesley, San Francisco, CA, 2011)

2011

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [62]

Khorasani, Applied Sciences7, 656 (2017)

S. Khorasani, Applied Sciences7, 656 (2017)

work page 2017

[2] [1]

Unlessclaimedotherwise, we will viewx as a parameter in the following contents as the two counterpropagating ﬁeldsˆe1,2(k) are deﬁned at the instantaneous position of the membrane

Input-output relation We start the rigorous derivation by writing down the input-output relations[23] for the coupling of incoming electromagnetic ﬁeldsˆa1,2(k) of wavenumberk to all in- tracavityﬁeldsshowninFig.6. Unlessclaimedotherwise, we will viewx as a parameter in the following contents as the two counterpropagating ﬁeldsˆe1,2(k) are deﬁned at the i...

[3] [2]

In this case no outside ﬁeld can couple in,i.e

Resonance structure To derive the Hamiltonian, we ﬁrst consider resonant cavity modes assuming a perfectly reﬂective front mir- ror M0. In this case no outside ﬁeld can couple in,i.e. Tin(k;x)≡ 0, and therefore Eq.(C4) becomes: Tc(k)ˆe(k) = ˆ0. (C7) The resonance condition can be obtained from the non- trivial solutions of Eq.(C7) which requires detTc = 0...

[4] [3]

Theoriginofthis z-coordinate isthefrontmirror M0 anditincreasesclockwisealongthe optical axis of the ring cavity

Electric ﬁeld standing wave distribution To quantitatively describe the electric ﬁeld distribu- tion, we construct a coordinate system inside the ring cavity, asshowninFIG.6. Theoriginofthis z-coordinate isthefrontmirror M0 anditincreasesclockwisealongthe optical axis of the ring cavity. It becomeszx =L/2+x at the instantaneous position of the membrane an...

[5] [4]

(C23) Substituting Eq.(C19) in, we can obtain: ˆHcav = ¯hω−ˆc† −ˆc− + ¯hω+ˆc† +ˆc+

Conservative cavity Hamiltonian The cavity Hamiltonian can be obtained from the total optical energy inside the ring cavity[75]: ˆHcav = 2Aϵ0 ∫ L 0 ˆE−(z;x) ˆE+(z;x)dz. (C23) Substituting Eq.(C19) in, we can obtain: ˆHcav = ¯hω−ˆc† −ˆc− + ¯hω+ˆc† +ˆc+. (C24) The cavity Hamiltonian doesn’t havex-dependence be- cause the ring cavity is a closed quantum syst...

[6] [5]

The cav- ity linewidthγ can be obtained from the imaginary part of the pole of Eq.(C5) and it only depends onL and front mirror transmittancet0: γ = ct2 0 2L

Interaction with the environment To complete the total Hamiltonian derivation and re- veal thex-dependence, we consider the coupling of the cavity modes to the outside continuum by assuming the front mirror to have low transmittancet0≪ 1. The cav- ity linewidthγ can be obtained from the imaginary part of the pole of Eq.(C5) and it only depends onL and fro...

[7] [6]

˙ˆc±≡ ∂ˆc±/∂t; ˆc±(in) are used to representˆc±(in)(0);C±≡⟨ ˆc±⟩ are used to represent the expectation value of optical modes, i.e

Coupled optical and mechanical equations of motion For notational convenience, in the following contents, all expressions without explicit arguments are by default in time domain with temporal argument t; all deriva- tives represented by dot are with respect tot, i.e. ˙ˆc±≡ ∂ˆc±/∂t; ˆc±(in) are used to representˆc±(in)(0);C±≡⟨ ˆc±⟩ are used to represent t...

[8] [7]

Based on the Hamiltonian in Eq.(41), the equations of motion forˆc± modes read: ˙ˆc− = 2ωskpˆxˆc+−γˆc− + √ 2γˆc−in, (D1a) ˙ˆc+ =−2iωsˆc+− 2ωskpˆxˆc−−γˆc+ + √ 2γˆc+in

We work in the ro- tating frame with pumping frequencyω−. Based on the Hamiltonian in Eq.(41), the equations of motion forˆc± modes read: ˙ˆc− = 2ωskpˆxˆc+−γˆc− + √ 2γˆc−in, (D1a) ˙ˆc+ =−2iωsˆc+− 2ωskpˆxˆc−−γˆc+ + √ 2γˆc+in. (D1b) The intracavity amplitudes of the two modes are given by the static solutions of Eq.(D1) withˆx = 0: C− = C−in √2γ γ = A1 √γ, ...