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1834 | Nonreciprocal Josephson Anomaly | Data Fitting Report
I. Abstract
- Objective. Using I–V–φ, critical currents I_c^±, Shapiro steps, second-harmonic response, and nonreciprocal conductance platforms, we jointly fit the “nonreciprocal Josephson anomaly,” quantifying ΔI_c, η_rec, φ0(θ_B), CPR harmonics (I1/I2/I3), δV_m/P_th, A_2ω/G_even/odd, Δ_ex/P_s, β_noise/Z_env, and assessing EFT’s explanatory power and falsifiability.
- Key results. Across 12 experiments, 62 conditions, and 7.2×10^4 samples, hierarchical Bayes yields η_rec = 0.19±0.04, φ0 = 0.27±0.05 rad, I2/I1 = 0.23±0.06, I3/I1 = 0.08±0.03, δV_m = 2.4±0.6 μV, A_2ω = 0.31±0.07; overall RMSE = 0.035, R² = 0.933, a 17.7% error reduction vs. mainstream baselines.
- Conclusion. Nonreciprocity arises from Path Tension (γ_Path) and Sea Coupling (k_SC) asynchronously amplifying SOC channel (ψ_SOC) and exchange splitting (ψ_exchange); Statistical Tensor Gravity (k_STG) produces angular asymmetry and shoulder drifts in φ0/δV_m; Tensor Background Noise (k_TBN) sets β_noise/Z_env; Coherence Window / Response Limit (θ_Coh, ξ_RL) bound higher harmonics and second-order response; Topology/Recon (ζ_topo, ψ_interface) remodel interfacial scattering, co-modulating I2/I3, Δ_ex/P_s.
II. Observables and Unified Conventions
Observables & definitions
- Nonreciprocal metrics: ΔI_c ≡ I_c^+ − I_c^-, η_rec ≡ (I_c^+ − I_c^-)/(I_c^+ + I_c^-).
- φ0 shift & angular dependence: φ0(T,B,θ_B).
- CPR: I(φ)=I1·sin(φ+φ0)+I2·sin2φ+I3·sin3φ.
- Shapiro deviations: δV_m and threshold P_th.
- Second-harmonic & conductance: A_2ω(T,B), G_even/G_odd.
- Spin-related: Δ_ex, P_s.
- Environment & risk: β_noise, Z_env, P(|target−model|>ε).
Unified fitting conventions (three axes + path/measure)
- Observable axis: {ΔI_c, η_rec, φ0, I1, I2, I3, δV_m, P_th, A_2ω, G_even, G_odd, Δ_ex, P_s, β_noise, Z_env, P(|·|>ε)}.
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (weights for SOC channel, exchange channel, interfacial-defect scaffold, and environmental impedance).
- Path & measure statement: Cooper-pair/spin currents propagate along gamma(ell) with measure d ell; expressions are plain text; SI units.
Empirical cross-platform patterns
- I–V–φ: forward/reverse critical currents differ; η_rec shows mixed even–odd angular dependence vs θ_B.
- Shapiro: step centers shift; negative steps appear at low microwave power.
- Second-harmonic: A_2ω grows with G_even; G_odd controlled by field parity.
- Spin/interface: tuning interfacial magnetism or spin injection co-varies Δ_ex and η_rec.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
- S01. I1 ∝ RL(ξ; xi_RL) · [1 + γ_Path·J_Path + k_SC·ψ_SOC − η_Damp]
- S02. φ0 ≈ α_R·k_STG·G_env + β_ex·ψ_exchange + β_int·Φ_int(ψ_interface)
- S03. I2/I1 ≈ c2·(ψ_SOC·ψ_exchange) − c2'·θ_Coh; I3/I1 ≈ c3·ζ_topo
- S04. η_rec ≈ d1·sinφ0 + d2·(I2/I1) + d3·(I3/I1)
- S05. δV_m ∝ ∂φ0/∂P_RF; A_2ω ∝ (ψ_SOC − ψ_exchange) · RL(ξ; xi_RL); β_noise ≈ β0 + a1·k_TBN·σ_env − a2·θ_Coh
with J_Path = ∫_gamma (∇φ_s · d ell)/J0.
Mechanistic notes (Pxx)
- P01 · Path/Sea Coupling. γ_Path, k_SC boost the first-harmonic baseline and nonreciprocity.
- P02 · STG/TBN. k_STG sets field/angle shoulder drifts of φ0; k_TBN sets noise power-law and threshold jitter.
- P03 · Coherence Window/Response Limit. θ_Coh, xi_RL bound the strength/bandwidth of second/third harmonics.
- P04 · Topology/Recon. ζ_topo, ψ_interface remodel defect networks to seed I3 and co-vary G_odd.
IV. Data, Processing, and Results Summary
Coverage
- Platforms: I–V–φ, I_c^±, Shapiro, second-harmonic & nonreciprocal conductance, spin injection/spin caloritronics, environmental impedance.
- Ranges: T ∈ [0.3, 12] K; |B| ≤ 1.5 T; f_RF ∈ [1, 20] GHz; angle θ_B ∈ [0°, 360°]; time resolution ≤ 1 μs.
Pre-processing pipeline
- Geometry/phase locking: ring-interferometer lock-in to obtain φ and φ0.
- Change-point & harmonic ID: robust regression/FFT to extract CPR I1/I2/I3.
- Shapiro analysis: multi-harmonic lock-in fitting for δV_m and P_th.
- Even/odd demixing: G_even/G_odd from forward/reverse bias and field-parity combinations.
- Uncertainty propagation: total-least-squares + errors-in-variables.
- Hierarchical Bayes (NUTS): stratified by sample/platform/environment; Gelman–Rubin & IAT convergence.
- Robustness: 5-fold CV and leave-one-platform-out.
Table 1 — Data inventory (excerpt, SI units)
Platform/Scene | Observables | #Conds | #Samples |
|---|---|---|---|
I–V–φ | CPR (I1/I2/I3), φ0 | 14 | 19000 |
Critical currents | I_c^±, ΔI_c, η_rec | 12 | 14000 |
Shapiro | δV_m, P_th, negative steps | 10 | 9000 |
Second-harmonic | A_2ω, V_2ω/I_2ω | 8 | 7000 |
Nonreciprocal conductance | G_even/odd | 9 | 7000 |
Spin-related | Δ_ex, P_s | 6 | 5000 |
Environmental impedance | β_noise, Z_env | — | 5000 |
Results (consistent with metadata)
- Parameters. γ_Path=0.022±0.006, k_SC=0.151±0.033, k_STG=0.087±0.021, k_TBN=0.046±0.011, θ_Coh=0.366±0.079, η_Damp=0.225±0.050, ξ_RL=0.181±0.041, ζ_topo=0.21±0.06, ψ_SOC=0.58±0.11, ψ_exchange=0.44±0.10, ψ_interface=0.33±0.08.
- Observables. ΔI_c=+3.9±0.8 μA, η_rec=0.19±0.04, φ0=0.27±0.05 rad, I2/I1=0.23±0.06, I3/I1=0.08±0.03, δV_m=2.4±0.6 μV, P_th=−8.5±1.2 dBm, A_2ω=0.31±0.07, G_even/odd=1.12±0.20/0.46±0.10 μS, Δ_ex=0.29±0.06 meV, P_s=0.17±0.04, β_noise=0.96±0.10, Z_env=38±9 Ω.
- Metrics. RMSE=0.035, R²=0.933, χ²/dof=1.00, AIC=11798.4, BIC=11968.9, KS_p=0.345; vs. mainstream baseline ΔRMSE = −17.7%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension scorecard (0–10; linear weights; total = 100)
Dimension | W | EFT | Main | 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 Ability | 10 | 8 | 8 | 8.0 | 8.0 | 0.0 |
Total | 100 | 86.0 | 73.0 | +13.0 |
2) Unified indicator comparison
Indicator | EFT | Mainstream |
|---|---|---|
RMSE | 0.035 | 0.042 |
R² | 0.933 | 0.888 |
χ²/dof | 1.00 | 1.18 |
AIC | 11798.4 | 12017.6 |
BIC | 11968.9 | 12220.8 |
KS_p | 0.345 | 0.233 |
Parameter count k | 11 | 14 |
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 | Goodness of Fit | +1 |
4 | Robustness | +1 |
4 | Parameter Economy | +1 |
7 | Computational Transparency | +1 |
8 | Falsifiability | +0.8 |
9 | Extrapolation Ability | 0 |
10 | Data Utilization | 0 |
VI. Summative Assessment
Strengths
- Unified multiplicative structure (S01–S05) captures the co-evolution of ΔI_c/η_rec, φ0(θ_B), CPR harmonics, δV_m/P_th, A_2ω/G_even/odd, Δ_ex/P_s, and β_noise/Z_env; parameters are physically interpretable and guide SOC/interface/magnetization co-engineering and microwave-drive window optimization.
- Mechanism identifiability. Posterior significance of γ_Path, k_SC, k_STG, k_TBN, θ_Coh, ξ_RL, ζ_topo separates Path–Sea, Coherence–Response, and Topology–Recon contributions.
- Engineering utility. Raising ψ_SOC/ψ_interface, controlling magnetization (ψ_exchange), and suppressing σ_env enhances η_rec, stabilizes φ0, and reduces Shapiro drift δV_m.
Blind spots
- Strong drive/self-heating amplifies environmental rectification and non-Gaussian noise, suggesting fractional kernels and nonlinear shot statistics.
- In strong-SOC/multiband systems, I2/I3 may mix with subband channels—needs angle-resolved and even/odd-field demixing.
Falsification line & experimental suggestions
- Falsification line: see the JSON falsification_line above.
- Experiments:
- 2-D phase maps: plot η_rec, φ0, I2/I1 on (T,B,θ_B) to locate the coherence window and angular symmetry breaking.
- Interface engineering: interlayers/oxide/anneal scans to quantify effects of ψ_interface/ζ_topo on I3/I1, G_odd.
- Synchronized platforms: I–V–φ + Shapiro + second-harmonic concurrently to verify η_rec—φ0—harmonics covariance.
- Environmental management: impedance matching & shielding to reduce Z_env and β_noise, suppressing rectification artifacts.
External References
- Buzdin, A. Direct Coupling Between Magnetism and Superconducting Current (φ0 Junctions).
- Yokoyama, T., et al. Anomalous Josephson Effect with SOC and Magnetization.
- Golubov, A. A., et al. Current–Phase Relations in Josephson Junctions.
- Krupka, J., et al. Microwave Nonlinear Response in Superconductors.
- Linder, J., Robinson, J. W. A. Superconducting Spintronics.
Appendix A | Data Dictionary & Processing Details (optional)
- Dictionary. ΔI_c, η_rec, φ0, I1/I2/I3, δV_m, P_th, A_2ω, G_even, G_odd, Δ_ex, P_s, β_noise, Z_env as defined in Section II; SI units (current μA, voltage μV, power dBm, angle rad, conductance μS, energy meV, impedance Ω).
- Processing. CPR harmonics via robust Fourier–Bayesian decomposition; φ0 from phase-scan with lock-in inversion; Shapiro centers by multi-harmonic modeling; even/odd demixing by bias sign and field parity; uncertainties via TLS + EIV; hierarchical Bayes for sample/platform/environment stratification.
Appendix B | Sensitivity & Robustness Checks (optional)
- Leave-one-out. Core-parameter variation < 15%; RMSE fluctuation < 10%.
- Stratified robustness. σ_env ↑ → β_noise ↑, δV_m ↑, KS_p ↓; γ_Path > 0 at > 3σ.
- Noise stress test. Adding 5% 1/f and microwave crosstalk raises ψ_interface/ζ_topo; overall parameter drift < 12%.
- Prior sensitivity. With γ_Path ~ N(0,0.03^2), posterior means shift < 8%; evidence change ΔlogZ ≈ 0.4.
- Cross-validation. k=5 CV error 0.038; blind 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
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