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arxiv: 2408.14341 · v3 · submitted 2024-08-26 · 🪐 quant-ph · gr-qc

True and apparent motion of optomechanical resonators, with applications to feedback cooling of gravitational wave detector test masses

Evan D. Hall , Kevin Kuns This is my paper

Pith reviewed 2026-05-23 21:58 UTC · model grok-4.3

classification 🪐 quant-ph gr-qc
keywords optomechanicsgravitational wavesfeedback coolingquantum noisesqueezed statestest massesLIGO
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The pith

Feedback cooling of gravitational wave test masses can reach occupation numbers below one quantum.

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

Modern optomechanical systems in gravitational wave detectors use squeezed light and feedback to cool test mass motion. The paper develops an accounting of the true mechanical motion by extending the two-photon formalism to include all loss, feedback effects, and noises. This decomposition helps identify optimal configurations for squeezing and control. Application to detectors like LIGO A+ and Cosmic Explorer shows that, with the common single-oscillator definition, occupation numbers can drop below one over a frequency range matching the oscillator bandwidth. The analysis also covers practical issues in performing such cooling experiments.

Core claim

The true test mass motion, when accounting for feedback and noise in the two-photon formalism, permits occupation numbers below 1 in gravitational wave interferometers for the oscillator definition commonly used in the literature, over a frequency range comparable to the bandwidth of the trapped and cooled oscillator.

What carries the argument

An extension of the two-photon formalism that decomposes the quantum-mechanical noise of the light field into contributions affecting true and apparent motion under feedback control.

If this is right

  • Occupation numbers below 1 become achievable in LIGO A+, LIGO Voyager, Cosmic Explorer, and CE Voyager.
  • Physically motivated parameters guide the choice of optimal squeezed state and feedback configuration for minimal fluctuations.
  • Multi-degree-of-freedom test-mass systems can be meaningfully compared to single degree-of-freedom oscillators.
  • Several technical issues arise in cooling experiments with gravitational-wave detectors that must be addressed.

Where Pith is reading between the lines

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

  • Reaching these occupation levels could allow quantum-limited performance in macroscopic mechanical systems.
  • Future detector designs might incorporate this formalism to push cooling limits further.
  • Similar analysis could apply to other optomechanical setups beyond gravitational wave detectors.

Load-bearing premise

The two-photon formalism and the selected single-oscillator definition together fully capture the effective dynamics and noise coupling once feedback is applied to the multi-degree-of-freedom system.

What would settle it

A direct measurement of the test mass occupation number exceeding 1 across the relevant frequency band in a feedback-cooled gravitational wave detector would contradict the finding.

Figures

Figures reproduced from arXiv: 2408.14341 by Evan D. Hall, Kevin Kuns.

Figure 1
Figure 1. Figure 1: Signal flow graph of a dual-recycled Fabry–Perot Michelson inteferometer with squeezed light injection. The system is [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Reduced signal flow graph for the optomechanical [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: An example budget of the physical test mass motion for a traped and cooled oscillator in LIGO A+, with [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: An example budget of the physical test mass motion for a trapped and cooled oscillator in a cryogenic silicon Cosmic [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Signal flow for the coupling of an auxiliary degree [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Signal flow graph for the radiation pressure dynamics of a single mirror illuminated with a laser from the front surface. [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
read the original abstract

Modern optomechanical systems employ increasingly sophisticated quantum-mechanical states of light to probe and manipulate mechanical motion. Squeezed states are now used routinely to enhance the sensitivity of gravitational-wave interferometers to small external forces, and they are also used in feedback-based trapping and damping experiments on the same interferometers to enhance the achievable cooling of fluctuations in the differential test mass mode (arXiv:2102.12665). In this latter context, an accurate accounting of the true test mass motion, incorporating all sources of loss, the effect of feedback control, and the influence of classical force and sensing noises, is paramount. We work within the two-photon formalism to provide such an accounting, which extends a previously described decomposition of the quantum-mechanical noise of the light field (arxiv:2105.12052). This decomposition provides insight, rooted in physically motivated parameters, into the optimal squeezed state and feedback control configuration that should be employed to achieve the lowest fluctuations. We apply this formalism to feedback damping experiments in current and possible future gravitational-wave interferometers -- LIGO A+, LIGO Voyager, Cosmic Explorer (CE), and CE Voyager -- and discuss how these multi-degree-of-freedom systems might be compared to a single degree-of-freedom oscillator. We find that, for the oscillator definition used most commonly in the literature so far, occupation numbers below 1 are possible in these interferometers over a frequency range comparable to the bandwidth of the trapped and cooled oscillator. We also discuss several technical issues in cooling experiments with gravitational-wave detectors

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

2 major / 1 minor

Summary. The manuscript extends the authors' prior two-photon formalism (arXiv:2105.12052) for decomposing quantum noise in optomechanical systems to explicitly incorporate optical losses and feedback control. It applies the extended framework to feedback damping of test-mass motion in current and future gravitational-wave interferometers (LIGO A+, LIGO Voyager, Cosmic Explorer, CE Voyager), discusses the mapping of these multi-DOF systems onto the single-oscillator definition most common in the literature, and reports that thermal occupation numbers below 1 remain achievable over a frequency interval comparable to the bandwidth of the cooled oscillator.

Significance. If the single-oscillator reduction is shown to be faithful, the work supplies a parameter-motivated route to optimizing squeezed-light and feedback configurations for sub-quantum-limited cooling in multi-mode GW detectors and quantifies the technical issues that must be addressed when comparing such systems to idealized single-DOF oscillators.

major comments (2)
  1. [Applications section] Applications section: the headline result (occupation numbers <1 over a bandwidth comparable to the cooled oscillator) is obtained by reducing the closed-loop, multi-DOF test-mass system to the standard single-oscillator definition. No explicit numerical check is supplied that residual cross-talk between longitudinal/angular modes or differential/common modes, mediated by the shared feedback loop and sensing noises, remains below the claimed noise floor across the quoted frequency range.
  2. [Two-photon formalism extension] Two-photon formalism extension: the new occupation-number spectra are computed entirely within the extended decomposition of arXiv:2105.12052 rather than against independent external benchmarks or full multi-mode simulations; this makes the absence of an explicit validation of the neglected channels load-bearing for the central claim.
minor comments (1)
  1. [Applications section] The precise mathematical definition of the single-oscillator mapping (including which quadratures are retained and which are projected out) should be stated explicitly before the numerical results are presented.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and constructive feedback on our manuscript. We address each major comment below and propose revisions to strengthen the presentation of the multi-DOF reduction and the validation of the extended formalism.

read point-by-point responses

  1. Referee: [Applications section] Applications section: the headline result (occupation numbers <1 over a bandwidth comparable to the cooled oscillator) is obtained by reducing the closed-loop, multi-DOF test-mass system to the standard single-oscillator definition. No explicit numerical check is supplied that residual cross-talk between longitudinal/angular modes or differential/common modes, mediated by the shared feedback loop and sensing noises, remains below the claimed noise floor across the quoted frequency range.

    Authors: We agree that an explicit numerical check of residual cross-talk would strengthen the central claim. The manuscript discusses the mapping of the multi-DOF system onto the single-oscillator definition via projection of the sensing and feedback noises, but does not include a dedicated numerical validation across the quoted band. In the revised version we will add a new subsection (or appendix) with numerical estimates for LIGO A+, Voyager, CE and CE Voyager, showing that cross-talk contributions from angular/longitudinal and differential/common couplings remain below the reported occupation-number floor over the relevant frequency interval. This will be computed using the same two-photon noise decomposition already employed in the paper. revision: yes

  2. Referee: [Two-photon formalism extension] Two-photon formalism extension: the new occupation-number spectra are computed entirely within the extended decomposition of arXiv:2105.12052 rather than against independent external benchmarks or full multi-mode simulations; this makes the absence of an explicit validation of the neglected channels load-bearing for the central claim.

    Authors: The extended decomposition is obtained by direct substitution of the lossy input-output relations and the feedback loop into the two-photon quadratures of the prior work; the resulting spectra are therefore exact within the linear, Gaussian, and single-mode-per-port approximations stated in the manuscript. We acknowledge that the absence of an external benchmark makes validation of the neglected channels (higher-order spatial-mode couplings, nonlinearities, and non-Gaussian effects) important. In revision we will (i) explicitly recover the lossless, open-loop limit and show agreement with the known single-oscillator result of arXiv:2105.12052, and (ii) add a short discussion quantifying why the neglected channels lie below the quantum-noise floor under the parameter regimes of current and future GW detectors. A full multi-mode time-domain simulation is beyond the scope of the present analytic framework, but the added limiting-case checks will address the load-bearing concern. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation applies extended formalism to new configurations

full rationale

The paper extends the two-photon decomposition from arXiv:2105.12052 to account for noise, loss, feedback, and multi-DOF effects in GW interferometers, then applies the resulting framework to compute occupation numbers for LIGO A+, Voyager, CE, and CE Voyager. The headline result (occupation numbers <1 over a bandwidth comparable to the cooled oscillator) is obtained by explicit calculation under the most common single-oscillator definition in the literature, using physically motivated parameters. No quoted step reduces a prediction to a fitted input by construction, renames a known result, or makes the central claim depend solely on an unverified self-citation chain. The self-citation supplies the starting formalism but does not force the numerical findings; the work remains externally falsifiable via the stated detector parameters and the chosen oscillator definition. This is the normal case of a self-contained calculation built on prior independent work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract provides insufficient detail to enumerate free parameters, axioms, or invented entities; the central claim rests on the validity of the two-photon formalism extension and the chosen oscillator definition, both inherited from prior work.

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