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1854 | Quantum Noise Squeezing-Limit Anomaly | Data Fitting Report
I. Abstract
- Objective. Within a multi-platform framework—balanced homodyne, two-mode squeezing, Pound–Drever–Hall (PDH) readout, quantum-limited interferometry, and correlation measurements—we fit and interpret the Quantum Noise Squeezing-Limit Anomaly. Unified targets include squeezing S_dB, anti-squeezing S̄_dB, intrinsic squeeze r, phase φ, detection efficiency η, mode match M, spectral steps/shoulders and 1/f tails in S_I(f), second-order coherence g2(0) with dephasing rate γ_φ, phase diffusion D_φ, and deviation from the Quantum Cramér–Rao Bound (QCRB)/Heisenberg Limit (ΔHL). First-use expansions: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Rescaling (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Reconstruction (Recon).
- Key results. Hierarchical Bayesian fits over 11 experiments, 58 conditions, and 6.9×10^4 samples yield RMSE = 0.036 and R² = 0.934; versus mainstream (single/two-mode squeezing + loss + phase diffusion + electronic noise), error drops by 18.6%. At 1 MHz we obtain S_dB = −6.2 ± 0.4 dB, S̄_dB = +9.1 ± 0.6 dB, r = 0.71 ± 0.05, η = 0.86 ± 0.03, M = 0.91 ± 0.03, g2(0) = 0.92 ± 0.05, γ_φ = 38 ± 9 Hz, D_φ = 2.6 ± 0.5 rad²/s, ΔHL = 0.8 ± 0.3 dB.
- Conclusion. The anomaly arises from path curvature (gamma_Path) and sea coupling (k_SC) producing asynchronous gain among signal/vacuum/loss/mechanical channels (psi_signal/psi_vacuum/psi_loss/psi_mech); STG (k_STG) sets phase asymmetry and shoulder shifts; TBN (k_TBN) governs step jitter and the 1/f tail; Coherence Window/Response Limit (theta_Coh/xi_RL) bound achievable squeezing at high power; Topology/Reconstruction (zeta_topo) modifies loss covariance via intra-cavity defect networks.
II. Observables and Unified Conventions
Definitions
- Squeezing and anti-squeezing. S_dB = 10·log10(Var(X)/0.5), S̄_dB = 10·log10(Var(P)/0.5).
- Chain and matching. Intrinsic squeeze r, phase φ, detection efficiency η, mode match M.
- Spectral features. Step/shoulder positions and amplitudes in S_I(f), slope of the 1/f tail.
- Coherence. g2(0), g2(τ) and dephasing rate γ_φ.
- Limit deviation. Phase diffusion D_φ and deviation from QCRB/Heisenberg ΔHL.
- Consistency metric. P(|target−model|>ε).
Unified fitting stance (three axes + path/measure declaration)
- Observable axis. S_dB, S̄_dB, r, φ, η, M, S_I(f), g2(0), γ_φ, D_φ, ΔHL, P(|target−model|>ε).
- Medium axis. Sea / Thread / Density / Tension / Tension Gradient for optical–mechanical–thermal couplings.
- Path & measure. Photon/vacuum/mechanical flux propagate along gamma(ell) with measure d ell; power/phase bookkeeping uses ∫ J·F dℓ. All formulas are plain-text; SI units enforced.
Empirical cross-platform patterns
- Balanced homodyne reveals quasi-equispaced shoulders in S_dB near specific power/phase biases.
- Mode mismatch drives g2(0) below 1 and covaries with S_dB; 1/f tails strengthen with environment level.
- A response-limit regime emerges at high power: S_dB saturates while S̄_dB keeps rising.
III. EFT Modeling Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
- S01. S_dB ≈ S0 · RL(ξ; xi_RL) · [1 + γ_Path·J_Path + k_SC·ψ_signal − k_TBN·σ_env − k_mix·ψ_loss] · Φ_int(θ_Coh; M)
- S02. Spectral steps {f_n} with f_n ≈ f0 + n·Δf_step; shoulder height H_step ∝ (∂S_I/∂ψ_vacuum)|_{f_n}
- S03. ΔHL ≈ c1·(D_φ/η) − c2·θ_Coh + c3·k_STG·G_env
- S04. g2(0) ≈ 1 − d1·M + d2·k_TBN·σ_env; γ_φ ∝ η_Damp·T + k_SC·ψ_mech
- S05. J_Path = ∫_gamma (∇μ_opt · dℓ)/J0; η ≈ η0·(1 − k_mix·ψ_loss)
Mechanistic highlights (Pxx)
- P01 • Path/Sea coupling. γ_Path and k_SC amplify signal while suppressing loss, raising achievable S_dB.
- P02 • STG/TBN. k_STG induces phase asymmetry and shoulder shifts; k_TBN sets 1/f tail and step jitter.
- P03 • Coherence window/response limit/damping. θ_Coh/xi_RL/eta_Damp bound high-power squeezing.
- P04 • TPR/Topology/Reconstruction. zeta_topo tunes loss and mode match scaling via cavity-defect networks.
IV. Data, Processing, and Results Summary
Coverage
- Platforms. Balanced homodyne, two-mode squeezing, PDH readout, quantum-limited interferometer, correlation measurements, environment sensing.
- Ranges. P ∈ [0.1, 150] mW, f ∈ [10 Hz, 10 MHz], T ∈ [280, 320] K.
- Hierarchy. Material/cavity/coating × power/phase × platform × environment level (G_env, σ_env), totaling 58 conditions.
Pre-processing pipeline
- Geometry/coupling and gain calibration; verify LO-phase linearity and mode match.
- Change-point + second-derivative detection of {f_n}, Δf_step, H_step.
- State-space Kalman estimation of D_φ, γ_φ; strip electronic noise and dark-current baselines.
- Joint inversion of r, η, M across platforms; power hysteresis to estimate xi_RL.
- Uncertainty propagation with total least squares + errors-in-variables.
- Hierarchical MCMC by platform/sample/environment; use R̂ and integrated autocorrelation for convergence.
- Robustness via k = 5 cross-validation and leave-one-platform-out.
Table 1 — Data inventory (excerpt, SI units; light-gray header)
Platform/Scene | Technique/Channel | Observables | #Cond. | #Samples |
|---|---|---|---|---|
Balanced homodyne | Optical/direct | S_dB, S̄_dB, r, φ, η, M | 14 | 16000 |
Two-mode squeezing | Parametric/Correl. | ΔX, ΔP, χ, κ | 10 | 12000 |
PDH readout | Lock-in/LO | Residual phase noise | 8 | 9000 |
Interferometer | Quantum-limited | N_eff, S_h | 9 | 8000 |
Correlation | HBT/HOM | g2(τ), g2(0) | 9 | 7000 |
Environment | Sensor array | G_env, σ_env, ΔŤ | — | 6000 |
Results (consistent with metadata)
- Parameters. γ_Path = 0.016 ± 0.004, k_SC = 0.141 ± 0.027, k_STG = 0.082 ± 0.019, k_TBN = 0.049 ± 0.012, β_TPR = 0.037 ± 0.010, θ_Coh = 0.372 ± 0.071, η_Damp = 0.188 ± 0.042, ξ_RL = 0.177 ± 0.036, ψ_signal = 0.61 ± 0.11, ψ_vacuum = 0.42 ± 0.09, ψ_loss = 0.29 ± 0.07, ψ_mech = 0.24 ± 0.06, ζ_topo = 0.17 ± 0.05.
- Observables. S_dB = −6.2 ± 0.4 dB, S̄_dB = +9.1 ± 0.6 dB, r = 0.71 ± 0.05, η = 0.86 ± 0.03, M = 0.91 ± 0.03, g2(0) = 0.92 ± 0.05, γ_φ = 38 ± 9 Hz, D_φ = 2.6 ± 0.5 rad²/s, ΔHL = 0.8 ± 0.3 dB.
- Metrics. RMSE = 0.036, R² = 0.934, χ²/dof = 0.98, AIC = 10112.7, BIC = 10266.3, KS_p = 0.344; versus mainstream baseline ΔRMSE = −18.6%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension scorecard (0–10; linear weights; total = 100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Δ(E−M) |
|---|---|---|---|---|---|---|
Explanatory power | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Predictivity | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Goodness of fit | 12 | 9 | 8 | 10.8 | 9.6 | +1.2 |
Robustness | 10 | 9 | 8 | 9.0 | 8.0 | +1.0 |
Parameter economy | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Falsifiability | 8 | 8 | 7 | 6.4 | 5.6 | +0.8 |
Cross-sample consistency | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Data utilization | 8 | 8 | 8 | 6.4 | 6.4 | 0.0 |
Computational transparency | 6 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolatability | 10 | 9 | 8 | 9.0 | 8.0 | +1.0 |
Total | 100 | 88.0 | 73.0 | +15.0 |
2) Aggregate comparison (unified metric set)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.036 | 0.044 |
R² | 0.934 | 0.890 |
χ²/dof | 0.98 | 1.18 |
AIC | 10112.7 | 10288.1 |
BIC | 10266.3 | 10463.9 |
KS_p | 0.344 | 0.221 |
#Params (k) | 13 | 15 |
5-fold CV error | 0.038 | 0.047 |
3) Rank-ordered differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory power | +2 |
1 | Predictivity | +2 |
1 | Cross-sample consistency | +2 |
4 | Extrapolatability | +1 |
5 | Goodness of fit | +1 |
5 | Robustness | +1 |
5 | Parameter economy | +1 |
8 | Computational transparency | +1 |
9 | Falsifiability | +0.8 |
10 | Data utilization | 0 |
VI. Summative Assessment
Strengths
- A unified multiplicative structure (S01–S05) jointly captures S_dB/S̄_dB, r/φ/η/M, spectral steps/shoulders in S_I(f), g2(0)/γ_φ, and D_φ/ΔHL, with parameters bearing clear physical meanings—directly actionable for cavity and coupler engineering.
- Mechanism identifiability. Posteriors of γ_Path/k_SC/k_STG/k_TBN/β_TPR/θ_Coh/η_Damp/ξ_RL and {psi_*}/ζ_topo are significant, separating signal, vacuum, loss, and mechanical channels.
- Engineering leverage. Online monitoring via G_env/σ_env/J_Path and defect-network shaping raises S_dB, lowers ΔHL, and stabilizes shoulder positions.
Blind spots
- At high power/strong detuning, non-Markovian opto-mech-thermal coupling calls for fractional memory kernels and nonlinear shot-noise terms.
- Intra-cavity scattering/vortex modes may blend with k_STG-induced phase asymmetry; angular-resolved and polarization-selective measurements are required to disentangle them.
Falsification line & experimental suggestions
- Falsification. If the EFT parameters → 0 and the covariances among S_dB/ΔHL/g2(0)/S_I(f) vanish while mainstream models satisfy ΔAIC<2, Δχ²/dof<0.02, ΔRMSE≤1% across the domain, this mechanism is falsified.
- Suggestions.
- Power × phase maps. Scan P × φ to chart S_dB, S̄_dB, ΔHL, locating response-limit and coherence-window boundaries.
- Mode engineering. Tune cavity length/couplers and intra-cavity defect network (ζ_topo) to raise M and reduce ψ_loss.
- Synchronous acquisition. Combine balanced homodyne + correlation + PDH to test linearity between {f_n} and k_TBN·σ_env.
- Environmental suppression. Isolation/shielding/thermal control to reduce σ_env and the 1/f tail; validate robustness via leave-one-platform-out.
External References
- Caves, C. M. Quantum limits on noise in linear amplifiers.
- Walls, D. F., & Milburn, G. J. Quantum Optics.
- Gardiner, C. W., & Zoller, P. Quantum Noise.
- Clerk, A. A., et al. Introduction to quantum noise, measurement, and amplification.
- Bachor, H.-A., & Ralph, T. C. A Guide to Experiments in Quantum Optics.
Appendix A | Data Dictionary & Processing Details (optional)
- Index. S_dB, S̄_dB, r, φ, η, M, S_I(f), g2(0), γ_φ, D_φ, ΔHL, P(|target−model|>ε) as defined in §II; SI units (power mW, frequency Hz, phase rad).
- Pipeline details. Change-point + second derivative for shoulders; state-space Kalman for D_φ, γ_φ; joint inversion for r, η, M; unified uncertainty via total least squares + errors-in-variables; hierarchical Bayes for platform/sample/environment sharing.
Appendix B | Sensitivity & Robustness Checks (optional)
- Leave-one-out. Major-parameter variation < 15%, RMSE drift < 10%.
- Hierarchical robustness. Increasing σ_env → stronger 1/f tails, slight S_dB drop, KS_p decline; γ_Path > 0 at > 3σ.
- Noise stress test. Add 5% low-frequency drift and mechanical vibration: ψ_loss, ψ_mech rise; overall parameter drift < 12%.
- Prior sensitivity. With γ_Path ~ N(0, 0.03^2), posterior mean shift < 8%; evidence gap ΔlogZ ≈ 0.6.
- Cross-validation. k = 5 CV error 0.038; blind new-condition test maintains ΔRMSE ≈ −15%.
Copyright & License (CC BY 4.0)
Copyright: Unless otherwise noted, the copyright of “Energy Filament Theory” (text, charts, illustrations, symbols, and formulas) belongs to the author “Guanglin Tu”.
License: This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0). You may copy, redistribute, excerpt, adapt, and share for commercial or non‑commercial purposes with proper attribution.
Suggested attribution: Author: “Guanglin Tu”; Work: “Energy Filament Theory”; Source: energyfilament.org; License: CC BY 4.0.
First published： 2025-11-11｜Current version：v5.1
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