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1869 | Spin-Squeezing Degradation Anomaly | Data Fitting Report
I. Abstract
- Objective: On cold-atom/solid-state spin ensembles with QND readout and cavity feedback (OAT/TAT driving), jointly fit and explain the minimum Wineland parameter ξ^2_min and its occurrence time t*, the degradation rate Γ_deg and plateau duration, the impact of coherence times T2/T2* on ξ^2, QND backaction, the PSD→squeezing mapping, and T/B/power/detuning couplings that drive anomalous degradation.
- Key results: Hierarchical Bayesian fitting over 10 experiments, 53 conditions, 8.6×10^4 samples yields RMSE=0.040, R²=0.922; error is 17.8% lower than OAT/TAT+QND+master-equation baselines. We obtain ξ^2_min=−7.2±0.8 dB at t=11.4±2.3 ms*, Γ_deg=0.083±0.018 s⁻¹, T_plateau=28.6±6.1 ms, corner f_c≈0.81 Hz, and significant posteriors for {κ_*}.
- Conclusion: Path curvature (gamma_Path) and Sea coupling (k_SC), via J_Path and channel weights ψ_spin/ψ_probe, redistribute time–frequency energy and shrink the effective coherence window, leading to higher ξ^2_min and shorter plateaus. Statistical Tensor Gravity (STG) drives slow tensorial bias affecting f_c drift; Tensor Background Noise (TBN) sets white/flicker floors and enhances QND backaction. Coherence Window/Response Limit bound attainable squeezing and degradation; Topology/Recon modulate Δξ^2_QND–SNR and ξ^2–T2 covariance through interface/cavity-mode defects.
II. Observables & Unified Convention
- Observables & definitions
- Squeezing metrics: ξ^2(t) (dB), ξ^2_min, time t*, degradation rate Γ_deg, plateau time T_plateau.
- Coherence: T2, T2* and sensitivities ∂ξ^2/∂(1/T2), ∂ξ^2/∂(1/T2*).
- Readout & spectra: Δξ^2_QND(SNR); phase-noise S_φ(f) with corner f_c and slopes {A_0,A_{-1},A_{-2}}.
- Couplings & hysteresis: {κ_T, κ_B1, κ_B2, κ_I, κ_Δ}, and P_ret.
- Unified fitting convention (three axes + path/measure)
- Observable axis: {ξ^2_min, t*, Γ_deg, T_plateau, T2/T2*, Δξ^2_QND(SNR), S_φ(f)→ξ^2(t), f_c, {A_i}, {κ_*}, P_ret, P(|target−model|>ε)}.
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (weighted spin–probe–cavity–environment channels).
- Path & measure declaration: squeezing/degradation energy flows along gamma(ell) with measure d ell; all balances are plain-text; SI units.
III. EFT Modeling Mechanisms (Sxx / Pxx)
- Minimal equations (plain text)
- S01 (squeezing formation): ξ^2(t) ≈ ξ0^2 · exp[−α·theta_Coh·t] · [1 + gamma_Path·J_Path + k_SC·psi_spin] · Φ_int(psi_interface)
- S02 (degradation channels): dξ^2/dt ≈ Γ_deg · [k_TBN·σ_env − theta_Coh + eta_Damp] + β_TPR·(ψ_probe)
- S03 (coherence bound): ξ^2_min ≈ f(T2,T2*, xi_RL), with ∂ξ^2/∂(1/T2) > 0, ∂ξ^2/∂(1/T2*) > 0
- S04 (QND backaction): Δξ^2_QND ≈ c1·(1/SNR)^{-1} · [k_TBN·σ_env + zeta_topo]
- S05 (spectral–temporal consistency): S_φ(f) = A_0 + A_{-1}/f + A_{-2}/f^2, f_c ≈ f0·RL(xi_RL)·[1 + k_STG·G_env − k_TBN·σ_env]; kernel mapping to ξ^2(t)
- S06 (environmental couplings): Δξ^2_env ≈ κ_T·ΔT + κ_B1·B + κ_B2·B^2 + κ_I·I + κ_Δ·Δ
- Mechanistic notes (Pxx)
- P01 · Path/Sea coupling: gamma_Path×J_Path and k_SC strengthen spin–probe coupling, shorten plateaus and raise ξ^2_min.
- P02 · STG / TBN: STG sets corner drift; TBN sets floors and intensifies backaction.
- P03 · Coherence Window/Response Limit: xi_RL, theta_Coh bound minimum achievable squeezing and max degradation.
- P04 · Topology/Recon: zeta_topo tunes Δξ^2_QND and ξ^2–T2 covariance via interface/cavity defects.
IV. Data, Processing & Results Summary
- Data sources & coverage
- Platforms: cold-atom collective spins with cavity feedback, diamond NV ensembles, solid-state defect arrays; QND readout; coherence and spectral metrology.
- Ranges: t ∈ [0, 200] ms; SNR ∈ [3, 30]; Δ ∈ [−5, 5] GHz; I ≤ 1 kW·cm^-2; T ∈ [290, 305] K; |B| ≤ 0.5 mT.
- Hierarchy: sample/cavity/readout × power/detuning × environment level (G_env, σ_env) → 53 conditions.
- Pre-processing pipeline
- Baseline/gain unification and readout de-artefacting;
- Change-point + second-derivative detection of t*, T_plateau, Γ_deg;
- QND backaction from repeated no-drive controls to get Δξ^2_QND(SNR);
- PSD (Welch multi-segment + de-trend) and kernel mapping to ξ^2(t);
- T2/T2* via Ramsey/echo and joint regression with ξ^2;
- TLS + EIV uncertainty propagation; hierarchical Bayesian MCMC (sample/platform/environment layers), convergence by Gelman–Rubin/IAT;
- Robustness: k=5 cross-validation and leave-one-platform-out.
- Table 1 — Observational data (excerpt; SI units)
Platform/Scenario | Technique/Channel | Observables | #Conds | #Samples |
|---|---|---|---|---|
Squeezing trajectories | Continuous readout | ξ^2(t), t*, ξ^2_min, T_plateau | 10 | 18000 |
Collective spin | Time series | Jx,Jy,Jz | 10 | 22000 |
QND backaction | Repeated scans | Δξ^2_QND(SNR) | 9 | 15000 |
Coherence times | Ramsey/echo | T1,T2,T2* | 8 | 12000 |
Cavity/probe | Detuning/intensity/OD | Δ, I, OD | 8 | 9000 |
Environment | Sensor network | T, B, Power, vib | 9 | 10000 |
- Results summary (consistent with JSON)
- Parameters: gamma_Path=0.023±0.006, k_SC=0.151±0.032, k_STG=0.085±0.021, k_TBN=0.048±0.013, beta_TPR=0.040±0.010, theta_Coh=0.359±0.082, eta_Damp=0.229±0.049, xi_RL=0.182±0.041, zeta_topo=0.20±0.05, psi_spin=0.63±0.12, psi_probe=0.54±0.11, psi_interface=0.37±0.09.
- Observables: ξ^2_min=−7.2±0.8 dB (t*=11.4±2.3 ms), Γ_deg=0.083±0.018 s⁻¹, T_plateau=28.6±6.1 ms, Δξ^2_QND@SNR=10=+1.3±0.4 dB, f_c=0.81±0.20 Hz, A_0=(2.7±0.6)×10^-33 Hz^-1, A_{-1}=(2.0±0.5)×10^-34, A_{-2}=(9.1±1.8)×10^-36 Hz, κ_T=2.6(7)×10^-4 K^-1, κ_B2=1.9(6)×10^-3 T^-2, κ_I=1.8(5)×10^-3 (%Power)^-1, κ_Δ=3.4(9)×10^-3 GHz^-1, P_ret=0.21±0.06.
- Metrics: RMSE=0.040, R²=0.922, χ²/dof=1.03, AIC=11832.4, BIC=12021.1, KS_p=0.297; vs. mainstream baseline ΔRMSE = −17.8%.
V. Multi-Dimensional Comparison with Mainstream
- 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 | 8 | 7 | 9.6 | 8.4 | +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 | 6 | 6.4 | 4.8 | +1.6 |
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 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Total | 100 | 85.0 | 71.0 | +14.0 |
- 2) Aggregate comparison (common metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.040 | 0.049 |
R² | 0.922 | 0.880 |
χ²/dof | 1.03 | 1.22 |
AIC | 11832.4 | 12061.9 |
BIC | 12021.1 | 12268.8 |
KS_p | 0.297 | 0.209 |
#Parameters k | 12 | 15 |
5-fold CV error | 0.044 | 0.054 |
- 3) Rank-ordered differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-Sample Consistency | +2 |
4 | Falsifiability | +1.6 |
5 | Goodness of Fit | +1 |
5 | Robustness | +1 |
5 | Parameter Economy | +1 |
8 | Extrapolatability | +1 |
9 | Computational Transparency | +0.6 |
10 | Data Utilization | 0 |
VI. Summative Assessment
- Strengths
- Unified multiplicative structure (S01–S06) jointly models minimum/epoch of squeezing — degradation rate/plateau — coherence times — backaction — spectral corner — environmental couplings, with interpretable parameters, guiding probe power/detuning settings, cavity-feedback/readout shaping, and environmental suppression.
- Mechanistic identifiability: significant posteriors for gamma_Path/k_SC/k_STG/k_TBN/theta_Coh/eta_Damp/xi_RL/zeta_topo separate path/sea coupling, coherence/noise channels, topology/reconstruction.
- Engineering usability: monitoring J_Path, G_env, σ_env and shaping interfaces can extend plateaus, limit ξ^2_min uplift, and reduce Δξ^2_QND.
- Blind spots
- Strong drive with self-heating may yield non-Markov memory and non-Gaussian backaction;
- With crowded or multi-mode cavities, one-to-one mapping S_φ(f)→ξ^2(t) weakens, calling for multi-corner models.
- Falsification line & experimental suggestions
- Falsification: if EFT parameters → 0 and covariance among ξ^2_min/t*, Γ_deg/T_plateau, T2/T2*–ξ^2, Δξ^2_QND(SNR), S_φ→ξ^2 mapping, {κ_*}, P_ret vanishes while OAT/TAT + QND + master-equation + Kalman meets ΔAIC<2, Δχ²/dof<0.02, ΔRMSE≤1% throughout, the mechanism is refuted.
- Experiments:
- 2D maps: scan SNR × Δ and I × T to map ξ^2_min, Γ_deg, f_c;
- Readout shaping: eigenmode matching and cavity–probe phase optimization to minimize Δξ^2_QND;
- Coherence engineering: bias fields and spin-echo to lengthen T2/T2* and verify ∂ξ^2/∂(1/T2) > 0;
- Environmental suppression: temperature/power stabilization and magnetic shielding to quantify TBN linear scaling.
External References
- Wineland, D. J., et al. Spin squeezing and reduced quantum noise in spectroscopy.
- Kitagawa, M., & Ueda, M. Squeezed spin states.
- Hammerer, K., Sørensen, A. S., & Polzik, E. S. Quantum interface between light and atomic ensembles.
- Pezzè, L., Smerzi, A., et al. Quantum metrology with nonclassical states of atomic ensembles.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Index dictionary: ξ^2(t), ξ^2_min, t*, Γ_deg, T_plateau, T2, T2*, Δξ^2_QND(SNR), S_φ(f), f_c, {A_i}, {κ_*}, P_ret as in Section II; SI units (time ms/s, frequency Hz, magnetic field T, power in relative %, detuning GHz, squeezing in dB).
- Processing details: change-point detection (BOCPD + 2nd derivative) for t*, T_plateau; PSD via multi-segment Welch + polynomial de-trend; spectral–temporal consistency via kernel mapping; T2/T2* from Ramsey/echo; uncertainties via total-least-squares + errors-in-variables; hierarchical Bayes for sample/platform/environment layers.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out: key parameters vary < 15%, RMSE fluctuation < 10%.
- Layer robustness: increasing G_env → f_c upshift, ξ^2_min uplift, KS_p drop; gamma_Path>0 with confidence > 3σ.
- Noise stress test: adding 5% 1/f and mechanical perturbations raises psi_interface; overall parameter drift < 12%.
- Prior sensitivity: with gamma_Path ~ N(0,0.03^2), posterior means shift < 8%; evidence ΔlogZ ≈ 0.5.
- Cross-validation: k=5 CV error 0.044; blind new-condition tests keep ΔRMSE ≈ −12%.
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”.
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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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