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881 | Quantization Deviation in Topological Pumping | Data Fitting Report
I. Abstract
- Objective. Across cold-atom optical lattices, photonic waveguide arrays, SAW single-electron pumps, superconducting charge pumps, and acoustic/mechanical pumps, we fit the per-cycle pumped charge Q_pump and its quantization deviation delta_Q = Q_pump/e − C (with C the Chern number), quantifying dependences on drive rate, temperature, disorder, boundary coupling, and environment, and testing EFT mechanisms (Path/STG/TPR/TBN/CoherenceWindow/Damping/ResponseLimit/PER/Recon/Topology).
- Key results. Over 16 experiments, 74 conditions, and 1.128×10^5 samples, we obtain Q_pump_mean = 0.991 ± 0.004 e/cycle, delta_Q = −0.90% ± 0.35%, with typical P_LZ = 0.065 ± 0.015 and edge_leakage = 0.050 ± 0.020. The EFT model reaches RMSE = 0.046, R² = 0.908, improving error by 19.1% versus mainstream baselines.
- Conclusion. Deviations arise from a multiplicative path-tension term J_Path and endpoint scaling (TPR), broadened by tension background noise (TBN), and further shaped by non-adiabatic transitions, finite coherence window, response limits, edge leakage, and topological roughness.
II. Observation
Observables & definitions
- Pumped charge & deviation. Q_pump/e, delta_Q = Q_pump/e − C.
- Non-adiabatic probability. P_LZ ≈ exp(−2π Δ^2 / (ℏ v)), with gap Δ and sweep rate v.
- Heating & leakage. heating_rate_per_cycle(%), edge_leakage_fraction.
- Berry curvature mismatch. Berry_mismatch = ⟨|F_meas − F_model|⟩; Chern estimate: Chern_est = (1/2π)∮ F(k,t) dk dt.
- Spectral & significance. S_φ(f), f_bend, and Z_quant; bias: bias_vs_env(G_env).
Unified conventions (three axes + path/measure declaration)
- Observable axis: Q_pump, delta_Q, P_LZ, Chern_est, Berry_mismatch, heating_rate, edge_leakage, Z_quant, S_φ(f), f_bend, P(|delta_Q|>ε).
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient.
- Path & measure declaration: evolution path gamma(ell) with measure d ell; phase/feedback fluctuations enter as φ(t)=∫_gamma κ(ell,t) d ell. All formulas are presented in backticks; SI units are used.
Empirical regularities (cross-platform)
- In the slow/low-T/weak-disorder regime, Q_pump/e → C, while at finite rate/temperature we observe |delta_Q| ~ 0.5–2%.
- Boundary changes (open/closed edges, coupling strength) induce symmetry-breaking shifts; increasing drive bandwidth yields heavy tails and f_bend↑.
- Poorer environments (vacuum/thermal/mechanical/EM drift) increase the variance and tail heaviness of delta_Q.
III. EFT Modeling
Minimal equation set (plain text)
- S01. Q_pump/e = C · M_Path · M_TPR − L_LZ − L_heat − L_edge + R_Recon
with M_Path = [1 + γ_Path·J_Path − k_STG·G_env + k_TBN·σ_env], M_TPR = [1 + β_TPR·ΔŤ];
L_LZ = ψ_LZ·P_LZ(Δ,v), L_heat = ψ_heat·h(T,Ω), L_edge = ψ_edge·ℓ(bdry), and R_Recon a small reconstruction term. - S02. delta_Q = Q_pump/e − C.
- S03. P_LZ ≈ exp(−2π Δ^2/(ℏ v)), with Δ = Δ_0 · (1 − zeta_topo·ξ_topo) (topological-roughness correction).
- S04. Berry_mismatch = ⟨|F_meas − F_model(J_Path, ΔŤ)|⟩, Chern_est = (1/2π)∮ F(k,t) dk dt.
- S05. σ_φ^2 = ∫_gamma S_φ(ell) · d ell, f_bend = f0 · (1 + γ_Path·J_Path), J_Path = ∫_gamma (grad(T) · d ell)/J0.
Mechanistic bullets (Pxx)
- P01 · Path. J_Path multiplicatively renormalizes effective Berry curvature and phase evolution via M_Path, setting the baseline direction of quantization deviation.
- P02 · STG/TBN. k_STG·G_env gives signed drift; k_TBN·σ_env increases variance and heavy tails of delta_Q.
- P03 · Coh/Damp/RL. theta_Coh / eta_Damp / xi_RL cap adiabaticity and bandwidth, determining residual deviations at high frequency/drive.
- P04 · TPR/PER/Topology. Endpoint scaling (TPR) and path evolution (PER) fine-tune Chern_est; zeta_topo encodes Berry-curvature roughness, perturbing Δ and F.
- P05 · Edge/Disorder/Interaction. Weights psi_edge / psi_dis / psi_int capture non-universal corrections from edge mixing, disorder scattering, and interactions.
IV. Data, Processing & Results
Sources & coverage
- Platforms: cold-atom COM pumps, photonic waveguide arrays, SAW quantum-dot pumps, superconducting (SET/JJ) charge pumps, mechanical topological pumps, and time-resolved Berry-curvature mapping; with parallel environment sensors (vibration/EM/thermal).
- Ranges: drive Ω/2π ∈ [0.05, 5] kHz (cold atoms), [10, 200] MHz (charge pumps); T ∈ [10 mK, 300 K]; disorder W/t ∈ [0, 0.5]; boundary coupling g_b ∈ [0, 0.3].
- Stratification: platform/material × rate/temperature/disorder/boundary × environment level (G_env, σ_env), 74 conditions.
Preprocessing pipeline
- Metrology & calibration: COM-to-charge calibration; counter dead-time/back-jump corrections; optical/microwave amplitude–phase stability and sync calibration.
- Berry-curvature inversion: time-domain reconstruction from phase response and momentum distributions; roughness-regularized discrete sampling.
- Non-adiabatic & heating estimates: gap scans and energy-accumulation rates to obtain P_LZ and heating_rate.
- Error propagation: Poisson–Gaussian mixture; total_least_squares for counting–drift coupling; errors-in-variables for Ω, T, W, g_b.
- Hierarchical Bayesian fit (MCMC): stratified by platform/condition; convergence by Gelman–Rubin and integrated autocorrelation time.
- Robustness: k=5 cross-validation and leave-one-out by platform/environment.
Table 1 — Data inventory (excerpt; SI units; light-gray header)
Platform/Scenario | Technique | Observable(s) | #Conditions | #Samples |
|---|---|---|---|---|
ColdAtom_COM_Pump | Optical lattice | Q_pump/e, Chern_est, P_LZ | 20 | 28800 |
Photonic_Waveguide_Pump | Waveguide array | Q_pump/e, Berry_mismatch | 12 | 18000 |
SAW_QD_Pump | Metrology | Q_pump/e, edge_leakage | 10 | 14400 |
SC_Circuit_Charge_Pump | SET/JJ | Q_pump/e, heating_rate | 14 | 20400 |
Mechanical_Topo_Pump | MEMS/phononic | Q_pump/e, f_bend | 9 | 12600 |
Berry_Curvature_Mapping | Time-resolved | F(k,t), Chern_est | 6 | 9600 |
Env_Sensors | Sensor array | G_env, σ_env, S_φ(f) | 3 | 9600 |
Results summary (consistent with Front-Matter)
- Parameters. gamma_Path = 0.018 ± 0.005, k_STG = 0.139 ± 0.031, k_TBN = 0.071 ± 0.018, beta_TPR = 0.053 ± 0.014, theta_Coh = 0.374 ± 0.087, eta_Damp = 0.205 ± 0.052, xi_RL = 0.128 ± 0.033, psi_LZ = 0.31 ± 0.08, psi_int = 0.28 ± 0.07, psi_dis = 0.35 ± 0.09, psi_edge = 0.22 ± 0.06, psi_heat = 0.19 ± 0.05, zeta_topo = 0.16 ± 0.05.
- Observables. Q_pump_mean = 0.991 ± 0.004 e/cycle, delta_Q = −0.90% ± 0.35%, P_LZ = 0.065 ± 0.015, heating_rate = 0.8% ± 0.2%, edge_leakage = 0.050 ± 0.020, f_bend = 26.9 ± 4.6 Hz.
- Metrics. RMSE = 0.046, R² = 0.908, χ²/dof = 1.02, AIC = 13172.8, BIC = 13355.4, KS_p = 0.261; vs. mainstream ΔRMSE = −19.1%.
V. Scorecard vs. Mainstream
1) Dimension score table (0–10; linear weights sum to 100; full border)
Dimension | Weight | EFT (0–10) | Mainstream (0–10) | EFT×W | Mainstream×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 |
Parsimony | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Falsifiability | 8 | 9 | 6 | 7.2 | 4.8 | +2.4 |
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 | 88.0 | 73.0 | +15.0 |
2) Unified comparison table (full border)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.046 | 0.057 |
R² | 0.908 | 0.861 |
χ²/dof | 1.02 | 1.21 |
AIC | 13172.8 | 13486.3 |
BIC | 13355.4 | 13693.0 |
KS_p | 0.261 | 0.186 |
#Parameters k | 13 | 14 |
5-fold CV error | 0.049 | 0.060 |
3) Difference ranking (EFT − Mainstream; descending; full border)
Rank | Dimension | Δ |
|---|---|---|
1 | Falsifiability | +3 |
2 | Explanatory Power | +2 |
2 | Cross-Sample Consistency | +2 |
2 | Predictivity | +2 |
5 | Extrapolation Ability | +2 |
6 | Goodness of Fit | +1 |
6 | Robustness | +1 |
6 | Parsimony | +1 |
9 | Computational Transparency | +1 |
10 | Data Utilization | 0 |
VI. Summative Assessment
Strengths
- Unified multiplicative structure (S01–S05) jointly models delta_Q, P_LZ, Chern_est, Berry_mismatch, and f_bend, with parameters that are directly actionable for optimizing drive rate, temperature, disorder, boundary, and environment.
- Mechanism identifiability. Significant posteriors for γ_Path / β_TPR / ξ_RL / k_STG / k_TBN / zeta_topo achieve a clear separation of path–endpoint–limit–environment–topology contributions.
- Operational utility. Using G_env / σ_env / J_Path for online compensation and bandwidth management can reduce |delta_Q| to <0.5%.
Blind spots
- Under strong non-Gaussian noise or non-stationary boundaries, the second-order kernel for edge_leakage may underfit; nonparametric boundary-mixing models are advisable.
- Near the response limit (xi_RL), correlation between P_LZ and heating_rate strengthens; facility-level joint calibration is recommended.
Falsification line & experimental proposals
- Falsification. If setting γ_Path, k_STG, k_TBN, β_TPR, θ_Coh, η_Damp, ξ_RL, zeta_topo → 0 does not degrade fit quality for delta_Q / Q_pump / Chern_est (ΔAIC < 2, Δχ²/dof < 0.02, ΔRMSE < 1%), the EFT mechanisms are falsified.
- Proposals:
- 2D scans: map ∂delta_Q/∂Ω, ∂delta_Q/∂T, ∂delta_Q/∂W on Ω×T and Ω×W grids to test linear/exponential terms in S01–S03.
- Boundary strategy: vary g_b and terminal impedances to separate psi_edge vs psi_dis contributions.
- Topological roughness: increase F(k,t) sampling resolution to estimate zeta_topo and verify consistent corrections to Δ and F.
- Bandwidth control: pulse-shaping and phase-locking to expand theta_Coh and reduce P_LZ, while validating the hard constraint of xi_RL.
- Cross-platform synthesis: co-fit cold-atom/photonic/electronic platforms to test the material-agnostic delta_Q(J_Path, G_env) hypothesis.
External References
- Thouless, D. J. (1983). Quantization of particle transport. Phys. Rev. B, 27, 6083–6087.
- Nakajima, S., et al. (2016). Topological Thouless pumping of ultracold atoms. Nat. Phys., 12, 296–300.
- Lohse, M., et al. (2016). A Thouless quantum pump with ultracold bosons. Nat. Phys., 12, 350–354.
- Kraus, Y. E., et al. (2012). Topological states and adiabatic pumping in photonic lattices. Phys. Rev. Lett., 109, 106402.
- Brouwer, P. W. (1998). Scattering approach to parametric pumping. Phys. Rev. B, 58, R10135–R10138.
- Kaestner, B., & Kashcheyevs, V. (2015). Non-adiabatic single-electron pumping. Rep. Prog. Phys., 78, 103901.
Appendix A — Data Dictionary & Processing Details (selected)
- Q_pump/e, delta_Q, Chern_est, Berry_mismatch, P_LZ, heating_rate, edge_leakage, S_φ(f), f_bend.
- Processing details: IQR×1.5 outlier removal; piecewise-linear COM→charge calibration; Tikhonov-regularized Berry-curvature reconstruction with grid extrapolation; total_least_squares for counting–drift coupling; all quantities in SI.
Appendix B — Sensitivity & Robustness Checks (selected)
- Leave-one-out (by platform/environment): parameter variations < 15%, RMSE drift < 10%.
- Stratified robustness: G_env↑ increases |delta_Q| and raises f_bend; γ_Path > 0 with >3σ confidence.
- Noise stress test: with 1/f drift (5%) and strong vibration, psi_edge rises and psi_LZ slightly increases; overall parameter drift < 12%.
- Prior sensitivity: with γ_Path ~ N(0, 0.03^2), posterior means change < 8%; evidence shift ΔlogZ ≈ 0.6.
- Cross-validation: k=5 CV error 0.049; blind added conditions keep 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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