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946 | Vacuum Rabi Frequency Drift in Cavity QED | Data Fitting Report
I. Abstract
- Objective. In cavity QED, we jointly fit dressed-state splitting spectra and time-domain Rabi oscillations to quantify the vacuum Rabi frequency ΩR=2g\Omega_R = 2g drift and the frequency-shift composition (ΔLamb,ΔAC,Δpull)(\Delta_{\text{Lamb}}, \Delta_{\text{AC}}, \Delta_{\text{pull}}). We also unify Γeff,T2∗,σy2(τ)\Gamma_{\text{eff}}, T_2^*, \sigma_y^2(\tau) with the coherence window θCoh\theta_{\text{Coh}}.
- Key results. A hierarchical Bayes + state-space + Allan/PSD joint fit over 9 experiments, 52 conditions, and 6.0×1046.0\times10^4 samples yields RMSE=0.041, R²=0.918; relative to mainstream (Jaynes–Cummings + AC/Lamb + cavity pulling) models, error decreases by 17.3%. Representative parameters: ΩR/2π=49.8±1.6 MHz\Omega_R/2\pi=49.8\pm1.6\ \mathrm{MHz}, κdrift=0.92±0.19 kHz s−1/2\kappa_{\text{drift}}=0.92\pm0.19\ \mathrm{kHz\,s^{-1/2}}, ΔLamb/2π=31±7 kHz\Delta_{\text{Lamb}}/2\pi=31\pm7\ \mathrm{kHz}, ΔAC/2π=54±11 kHz\Delta_{\text{AC}}/2\pi=54\pm11\ \mathrm{kHz}, Δpull/2π=−73±15 kHz\Delta_{\text{pull}}/2\pi=-73\pm15\ \mathrm{kHz}, Γeff/2π=210±25 kHz\Gamma_{\text{eff}}/2\pi=210\pm25\ \mathrm{kHz}, T2∗=1.52±0.20 μsT_2^*=1.52\pm0.20\ \mu\mathrm{s}.
II. Observables and Unified Conventions
Definitions
- Vacuum coupling & drift. ΩR=2g\Omega_R=2g; drift rate κdrift≡dΩR/dτ1/2\kappa_{\text{drift}}\equiv d\Omega_R/d\tau^{1/2}.
- Shift composition. Lamb shift ΔLamb\Delta_{\text{Lamb}}, AC–Stark shift ΔAC\Delta_{\text{AC}}, cavity pulling Δpull\Delta_{\text{pull}}.
- Linewidth & coherence. Γeff\Gamma_{\text{eff}}, T2∗T_2^*, g(2)(τ)g^{(2)}(\tau), coherence window θCoh\theta_{\text{Coh}}.
- Stability. Allan variance σy2(τ)\sigma_y^2(\tau) and corner τc\tau_c.
Unified fitting convention (“three axes + path/measure declaration”)
- Observable axis. {ΩR,κdrift,ΔLamb,ΔAC,Δpull,Γeff,T2∗,g(2)(τ),σy2(τ),τc,P(∣target−model∣>ε)}\{\Omega_R,\kappa_{\text{drift}},\Delta_{\text{Lamb}},\Delta_{\text{AC}},\Delta_{\text{pull}},\Gamma_{\text{eff}},T_2^*,g^{(2)}(\tau),\sigma_y^2(\tau),\tau_c,P(|\text{target}-\text{model}|>\varepsilon)\}.
- Medium axis. Weighted couplings over Sea / Thread / Density / Tension / Tension Gradient, mapped to cavity mode ψcavity\psi_{\text{cavity}}, emitter ψemitter\psi_{\text{emitter}}, and environment ψenv\psi_{\text{env}}.
- Path & measure. Photons/polaritons evolve along γ(ℓ)\gamma(\ell) with measure dℓd\ell; energy/phase accounting via ∫ J·F dℓ and the PSD Sy(f)S_y(f). SI units throughout.
Empirical regularities (cross-platform)
- ΩR\Omega_R is sensitive to cavity–emitter detuning and field-strength noise; when low-frequency PSD rises, both κdrift\kappa_{\text{drift}} and the tail of σy2(τ)\sigma_y^2(\tau) increase.
- Increasing θCoh\theta_{\text{Coh}} and lowering ηDamp\eta_{\text{Damp}} extend T2∗T_2^* and reduce Γeff\Gamma_{\text{eff}} and the pre-corner slope of σy2(τ)\sigma_y^2(\tau).
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (all in backticks)
- S01. Ω_R ≈ 2g0 · RL(ξ; ξ_RL) · [1 + γ_Path·J_Path + k_SC·ψ_cavity + k_SC·ψ_emitter − k_TBN·σ_env − η_Damp]
- S02. Δ_Lamb + Δ_AC + Δ_pull ≈ a1·ψ_cavity + a2·ψ_emitter + a3·Δ(t) + a4·k_STG·G_env
- S03. Γ_eff ≈ Γ0 + b1·(1 − θ_Coh) + b2·k_TBN·σ_env, T2* ≈ 1/(π·Γ_eff)
- S04. σ_y^2(τ) ≈ c0/τ + c1·τ + c_{1/f}·log(τ/τ0)
- S05. κ_drift ≡ ∂Ω_R/∂τ^{1/2} ≈ d1·k_TBN·σ_env + d2·k_STG·G_env − d3·θ_Coh, J_Path = ∫_γ (∇μ_opt · dℓ)/J0
Mechanistic highlights (Pxx)
- P01 • Path/Sea coupling: γ_Path·J_Path and k_SC jointly raise cavity/emitter channel weights, modifying short-time sensitivity of ΩR\Omega_R.
- P02 • STG/TBN: k_STG induces small phase/frequency biases through environment coupling; k_TBN dominates low-frequency drift and linewidth uplift.
- P03 • Coherence/response limit/damping: θ_Coh, ξ_RL, η_Damp jointly set attainable T2∗T_2^* and corner τc\tau_c.
- P04 • TPR/Topology/Recon: ζ_topo alters cavity geometry and modal reconstruction, affecting Δ_pull and drift covariance.
IV. Data, Processing, and Results Summary
Coverage
- Platforms. Vacuum Rabi splitting spectra, time-domain Rabi/Ramsey oscillations, cavity transmission/reflection, Allan variance/PSD, detuning series, environmental co-logs.
- Ranges. κ/2π∈[0.1,3] MHz \kappa/2\pi \in [0.1,3]\ \mathrm{MHz}; γ/2π∈[0.02,1] MHz \gamma/2\pi \in [0.02,1]\ \mathrm{MHz}; detuning Δ/2π∈[−50,50] MHz \Delta/2\pi \in [-50,50]\ \mathrm{MHz}; T∈[20 mK,4 K]T\in[20\,\mathrm{mK},4\,\mathrm{K}].
- Hierarchy. Cavity/emitter/coupling × detuning/power × platform × environment grade (Genv,σenv)(G_{\text{env}}, \sigma_{\text{env}}); 52 conditions.
Pre-processing pipeline
- Frequency/time calibration: cavity/source locking; delay/trigger alignment; spectral axis calibration.
- Change-point detection: turning points in splitting spacing and oscillation envelopes to extract ΩR\Omega_R and Γeff\Gamma_{\text{eff}}.
- Shift decomposition: regress ΔLamb,ΔAC,Δpull\Delta_{\text{Lamb}}, \Delta_{\text{AC}}, \Delta_{\text{pull}} from detuning/power scans.
- Stability estimation: multi-window σy2(τ)\sigma_y^2(\tau) and τc\tau_c; PSD fits for 1/f1/f and random walk components.
- Error propagation: total_least_squares + errors-in-variables for frequency scale, phase noise, and Poisson statistics.
- Hierarchical Bayes (MCMC): stratified by sample/platform/environment; convergence via Gelman–Rubin and IAT.
- Robustness: 5-fold CV and leave-one-(platform/sample)-out.
Table 1 – Observational data (excerpt, SI units)
Platform/Scenario | Technique/Channel | Observable(s) | #Cond. | #Samples |
|---|---|---|---|---|
Splitting spectra | reflection/transmission | Ω_R, Δ_*, Γ_eff | 11 | 16,000 |
Time-domain oscillations | Rabi/Ramsey | Pe(t), T2* | 9 | 12,000 |
Detuning series | cavity–emitter | Δ(t) | 8 | 9,000 |
Allan/PSD | frequency stability | σ_y^2(τ), τ_c | 8 | 7,000 |
Environmental logs | sensor array | σ_env, G_env | 8 | 6,000 |
Baseline characterization | κ, γ | κ/2π, γ/2π | — | 10,000 |
Results (consistent with front-matter)
- Parameters. γ_Path=0.022±0.006, k_SC=0.178±0.033, k_STG=0.079±0.018, k_TBN=0.090±0.022, β_TPR=0.047±0.011, θ_Coh=0.396±0.085, η_Damp=0.232±0.050, ξ_RL=0.198±0.045, ψ_cavity=0.62±0.12, ψ_emitter=0.58±0.11, ψ_env=0.56±0.11, ζ_topo=0.20±0.05.
- Observables. Ω_R/2π=49.8±1.6 MHz, κ_drift=0.92±0.19 kHz·s^{-1/2}, Δ_Lamb/2π=31±7 kHz, Δ_AC/2π=54±11 kHz, Δ_pull/2π=-73±15 kHz, Γ_eff/2π=210±25 kHz, T2*=1.52±0.20 μs, σ_y(τ_c)=2.1×10^{-4}±0.3×10^{-4}, τ_c=9.6±1.8 ms.
- Metrics. RMSE=0.041, R²=0.918, χ²/dof=1.04, AIC=10568.4, BIC=10718.9, KS_p=0.295; vs. mainstream baseline ΔRMSE=−17.3%.
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 | 8 | 7 | 9.6 | 8.4 | +1.2 |
Robustness | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
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 | 6 | 6 | 3.6 | 3.6 | 0.0 |
Extrapolation Ability | 10 | 9 | 7 | 9.0 | 7.0 | +2.0 |
Total | 100 | 85.0 | 71.0 | +14.0 |
2) Aggregate Comparison (Unified Metric Set)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.041 | 0.050 |
R² | 0.918 | 0.871 |
χ²/dof | 1.04 | 1.22 |
AIC | 10568.4 | 10762.9 |
BIC | 10718.9 | 10973.2 |
KSp_p | 0.295 | 0.206 |
#Parameters kk | 12 | 15 |
5-fold CV error | 0.045 | 0.055 |
3) Rank-Ordered Differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-Sample Consistency | +2 |
4 | Extrapolation Ability | +2 |
5 | Goodness of Fit | +1 |
5 | Robustness | +1 |
5 | Parameter Parsimony | +1 |
8 | Falsifiability | +0.8 |
9 | Computational Transparency | 0 |
10 | Data Utilization | 0 |
VI. Summative Assessment
Strengths
- Unified multiplicative structure (S01–S05) captures joint evolution among ΩR/κdrift\Omega_R/\kappa_{\text{drift}}, ΔLamb/ΔAC/Δpull\Delta_{\text{Lamb}}/\Delta_{\text{AC}}/\Delta_{\text{pull}}, and Γeff/T2∗/σy2(τ)\Gamma_{\text{eff}}/T_2^*/\sigma_y^2(\tau). Parameters (γ_Path, k_SC, k_STG, k_TBN, θ_Coh, η_Damp, ξ_RL, ψ_cavity, ψ_emitter, ψ_env, ζ_topo) are physically interpretable and engineerable.
- Mechanistic identifiability separates contributions from cavity/emitter channels, tensor background noise, and coherence-window limits to frequency drift and stability.
- Engineering usability: raising θ_Coh, optimizing cavity geometry/coupling (↑ψ_cavity, ζ_topo), and suppressing σ_env jointly lower κ_drift/Γ_eff and extend T2*.
Blind Spots
- Under strong drive/nonlinearity or many-emitter ensembles, Tavis–Cummings and saturation corrections are required.
- With ultra-low temperature yet strong 1/f1/f flicker noise, slope estimates of σy2(τ)\sigma_y^2(\tau) are window-sensitive; multi-window robust evaluation is recommended.
Falsification Line & Experimental Suggestions
- Falsification. If EFT parameters → 0 and the covariance among {Ω_R, κ_drift, Δ_* , Γ_eff, T2*, σ_y^2(τ)} is fully reproduced by mainstream models with global ΔAIC<2, Δ(χ²/dof)<0.02, and ΔRMSE≤1%, the mechanism is refuted.
- Suggestions.
- Detuning–power map: plot (Δ×P)(\Delta \times P) with iso-contours of Ω_R and κ_drift.
- Cavity-pulling calibration: sweep cavity length/thermal drift to separate Δ_pull from Δ_AC.
- Environmental suppression: tri-axis isolation (magnetic/vibration/thermal) to reduce σ_env, validating linear k_TBN.
- Coherence-window engineering: pulse shaping and filtering to increase θ_Coh, extending T2* and reducing Γ_eff.
External References
- Reviews on Jaynes–Cummings physics and vacuum Rabi splitting in cQED.
- Methods for AC–Stark and Lamb shift corrections in cQED.
- Texts on cavity pulling, Allan variance, and frequency-drift modeling.
- Surveys of dephasing and spectral diffusion in spins/quantum dots/atoms.
- Advances in noise spectra and stabilization for low-temperature microwave/optical cQED.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Dictionary. Ω_R/2π [MHz], κ_drift [kHz·s−1/2^{-1/2}], Δ_Lamb/2π [kHz], Δ_AC/2π [kHz], Δ_pull/2π [kHz], Γ_eff/2π [kHz], T2* [μs], σ_y(τ) [–], τ_c [ms].
- Processing. Frequency/time calibration; spectral/time-domain joint decomposition of shifts; errors-in-variables propagation; hierarchical MCMC convergence and prior sensitivity; multi-window Allan + PSD constraints for random-walk and 1/f1/f components.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out. Parameter variation < 15%; RMSE fluctuation < 10%.
- Hierarchical robustness. σ_env↑ → κ_drift↑, Γ_eff↑, T2*↓; evidence for γ_Path>0 exceeds 3σ.
- Noise stress test. +5% 1/f1/f and mechanical perturbations raise ψ_env and the tail of σ_y^2(τ); overall parameter drift < 12%.
- Prior sensitivity. With γ_Path ~ N(0,0.04^2), posterior means shift < 9%; evidence difference ΔlogZ ≈ 0.6.
- Cross-validation. k=5 CV error 0.045; blinded new-condition tests maintain ΔRMSE ≈ −14%.
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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