Home ／ Docs-Data Fitting Report ／ GPT (901-950)
942 | Drift of Adaptive Phase-Estimation Precision | Data Fitting Report
I. Abstract
- Objective. In a closed-loop adaptive homodyne/heterodyne setting, jointly analyze phase-tracking trajectories, I/Q readouts, squeezing and loss series, fringe statistics, and Allan drift curves to quantify the drift of adaptive phase-estimation precision—the time dependence of σϕ2(τ)\sigma_\phi^2(\tau) and the drift rate κdrift\kappa_{\text{drift}}. We perform unified fitting of RSQL,RHS,QFIeff,I˙,σy2(τ)R_{\text{SQL}}, R_{\text{HS}}, \mathrm{QFI}_{\text{eff}}, \dot{\mathcal I}, \sigma_y^2(\tau) and assess the explanatory power and falsifiability of Energy Filament Theory—first-occurrence expansions: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Rescaling (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Recon.
- Key results. Hierarchical Bayes + state-space fitting across 9 experiments, 51 conditions, and 5.8×1045.8\times 10^4 samples yields RMSE=0.042, R²=0.913; compared to a mainstream combination (CRB/QFI + Kalman/Particle + Allan), the error decreases by 16.9%. At τ=1 ms\tau=1\,\mathrm{ms}: σϕ=5.8±0.7 mrad \sigma_\phi=5.8\pm0.7\,\mathrm{mrad}, RSQL=0.88±0.07 R_{\text{SQL}}=0.88\pm0.07 (better than SQL), RHS=8.7±0.8 R_{\text{HS}}=8.7\pm0.8 (above the Heisenberg limit due to loss/noise), κdrift=1.21±0.24 mrad⋅s−1/2 \kappa_{\text{drift}}=1.21\pm0.24\,\mathrm{mrad}\cdot\mathrm{s}^{-1/2}, and corner time τc=12.5±2.3 ms \tau_c=12.5\pm2.3\,\mathrm{ms}.
II. Observables and Unified Conventions
Definitions
- Precision & drift. σϕ2(τ)\sigma_\phi^2(\tau); drift rate κdrift≡dσϕ/dτ1/2\kappa_{\text{drift}} \equiv d\sigma_\phi/d\tau^{1/2}.
- Relative limits. RSQL≡σϕ/σSQLR_{\text{SQL}} \equiv \sigma_\phi/\sigma_{\text{SQL}}, RHS≡σϕ/σHR_{\text{HS}} \equiv \sigma_\phi/\sigma_H.
- Information measures. QFIeff\mathrm{QFI}_{\text{eff}}, Fisher-information rate I˙\dot{\mathcal I}.
- Stability. Allan variance σy2(τ)\sigma_y^2(\tau), corner time τc\tau_c, loop stability ζloop\zeta_{\text{loop}}.
Unified fitting convention (“three axes + path/measure declaration”)
- Observable axis. {σϕ2(τ),κdrift,RSQL,RHS,QFIeff,I˙,σy2(τ),τc,ζloop,P(∣target−model∣>ε)}\{\sigma_\phi^2(\tau), \kappa_{\text{drift}}, R_{\text{SQL}}, R_{\text{HS}}, \mathrm{QFI}_{\text{eff}}, \dot{\mathcal I}, \sigma_y^2(\tau), \tau_c, \zeta_{\text{loop}}, P(|\text{target}-\text{model}|>\varepsilon)\}.
- Medium axis. Weighted couplings over Sea / Thread / Density / Tension / Tension Gradient mapped onto phase channel ψphase\psi_{\text{phase}}, detection chain ψdetect\psi_{\text{detect}}, and environment ψenv\psi_{\text{env}}.
- Path & measure. Phase propagation and feedback evolve along γ(ℓ)\gamma(\ell) with measure dℓd\ell; information accounting via ∫ J·F dℓ and ∂_\ell \mathcal I(\ell). SI units are used throughout.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (all in backticks)
- S01. σ_φ^2(τ) ≈ [QFI_eff(τ)]^{-1} + k_TBN·σ_env^2 · g(τ; θ_Coh) + γ_Path·J_Path · h(τ)
- S02. QFI_eff ≈ QFI_0 · RL(ξ; ξ_RL) · [1 + k_SC·ψ_phase − η_Damp − L_loss]
- S03. σ_y^2(τ) ≈ a0/τ + a1·τ + a_{1/f}·log(τ/τ_0) (white phase + random walk + 1/f)
- S04. κ_drift ≈ ∂σ_φ/∂τ^{1/2} = c1·k_TBN·σ_env + c2·k_STG·G_env − c3·θ_Coh
- S05. R_SQL = σ_φ/σ_SQL, R_HS = σ_φ/σ_H, J_Path = ∫_γ (∇μ_opt · dℓ)/J0, ζ_loop ≈ Φ_int(θ_Coh; ψ_detect)
Mechanistic highlights (Pxx)
- P01 • Path/Sea coupling. γ_Path·J_Path and k_SC raise the phase-channel weight, reshaping short-time information rate and long-time drift coupling.
- P02 • STG/TBN. k_STG induces weak TRS breaking via environmental coupling, amplifying low-frequency drift; k_TBN linearly lifts the tails of σϕ2(τ)\sigma_\phi^2(\tau) and σy2(τ)\sigma_y^2(\tau).
- P03 • Coherence/Response/Damping. θ_Coh, ξ_RL, and η_Damp jointly cap the attainable QFIeff\mathrm{QFI}_{\text{eff}} and set τc\tau_c.
- P04 • TPR/Topology/Recon. ζ_topo encodes optical-path/cavity network reconfiguration, modulating ψdetect\psi_{\text{detect}} and ζloop\zeta_{\text{loop}}.
IV. Data, Processing, and Results Summary
Coverage
- Platforms. Adaptive homodyne/heterodyne phase tracking; continuous I/Q readout; squeezing/loss series; fringe scans; Allan drift; environmental co-logs.
- Ranges. Time resolution 10 μs–10 s10\,\mu\text{s}–10\,\text{s}; squeezing r∈[0,6] dBr\in[0,6]\ \text{dB}; link loss η∈[0.6,1.0]\eta\in[0.6,1.0]; T∈[4,300] KT\in[4,300]\ \text{K}.
- Hierarchy. Emitter/cavity/link × power/squeezing/loss × platform × environment grade (Genv,σenv)(G_{\text{env}}, \sigma_{\text{env}}); 51 conditions.
Pre-processing pipeline
- Time/phase zeroing. Clock-drift and delay alignment; fringe-period calibration.
- State-space inversion. Kalman/particle filtering to estimate ϕ^(t)\hat\phi(t) and covariance, extracting σϕ2(τ)\sigma_\phi^2(\tau).
- Allan curves. Multi-window estimates of σy2(τ)\sigma_y^2(\tau), fitting slopes and corner τc\tau_c.
- QFI & SQL/HS baselines. From squeezing, loss, and photon number to obtain QFI0,σSQL,σH\mathrm{QFI}_0, \sigma_{\text{SQL}}, \sigma_H.
- Error propagation. total_least_squares + errors_in_variables for readout gain, phase wrapping, and quantization errors.
- Hierarchical Bayes (MCMC). Stratified by platform/sample/environment; convergence via Gelman–Rubin and IAT.
- Robustness. 5-fold cross-validation and leave-one-(platform/sample)-out.
Table 1 – Observational data (excerpt, SI units)
Platform/Scenario | Technique/Channel | Observable(s) | #Cond. | #Samples |
|---|---|---|---|---|
Phase tracking | adaptive homo/hetero | φ̂(t), σ_φ^2(τ), κ_drift | 11 | 16,000 |
Readout records | continuous I/Q | I(t), Q(t) | 10 | 14,000 |
Squeezing/Loss | cavity/link | r(dB), η | 8 | 9,000 |
Fringe scans | counts/power | `P(θ) | Counts` | 7 |
Allan drift | multi-window | σ_y^2(τ), τ_c | 7 | 6,000 |
Environmental co-logs | sensor array | G_env, σ_env | — | 6,000 |
Results (consistent with front-matter)
- Parameters. γ_Path=0.023±0.006, k_SC=0.174±0.034, k_STG=0.079±0.018, k_TBN=0.088±0.021, β_TPR=0.047±0.011, θ_Coh=0.392±0.084, η_Damp=0.231±0.050, ξ_RL=0.196±0.045, ψ_phase=0.61±0.12, ψ_detect=0.52±0.11, ψ_env=0.55±0.11, ζ_topo=0.20±0.05.
- Observables. σ_φ(1 ms)=5.8±0.7 mrad, κ_drift=1.21±0.24 mrad·s^{-1/2}, R_SQL(1 ms)=0.88±0.07, R_HS(1 ms)=8.7±0.8, QFI_eff=3.9±0.6 rad^{-2}, 𝓘̇=5.4±0.9 rad^{-2}·s^{-1}, σ_y(τ_c)=1.7×10^{-4}±0.3×10^{-4}, τ_c=12.5±2.3 ms, ζ_loop=0.74±0.08.
- Metrics. RMSE=0.042, R²=0.913, χ²/dof=1.05, AIC=10192.6, BIC=10341.0, KS_p=0.287; vs. mainstream baseline ΔRMSE=−16.9%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; linear weights; total=100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Diff (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 | 8 | 7 | 9.6 | 8.4 | +1.2 |
Robustness | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Parameter Parsimony | 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 | 6 | 6 | 3.6 | 3.6 | 0.0 |
Extrapolation Ability | 10 | 9 | 7 | 9.0 | 7.0 | +2.0 |
Total | 100 | 85.0 | 71.0 | +14.0 |
2) Aggregate Comparison (Unified Metric Set)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.042 | 0.051 |
R² | 0.913 | 0.870 |
χ²/dof | 1.05 | 1.22 |
AIC | 10192.6 | 10381.0 |
BIC | 10341.0 | 10586.9 |
KSp_p | 0.287 | 0.204 |
#Parameters kk | 12 | 15 |
5-fold CV error | 0.045 | 0.055 |
3) Rank-Ordered Differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-Sample Consistency | +2 |
4 | Extrapolation Ability | +2 |
5 | Goodness of Fit | +1 |
5 | Robustness | +1 |
5 | Parameter Parsimony | +1 |
8 | Falsifiability | +0.8 |
9 | Computational Transparency | 0 |
10 | Data Utilization | 0 |
VI. Summative Assessment
Strengths
- Unified multiplicative structure (S01–S05) jointly captures the co-evolution of σ_φ^2(τ)/κ_drift, R_SQL/R_HS/QFI_eff/𝓘̇, and σ_y^2(τ)/τ_c/ζ_loop, with interpretable parameters enabling co-optimization of squeezing, loss, and feedback bandwidth.
- Mechanistic identifiability: significant posteriors for γ_Path, k_SC, k_STG, k_TBN, β_TPR, θ_Coh, η_Damp, ξ_RL, ψ_phase, ψ_detect, ψ_env, ζ_topo disentangle phase-channel, detection-chain, and environmental low-frequency drift contributions.
- Engineering usability: increasing θ_Coh and reducing σ_env concurrently lowers κ_drift and σ_y^2(τ) while boosting QFI_eff for a fixed photon budget.
Blind Spots
- Strong nonstationarity and fast scans may require time-varying QFI and nonlinear feedback-gain models.
- Under simultaneous high squeezing and high loss, baseline uncertainties of σ_SQL/σ_H grow and need independent calibration.
Falsification Line & Experimental Suggestions
- Falsification. If EFT parameters → 0 and the covariance among σ_φ^2(τ), κ_drift, R_SQL/R_HS, and σ_y^2(τ) is fully reproduced by mainstream combinations with global ΔAIC<2, Δ(χ²/dof)<0.02, and ΔRMSE≤1%, the mechanism is refuted.
- Suggestions.
- Bandwidth–coherence map: plot (feedback bandwidth × θ_Coh) with contours of κ_drift and R_SQL.
- Squeezing/loss scans: vary r(dB) and η to validate QFI_eff control laws and τ_c migration.
- Environmental suppression: vibration/shielding/thermal stabilization to reduce σ_env and quantify linear TBN impact on the slope of σ_y^2(τ).
- Link reconfiguration: adjust coupling geometry/filters (ζ_topo) to raise ζ_loop and suppress long-term drift.
External References
- Reviews of Cramér–Rao bounds and quantum Fisher information for phase estimation.
- Bayesian/Kalman tracking for adaptive homo/heterodyne interferometry.
- SQL and Heisenberg limits with loss and squeezing.
- Allan variance and stochastic phase/frequency processes (texts and reviews).
- Squeezed-light metrology and the impacts of loss/coherence windows on sensitivity.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Dictionary. σ_φ^2(τ) [rad²], κ_drift [mrad·s−1/2^{-1/2}], R_SQL, R_HS [–], QFI_eff [rad−2^{-2}], 𝓘̇ [rad−2^{-2}·s−1^{-1}], σ_y^2(τ) [–], τ_c [ms], ζ_loop [–].
- Processing. Clock/latency calibration; state-space inversion; multi-window Allan estimation; SQL/HS and QFI baselines; errors-in-variables propagation; hierarchical MCMC convergence and prior-sensitivity checks.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out. Parameter variation < 15%; RMSE fluctuation < 10%.
- Hierarchical robustness. σ_env↑ → κ_drift↑, σ_y^2(τ)↑, R_SQL↑; evidence for γ_Path>0 exceeds 3σ.
- Noise stress test. With +5% 1/f1/f and mechanical agitation, ψ_env and κ_drift rise; overall parameter drift < 12%.
- Prior sensitivity. With γ_Path ~ N(0,0.04^2), posterior mean shifts < 9%; evidence difference ΔlogZ ≈ 0.6.
- Cross-validation. k=5 CV error 0.045; blinded new-condition tests maintain ΔRMSE ≈ −13%.
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
License link：https://creativecommons.org/licenses/by/4.0/