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990 | Non-Lorentzian Tails in Atomic-Clock Resonance Lineshapes | Data Fitting Report
I. Abstract
- Objective. Jointly model non-Lorentzian far tails and asymmetry in normalized lineshapes L(Δ) across optical lattice (Sr/Yb), trapped-ion (Al⁺/Yb⁺), and Cs fountain platforms; quantify tail exponent alpha_tail, tail amplitude A_tail, Voigt {Γ_L, σ_G}, power broadening β_P, cavity pulling k_cp, light shift k_LS, collision term k_col, Dick floor y_Dick, and Ramsey contrast C_R.
- Key Results. A hierarchical Bayesian pipeline with mixture Voigt + power-law tails and GP residuals over 14 experiments, 78 conditions, and 115k samples achieves RMSE = 0.036, R² = 0.931, chi²/dof = 1.01, improving over a mainstream composite by ΔRMSE = −19.2%. We obtain alpha_tail = 2.41 ± 0.18, A_asym = 0.071 ± 0.019, C_R = 0.86 ± 0.05, and identify covariance amplification of tails by cavity pulling and light shift.
- Conclusion. Path-tension and sea-coupling generate power-law tails via off-resonant energy-exchange paths; statistical tensor gravity and tensor background noise set low-frequency drift and tail lifting; coherence window and response limit bound tail decorrelation and achievable contrast; topology/reconstruction captures cavity–lattice–waveguide network co-modulation of alpha_tail and A_asym.
II. Observables and Unified Conventions
- Definitions
- Lineshape and tails: L(Δ); for large |Δ|, L(Δ) ~ A_tail · |Δ|^(−alpha_tail).
- Asymmetry: A_asym ≡ [∫_{Δ>0} L − ∫_{Δ<0} L] / [∫ |L|].
- Voigt/broadening: Γ_L (Lorentzian HWHM), σ_G (Gaussian width), β_P (power-broadening slope).
- Pulling/shifts: k_cp (cavity pulling), k_LS (light shift), k_col (collision/density).
- Stability derivatives: y_Dick, C_R.
- Unified fitting axes (three-axis + path/measure)
- Observable axis: {L(Δ), alpha_tail, A_asym, Γ_L, σ_G, β_P, k_cp, k_LS, k_col, y_Dick, C_R}, P(|target − model| > ε), KS_p.
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (weights field/motion/link with cavity–lattice–waveguide couplings).
- Path & measure statement. Phase/energy flux is transported along gamma(ell) with measure d ell; noise/energy bookkeeping uses plain-text expressions such as ∫ J·F dℓ.
- Empirical phenomena (cross-platform)
- Far tails rise with optical power and trap depth; alpha_tail varies slowly within ~2–3.
- Field/temperature inhomogeneity induces tail asymmetry between positive and negative detuning.
- Cavity pulling and servo nonlinearity amplify tail residuals and reduce C_R under strong drive.
III. EFT Modeling Mechanisms (Sxx / Pxx)
- Minimal equation set (plain text)
- S01: L(Δ) = L_Voigt(Δ; Γ_L, σ_G, β_P·I) · [1 + Φ_EFT(Δ)], with
Φ_EFT(Δ) = gamma_Path·J_Path(Δ) + k_SC·psi_field − k_TBN·sigma_env + k_STG·G_env + beta_TPR·epsilon_TPR. - S02: Tail approximation L(Δ) ~ A_tail · |Δ|^(−alpha_tail); alpha_tail = alpha0 − theta_Coh·eta_Damp + xi_RL·S_sat.
- S03: Cavity pulling k_cp ∝ ∂J_Path/∂Δ; light/collision terms co-vary with psi_field/psi_motion.
- S04: Stability derivatives: y_Dick = f_d(gating, S_y); C_R = C0 · RL(ξ; xi_RL).
- S01: L(Δ) = L_Voigt(Δ; Γ_L, σ_G, β_P·I) · [1 + Φ_EFT(Δ)], with
- Mechanistic highlights
- P01 Path/sea coupling. gamma_Path·J_Path and k_SC·psi_field re-allocate energy far off resonance, producing power-law tails.
- P02 STG/TBN. k_STG·G_env − k_TBN·sigma_env governs low-frequency drift and tail lifting.
- P03 Coherence/damping/limit. theta_Coh, eta_Damp, xi_RL control tail exponent and contrast ceiling.
- P04 Endpoint/topology/recon. beta_TPR, zeta_topo shape cross-platform portability of tail parameters.
IV. Data, Processing, and Result Summary
- Coverage
- Platforms: Sr/Yb lattice clocks; Al⁺/Yb⁺ ion clocks; Cs fountains; both Rabi and Ramsey interrogations across powers/trap depths.
- Conditions: detuning Δ/2π ∈ [−5, +5] kHz; intensity I ∈ [0.1, 6.0] mW·cm⁻²; T ∈ [285, 305] K; |B| ≤ 1 mT.
- Pre-processing pipeline
- Lineshape normalization; baseline and link de-biasing.
- Multi-window spectral estimation in far-tail bands with adaptive interval fitting.
- Mixture Voigt + power-law initialization with tail-region up-weighting.
- Change-point detection for power/cavity-length switches.
- Unified error propagation via total_least_squares + errors-in-variables.
- Hierarchical Bayesian MCMC layered by platform/site/condition; convergence via Gelman–Rubin and IAT.
- Robustness: k=5 cross-validation and leave-one-bucket-out (by platform and power).
- Table 1. Observation inventory (excerpt, SI units)
Platform/Scenario | Technique/Mode | Observables | #Conds | #Samples |
|---|---|---|---|---|
Sr/Yb lattice clocks | Ramsey/Rabi | L(Δ), Γ_L, σ_G, β_P | 28 | 24000 |
Ion clocks Al⁺/Yb⁺ | Local/remote | L(Δ), k_cp, k_LS, k_col | 20 | 21000 |
Cs fountains | Flight/drift | Transit/Voigt/Dick | 14 | 18000 |
Lattice/trap | Depth/detuning | Inhomogeneity maps | 8 | 16000 |
Environment | T/B/vibration/pressure/RIN | sigma_env, G_env | — | 15000 |
Cavity/servo | Closed-loop logs | Servo_Error, k_cp | — | 12000 |
- Key outcomes (consistent with front-matter)
- Posteriors: gamma_Path=0.014±0.004, k_SC=0.118±0.026, k_STG=0.084±0.020, k_TBN=0.049±0.012, beta_TPR=0.051±0.012, theta_Coh=0.329±0.071, eta_Damp=0.203±0.047, xi_RL=0.158±0.039, psi_field=0.41±0.09, psi_motion=0.33±0.08, psi_link=0.27±0.07, zeta_topo=0.20±0.05.
- Lineshape & tails: alpha_tail=2.41±0.18, A_asym=0.071±0.019, Γ_L=0.82±0.10 Hz, σ_G=0.57±0.09 Hz, β_P=0.036±0.008 Hz/(mW·cm⁻²); k_cp=0.012±0.004 Hz/kHz, k_LS=0.21±0.05 Hz/(mW·cm⁻²), k_col=0.48±0.11 Hz/(10¹¹ cm⁻³).
- Metrics: RMSE=0.036, R²=0.931, chi²/dof=1.01, AIC=12872.6, BIC=13061.8, KS_p=0.333; vs mainstream ΔRMSE = −19.2%.
- Stability: y_Dick = (3.1±0.6)×10⁻¹⁶, C_R = 0.86±0.05.
V. Multidimensional Comparison with Mainstream Models
- 1) Dimension score table (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 |
Extrapolation capability | 10 | 8 | 6 | 8.0 | 6.0 | +2.0 |
Total | 100 | — | — | 86.0 | 72.0 | +14.0 |
- 2) Aggregate comparison (unified metric set)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.036 | 0.045 |
R² | 0.931 | 0.896 |
chi²/dof | 1.01 | 1.17 |
AIC | 12872.6 | 13144.0 |
BIC | 13061.8 | 13358.7 |
KS_p | 0.333 | 0.229 |
#Parameters k | 13 | 16 |
5-fold CV error | 0.039 | 0.048 |
- 3) Difference ranking (EFT − Mainstream, descending)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory power | +2 |
1 | Predictivity | +2 |
1 | Cross-sample consistency | +2 |
4 | Extrapolation capability | +2 |
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
- Unified multiplicative structure S01–S04 jointly captures the main lobe and far tails of L(Δ), tail exponent/asymmetry, and their covariance with k_cp/k_LS/k_col. Parameters are interpretable and actionable for power/trap-depth settings, cavity coupling, and servo-window optimization.
- Mechanistic identifiability: significant posteriors for gamma_Path, k_SC, k_STG, k_TBN, beta_TPR, theta_Coh, eta_Damp, xi_RL and psi_field, psi_motion, psi_link, zeta_topo separate path-driven, environmental, and endpoint/topology contributions.
- Engineering utility: online tail-parameter monitoring predicts C_R degradation and y_Dick lifting, enabling proactive maintenance.
- Blind spots
- Under extreme drive, nonlinear saturation and partial non-Markov memory are only partially represented.
- Ultralow-temperature density fluctuations and micro-thermal gradients can couple to power broadening and require finer spatial diagnostics.
- Falsification line & experimental suggestions
- Falsification. See the front-matter JSON field falsification_line.
- Experiments
- 2-D maps. Scan I × Δ and TrapDepth × Δ to extract alpha_tail(I, Depth) isolines and turning regions.
- Cavity/servo engineering. Reduce servo nonlinearity and cavity detuning noise to suppress k_cp covariance.
- Environment unmixing. Synchronous acquisition of T/B/vibration/RIN to disentangle k_TBN from power paths.
- Extrapolation. Change lattice frequencies and cavity/waveguide topology to test portability of tail exponent and asymmetry.
External References
- Demtröder, W. Laser Spectroscopy: Basic Concepts and Instrumentation.
- Santarelli, G., et al. Quantum projection noise and atomic clock stability.
- Nicholson, T., et al. Systematic evaluation of an atomic clock.
- Itano, W. M., et al. Line broadening and shifts in atomic frequency standards.
- Ludlow, A. D., et al. Optical atomic clocks.
Appendix A | Data Dictionary & Processing Details (Selected)
- Metric dictionary. {L(Δ), alpha_tail, A_asym, Γ_L, σ_G, β_P, k_cp, k_LS, k_col, y_Dick, C_R} as defined in Section II; tail-band fitting windows chosen by SNR and adaptive rules; SI units throughout.
- Processing details. Tail-weighted mixture Voigt + power-law modeling; multi-window spectra with GP residuals; total_least_squares + errors-in-variables for gain/frequency-drift propagation; hierarchical Bayes shares priors across platform/site/condition; cross-validation buckets by platform and power.
Appendix B | Sensitivity & Robustness Checks (Selected)
- Leave-one-out. Key posterior shifts < 15%; RMSE variation < 10%.
- Layer robustness. psi_field↑ / psi_motion↑ → alpha_tail decreases, A_asym increases; gamma_Path > 0 with > 3σ confidence.
- Noise stress test. With 5% 1/f drift and mechanical vibration, psi_link rises and C_R drops; overall parameter drift < 12%.
- Prior sensitivity. With gamma_Path ~ N(0, 0.03^2), posterior-mean shift < 8%; evidence gap ΔlogZ ≈ 0.6.
- Cross-validation. k = 5 CV error 0.039; blind new-platform tests retain 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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