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arxiv: 2508.11098 · v2 · submitted 2025-08-14 · 🌀 gr-qc · astro-ph.IM· math-ph· math.MP· quant-ph

Theoretical Detailed Analyses for DC readout and a Fabri-P\'erot gravitational-wave detector

Kouji Nakamura This is my paper

Pith reviewed 2026-05-18 22:12 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.IMmath-phmath.MPquant-ph
keywords gravitational wave detectorDC readoutFabry-Pérot interferometershot noiseradiation pressurequantum noiseHeisenberg equations
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The pith

In DC readout for a Fabry-Pérot gravitational-wave detector, shot noise does not fall with rising laser power because classical radiation pressure from the carrier leaks to the output port.

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

The paper derives the quantum expectation values and stationary noise spectral density for a Fabry-Pérot interferometer operated with DC readout, using only quantum electrodynamics of lasers and the Heisenberg equations for the mirrors. It shows that the usual reduction of high-frequency shot noise with increased injected power fails to appear. The failure occurs because the carrier field, which serves as the reference in DC readout, leaks classical radiation-pressure forces into the output port; these forces act as a constant load that shifts the mirrors away from their equilibrium positions. The analysis demonstrates that initial mirror conditions are confined to the pendulum frequency and therefore do not affect the observation band, and it examines how incomplete adjustment of the tuning point produces the observed non-ideal behavior.

Core claim

Using the Heisenberg equations of motion for the mirrors together with the quantum electrodynamics of the laser fields, the stationary noise spectral density is obtained. In the derived spectrum the high-frequency shot-noise term remains independent of injected laser power. This independence arises because classical radiation-pressure forces generated by the carrier field leak through to the output port; since the carrier is used as the local reference in the DC scheme, these forces appear as a constant offset that displaces the mirrors from the exact equilibrium point required for ideal cancellation.

What carries the argument

Leakage of classical radiation-pressure forces from the carrier field to the output port, which shifts the pendulum equilibrium and prevents the expected power-dependent reduction of shot noise in the DC readout scheme.

If this is right

  • Tuning the interferometer so that the mirrors sit exactly at their radiation-pressure-shifted equilibrium positions restores the ideal reduction of shot noise with laser power.
  • Any residual offset from equilibrium produces a shot-noise floor that no longer improves with higher power.
  • The maximum mirror displacement from equilibrium that still allows near-ideal performance can be calculated from the incomplete-tuning analysis.

Where Pith is reading between the lines

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

  • Current gravitational-wave detectors that rely on DC readout may need auxiliary control loops to keep the mirrors at the radiation-pressure-adjusted equilibrium if they are to reach the theoretical shot-noise limit.
  • The same leakage effect could appear in any interferometer that uses a strong carrier as both reference and signal carrier.

Load-bearing premise

The initial conditions of the mirrors' motion remain concentrated around the pendulum's fundamental frequency and do not contribute to the frequency band of interest.

What would settle it

Measure the high-frequency shot-noise contribution in an actual DC-readout Fabry-Pérot interferometer while deliberately detuning the operating point away from the equilibrium position; if the noise level stays constant with increasing laser power, the leakage mechanism is confirmed.

Figures

Figures reproduced from arXiv: 2508.11098 by Kouji Nakamura.

Figure 1
Figure 1. Figure 1: FIG. 1. The setup for the Fabry-P´erot interferometer inclu [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Arm propagation in the Fabry-P´erot interferometer [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The ratio [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The ratio [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The square root of the noise spectral density [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The square root of the noise spectral density [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The square root ratio of the noise spectral density [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The schematic picture of the incomplete equilibrium [PITH_FULL_IMAGE:figures/full_fig_p034_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The square-root of the explicit signal referred noi [PITH_FULL_IMAGE:figures/full_fig_p035_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The square-root of the explicit signal-referred no [PITH_FULL_IMAGE:figures/full_fig_p037_11.png] view at source ↗
read the original abstract

The quantum expectation value and the stationary noise spectral density for a Fabry-P'erot gravitational-wave detector with a DC readout scheme are discussed in detail only through the quantum electrodynamics of lasers and the Heisenberg equations of mirrors' motion. We demonstrate that the initial conditions of the mirrors' motion concentrate around the fundamental frequency of the pendulum and are not related to the frequency range of our interest. Although, in the ideal case, there is consensus that the shot-noise contribution from the laser to the high-frequency range of the signal-referred noise spectral density decreases as the injected laser power increases, our derived noise spectral density shows that the shot-noise contribution does not decrease. This is due to leakage of classical radiation pressure forces from the carrier field to the output port, and the carrier field is used as the reference in the DC readout scheme. Since classical radiation pressure acts as a constant force, it shifts the pendulum's equilibrium point of the mirrors' motion. To recover the ideal case, we must consider adjusting the interferometer's tuning point to place the mirrors at their equilibrium positions. We investigate the case where the equilibrium tuning is incomplete and show that the behavior of the above shot noise is due to this incompleteness. We also discuss the maximum deviation of the mirror displacements from the equilibrium point during incomplete tuning to recover a near-ideal case.

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 / 2 minor

Summary. The manuscript derives the quantum expectation value and stationary noise spectral density for a Fabry-Pérot gravitational-wave detector with DC readout using quantum electrodynamics and the Heisenberg equations of mirror motion. It shows that mirror initial conditions concentrate around the pendulum frequency and are unrelated to the GW band of interest. The central claim is that, contrary to the ideal-case consensus, the shot-noise contribution to the high-frequency signal-referred noise spectral density does not decrease with increasing laser power; this is attributed to leakage of classical radiation pressure forces from the carrier field to the output port when equilibrium tuning is incomplete. The paper discusses recovery of near-ideal behavior by adjusting the tuning point and quantifies maximum mirror deviations under incomplete tuning.

Significance. If the result is correct, it would indicate that incomplete tuning in DC readout schemes can eliminate the expected 1/sqrt(P) improvement in shot noise, with direct implications for noise budgeting and tuning tolerances in current and future gravitational-wave detectors. The first-principles QED-plus-Heisenberg approach and the explicit treatment of incomplete-tuning deviations are strengths that could make the work useful for detector modeling if the frequency-domain mechanism is clarified.

major comments (2)
  1. [section presenting the derived noise spectral density] The manuscript asserts that the derived noise spectral density shows shot noise failing to decrease with power due to carrier leakage, yet the explicit NSD expression and the algebraic steps converting the constant classical force into a high-frequency contribution are not isolated. A static force F_class ~ P produces only a static displacement x_0 = F_class/k_pendulum and a δ(f) term; it is unclear how this modifies the shot-noise scaling at frequencies ≫ pendulum resonance without altering the GW signal transfer function or the effective carrier amplitude at the dark port.
  2. [section on initial conditions of mirrors' motion] The claim that initial conditions concentrate around the pendulum frequency and are independent of the GW frequency range is stated as a result of the Heisenberg equations, but the explicit solution, boundary conditions, or frequency-domain filtering that demonstrates this independence is not shown. This assumption is load-bearing for separating the classical leakage effect from the quantum fluctuation spectrum of interest.
minor comments (2)
  1. [Title] Title: 'Fabri-Pérrot' is misspelled; the standard term is 'Fabry-Pérot'.
  2. [Abstract] The abstract refers to 'our derived noise spectral density' without an equation number; adding a reference to the relevant equation would improve traceability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments identify points where additional explicit derivations and clarifications would strengthen the presentation. We respond to each major comment below and will incorporate the requested details in a revised version.

read point-by-point responses

  1. Referee: [section presenting the derived noise spectral density] The manuscript asserts that the derived noise spectral density shows shot noise failing to decrease with power due to carrier leakage, yet the explicit NSD expression and the algebraic steps converting the constant classical force into a high-frequency contribution are not isolated. A static force F_class ~ P produces only a static displacement x_0 = F_class/k_pendulum and a δ(f) term; it is unclear how this modifies the shot-noise scaling at frequencies ≫ pendulum resonance without altering the GW signal transfer function or the effective carrier amplitude at the dark port.

    Authors: We agree that the explicit NSD expression and the algebraic steps linking the classical force to the high-frequency noise were not isolated with sufficient clarity. The radiation-pressure force shifts the mirror equilibrium; when the interferometer is not retuned to this new position, a residual carrier leaks to the output port. This leakage alters the local oscillator amplitude at the dark port in a power-dependent manner, producing an additive noise term whose scaling cancels the usual 1/sqrt(P) improvement in shot noise at GW frequencies. The static displacement itself contributes only a DC term, but the resulting detuning modifies the readout quadrature without changing the GW signal transfer function. In the revised manuscript we will add a dedicated subsection that isolates the full NSD expression, shows the algebraic conversion from the constant force to the high-frequency contribution, and explicitly demonstrates the frequency-domain mechanism through the altered carrier amplitude. revision: yes

  2. Referee: [section on initial conditions of mirrors' motion] The claim that initial conditions concentrate around the pendulum frequency and are independent of the GW frequency range is stated as a result of the Heisenberg equations, but the explicit solution, boundary conditions, or frequency-domain filtering that demonstrates this independence is not shown. This assumption is load-bearing for separating the classical leakage effect from the quantum fluctuation spectrum of interest.

    Authors: The referee correctly notes that the explicit solution of the Heisenberg equations and the demonstration of frequency independence were not displayed. Solving the mirror equations with initial conditions corresponding to the ground (or thermal) state at t = 0 yields a displacement spectrum whose power is concentrated near the pendulum resonance; the mechanical susceptibility suppresses contributions at frequencies well above resonance. Consequently, the initial-condition transients do not affect the GW band. In the revision we will insert the explicit time-domain solution, state the boundary conditions, and show the frequency-domain filtering that isolates the quantum noise spectrum from the classical leakage term. revision: yes

Circularity Check

0 steps flagged

Derivation from QED and Heisenberg equations is self-contained with no circular reduction

full rationale

The paper explicitly starts from quantum electrodynamics of lasers combined with the Heisenberg equations for mirror motion to obtain both the quantum expectation value and the stationary noise spectral density. The statement that initial conditions concentrate around the pendulum frequency is derived as a direct consequence of those equations and is not introduced as a fitted parameter or ansatz tuned to the target high-frequency spectrum. The subsequent analysis of shot-noise behavior under incomplete equilibrium tuning follows from the same first-principles model without any self-definitional loop, fitted-input renaming, or load-bearing self-citation. No step reduces the claimed NSD result to an input by construction; the derivation therefore remains independent of the final noise expressions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis rests on standard quantum electrodynamics for the laser field and classical mechanics for the mirrors, plus one domain assumption about the frequency content of initial mirror motion.

axioms (1)
  • domain assumption Initial conditions of the mirrors' motion concentrate around the fundamental frequency of the pendulum and are not related to the frequency range of interest
    Invoked to justify that pendulum-frequency transients do not contaminate the high-frequency noise spectrum of interest.

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