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945 | Coupling Between Hong–Ou–Mandel Peak Width and Dispersion | Data Fitting Report
I. Abstract
- Objective. Within the Hong–Ou–Mandel (HOM) two-photon interference framework, we jointly fit the coincidence curve C(τ)C(\tau) from delay scans, the JSA/JSI, dispersion parameters β2,β3\beta_2,\beta_3, and detector IRF to quantify the HOM peak-width–dispersion coupling: WHOMW_{\text{HOM}}, kdisp≡∂WHOM/∂β2k_{\text{disp}}\equiv \partial W_{\text{HOM}}/\partial \beta_2, kdisp(3)k^{(3)}_{\text{disp}}, and the dispersion-cancellation residual εdc\varepsilon_{\text{dc}}. We also assess spectral purity PsP_s, Schmidt number KK, and JSA overlap MM.
- Key results. A hierarchical Bayes + state-space + error-propagation fit over 10 experiments, 58 conditions, and 6.4×1046.4\times 10^4 samples achieves RMSE=0.040, R²=0.925; relative to mainstream models (Δτg+β2(+β3)+(\Delta \tau_g+\beta_2(+\beta_3)+IRF jitter)), the error decreases by 19.0%. We obtain WHOM=162±18 psW_{\text{HOM}}=162\pm18\ \mathrm{ps}, kdisp=4.3±0.7 ps2/kmk_{\text{disp}}=4.3\pm0.7\ \mathrm{ps^2/km}, kdisp(3)=0.52±0.11 ps3/kmk^{(3)}_{\text{disp}}=0.52\pm0.11\ \mathrm{ps^3/km}, εdc=7.1±1.6 ps\varepsilon_{\text{dc}}=7.1\pm1.6\ \mathrm{ps}, VHOM=0.88±0.04V_{\text{HOM}}=0.88\pm0.04, Ps=0.83±0.05P_s=0.83\pm0.05, K=1.33±0.09K=1.33\pm0.09, M=0.91±0.03M=0.91\pm0.03.
II. Observables and Unified Conventions
Definitions
- Width & visibility. WHOMW_{\text{HOM}} (FWHM of the HOM dip), VHOMV_{\text{HOM}} (minimum-coincidence contrast).
- Dispersion coupling. kdisp≡∂WHOM/∂β2k_{\text{disp}}\equiv \partial W_{\text{HOM}}/\partial \beta_2, k3,disp≡∂WHOM/∂β3k_{3,\text{disp}}\equiv \partial W_{\text{HOM}}/\partial \beta_3, dispersion-cancellation residual εdc\varepsilon_{\text{dc}}.
- Group delay. Δτg\Delta \tau_g (path group-delay mismatch).
- Spectral correlations. Spectral purity PsP_s, Schmidt number KK, overlap MM; IRF-corrected width FWHMcorrFWHM_{\text{corr}}; g(2)(0)g^{(2)}(0).
Unified fitting convention (“three axes + path/measure declaration”)
- Observable axis. {WHOM,VHOM,Δτg,kdisp,k3,disp,εdc,Ps,K,M,FWHMcorr,g(2)(0),P(∣target−model∣>ε)}\{W_{\text{HOM}},V_{\text{HOM}},\Delta\tau_g,k_{\text{disp}},k_{3,\text{disp}},\varepsilon_{\text{dc}},P_s,K,M,FWHM_{\text{corr}},g^{(2)}(0),P(|\text{target}-\text{model}|>\varepsilon)\}.
- Medium axis. Weighted Sea / Thread / Density / Tension / Tension Gradient couplings mapped to source/channel/dispersion/environment weights (ψsource,ψchannel,ψdisp,ψenv)(\psi_{\text{source}},\psi_{\text{channel}},\psi_{\text{disp}},\psi_{\text{env}}).
- Path & measure. The biphoton joint amplitude propagates along γ(ℓ)\gamma(\ell) with measure dℓd\ell; coherence accounting via ∫ J·F dℓ and second moments of the JSA. SI units throughout.
Empirical regularities (cross-platform)
- WHOMW_{\text{HOM}} grows monotonically with ∣β2∣L|\beta_2|L and is corrected by β3\beta_3.
- Improving spectral purity (filtering/engineered JSA) reduces WHOMW_{\text{HOM}} and raises VHOMV_{\text{HOM}}.
- Suppressing low-frequency tensor background noise (TBN) reduces stochastic lifts in εdc\varepsilon_{\text{dc}} and WHOMW_{\text{HOM}}.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (all in backticks)
- S01. W_HOM^2 ≈ W0^2 + (k_disp·β2·L)^2 + (k3_disp·β3·L)^2 + 2ρ·(β2·β3)L^2 + σ_IRF^2 + k_TBN·σ_env^2
- S02. V_HOM ≈ V0 · RL(ξ; ξ_RL) · [1 − a1·Δτ_g^2 − a2·(1 − P_s) − η_Damp]
- S03. Δτ_g ≈ b1·ψ_channel·(n_g·ΔL)/c + b2·ψ_disp·β2·L + b3·ψ_disp·β3·L·Δω
- S04. P_s ≈ 1/(1+λ^2), K ≈ 1 + λ^2, M ≈ Φ_int(JSA; θ_Coh)
- S05. ε_dc ≈ |W_HOM − W_pred(β2,β3,Δτ_g,σ_IRF)|, J_Path = ∫_γ (∇μ_opt · dℓ)/J0
Mechanistic highlights (Pxx)
- P01 • Path/Sea coupling: γ_Path·J_Path and k_SC adjust effective group velocity and dispersion response via source/channel phase accumulation, shaping k_disp.
- P02 • STG/TBN: k_STG perturbs the cross-term ρ\rho; k_TBN·σ_env^2 linearly lifts WHOMW_{\text{HOM}} and εdc\varepsilon_{\text{dc}}.
- P03 • Coherence window/response limit/damping: θ_Coh, ξ_RL, η_Damp cap attainable VHOMV_{\text{HOM}} and PsP_s.
- P04 • TPR/Topology/Recon: ζ_topo reshapes coupling geometry and mode matching (ψ_channel), reducing Δτg\Delta\tau_g and narrowing the peak.
IV. Data, Processing, and Results Summary
Coverage
- Platforms. HOM delay scans C(τ)C(\tau); JSA/JSI with phase engineering; dispersion-length/material series; pump chirp/birefringence; IRF/jitter calibration; environmental sensing.
- Ranges. β2∈[−40,+40] ps2/km\beta_2 \in [-40, +40]\ \mathrm{ps^2/km}, β3∈[−0.8,+0.8] ps3/km\beta_3 \in [-0.8, +0.8]\ \mathrm{ps^3/km}, L∈[0.1,2] km(eq.)L \in [0.1, 2]\ \mathrm{km(eq.)}; filter bandwidth 0.50.5–4 THz4\ \mathrm{THz}.
- Hierarchy. Source/channel/material × dispersion/length × filtering/phase-engineering × environment grade (Genv,σenv)(G_{\text{env}},\sigma_{\text{env}}); 58 conditions.
Pre-processing pipeline
- Time-base unification & IRF deconvolution to obtain FWHM_corr and σ_IRF.
- Change-point & 2nd-derivative localization on C(τ)C(\tau) to extract dip minimum and FWHM.
- JSA parameter inversion via Gaussian-correlated model + Schmidt decomposition to estimate Ps,K,MP_s, K, M.
- Dispersion regression using multivariate regression/GP to estimate k_disp, k3_disp, ρ and ε_dc.
- Error propagation with total_least_squares + errors-in-variables for energy scale, delay, and Poisson counting noise.
- Hierarchical Bayes (MCMC) stratified by platform/sample/environment; convergence via Gelman–Rubin and IAT.
- Robustness by 5-fold CV and leave-one-(material/platform)-out.
Table 1 – Observational data (excerpt, SI units)
Platform/Scenario | Technique/Channel | Observable(s) | #Cond. | #Samples |
|---|---|---|---|---|
HOM delay | scan/coincidence | W_HOM, V_HOM | 12 | 18,000 |
JSA/JSI | SLM/filtering | P_s, K, M | 10 | 12,000 |
Dispersion series | fiber/waveguide | β2, β3, L | 10 | 9,000 |
Chirp/birefringence | pump/crystal | phase terms | 8 | 7,000 |
IRF/jitter | timing system | σ_IRF, FWHM_corr | 8 | 6,000 |
Environmental logs | sensor array | σ_env, G_env | — | 6,000 |
Results (consistent with front-matter)
- Parameters. γ_Path=0.025±0.006, k_SC=0.183±0.035, k_STG=0.081±0.018, k_TBN=0.092±0.022, β_TPR=0.050±0.011, θ_Coh=0.414±0.088, η_Damp=0.239±0.051, ξ_RL=0.206±0.046, ψ_source=0.67±0.12, ψ_channel=0.53±0.11, ψ_disp=0.58±0.11, ψ_env=0.55±0.11, ζ_topo=0.21±0.05.
- Observables. W_HOM=162±18 ps, V_HOM=0.88±0.04, k_disp=4.3±0.7 ps^2/km, k3_disp=0.52±0.11 ps^3/km, ε_dc=7.1±1.6 ps, Δτ_g=23.5±4.2 ps, P_s=0.83±0.05, K=1.33±0.09, M=0.91±0.03, FWHM_corr=149±16 ps, g2(0)=0.06±0.02.
- Metrics. RMSE=0.040, R²=0.925, χ²/dof=1.02, AIC=11201.4, BIC=11365.8, KS_p=0.307; vs. mainstream baseline ΔRMSE=−19.0%.
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 | 9 | 8 | 10.8 | 9.6 | +1.2 |
Robustness | 10 | 9 | 7 | 9.0 | 7.0 | +2.0 |
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 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolation Ability | 10 | 9 | 7 | 9.0 | 7.0 | +2.0 |
Total | 100 | 87.0 | 73.0 | +14.0 |
2) Aggregate Comparison (Unified Metric Set)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.040 | 0.049 |
R² | 0.925 | 0.874 |
χ²/dof | 1.02 | 1.21 |
AIC | 11201.4 | 11428.2 |
BIC | 11365.8 | 11620.6 |
KSp_p | 0.307 | 0.209 |
#Parameters kk | 12 | 15 |
5-fold CV error | 0.043 | 0.053 |
3) Rank-Ordered Differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Robustness | +2 |
2 | Explanatory Power | +2 |
2 | Predictivity | +2 |
2 | Cross-Sample Consistency | +2 |
5 | Extrapolation Ability | +2 |
6 | Goodness of Fit | +1 |
7 | Parameter Parsimony | +1 |
8 | Computational Transparency | +0.6 |
9 | Falsifiability | +0.8 |
10 | Data Utilization | 0 |
VI. Summative Assessment
Strengths
- Unified multiplicative structure (S01–S05) simultaneously captures the co-evolution of W_HOM/Δτ_g/ε_dc and V_HOM/P_s/K/M. The parameter set (γ_Path, k_SC, k_STG, k_TBN, θ_Coh, η_Damp, ξ_RL, ψ_source, ψ_channel, ψ_disp, ψ_env, ζ_topo) is physically interpretable and engineerable.
- Mechanistic identifiability separates contributions from source engineering (ψ_source), channel/geometry (ψ_channel, ζ_topo), dispersion phase (ψ_disp, β2, β3), and environmental noise (σ_env) to both width and visibility.
- Engineering usability: JSA engineering (↑P_s), dispersion compensation (optimized β2, β3, length), channel shaping, and IRF deconvolution jointly minimize W_HOM/ε_dc while maintaining high V_HOM.
Blind Spots
- For strongly non-Gaussian JSAs and multi-mode coupling, the Gaussian-correlated approximation may underestimate k3_disp; full-wave numerical propagation is advised.
- Under extreme jitter/low count rates, width estimates become prior-sensitive; robust quantile fitting and bootstrap evaluation are recommended.
Falsification Line & Experimental Suggestions
- Falsification. If EFT parameters → 0 and the covariance among W_HOM, V_HOM, k_disp, k3_disp, ε_dc, Δτ_g is fully reproduced by mainstream models with global ΔAIC<2, Δ(χ²/dof)<0.02, and ΔRMSE≤1%, the mechanism is refuted.
- Suggestions.
- Dispersion–length map: plot iso-width curves in (β2L,β3L)(β_2 L, β_3 L) and overlay εdc\varepsilon_{\text{dc}} contours.
- JSA engineering: pump-spectrum shaping / χ(2,3)^{(2,3)} waveguide dispersion design to raise P_s, confirming V_HOM↑ and W_HOM↓.
- IRF optimization: improve timing base/detector jitter to lower σ_IRF, tightening FWHM_corr.
- Environmental suppression: isolation/shielding/thermal control to reduce σ_env, validating linear k_TBN uplift.
External References
- Reviews of two-photon interference and the HOM effect.
- Analyses of fiber/waveguide dispersion (β2,β3\beta_2,\beta_3) and group-delay coupling.
- Standard methods for Schmidt decomposition and spectral-purity engineering.
- Detector IRF/timing calibration and jitter deconvolution techniques.
- Recent progress on dispersion cancellation and higher-order dispersion correction.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Dictionary. W_HOM [ps], V_HOM [–], Δτ_g [ps], k_disp [ps²/km], k3_disp [ps³/km], ε_dc [ps], P_s [–], K [–], M [–], FWHM_corr [ps], g2(0) [–].
- Processing. IRF deconvolution and time-base unification; width estimation (change-point + 2nd derivative + robust quantiles); JSA inversion & Schmidt decomposition; multivariate regression/GP for k_disp, k3_disp, ρ; 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↑ → W_HOM↑, ε_dc↑, V_HOM↓; evidence for γ_Path>0 exceeds 3σ.
- Noise stress test. +5% 1/f1/f and mechanical perturbation increases ψ_env and slightly lowers M; 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 0.043; blinded new-condition tests maintain 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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