42-EFT.WP.Heat.Decoherence v1.0 | Chapter 5: Fluctuation–Dissipation & Spectral Estimation — FDT / Johnson / 1/f / TLS

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Chapter 5: Fluctuation–Dissipation & Spectral Estimation — FDT / Johnson / 1/f / TLS

I. Objectives & Applicability

• Under a unified metrology convention, establish a minimal usable framework for the Fluctuation–Dissipation relation (FDT) and spectral estimation (Welch/Multitaper/Bayesian), covering Johnson–Nyquist thermal noise, 1/f, and TLS (two-level systems) noise; make explicit the relations among S_{xx}(ω), J(ω), and the response χ(ω), and supply computable inputs for Γ_1, Γ_φ in Chapters 4 and 7.
• All formulas/symbols/definitions are in English and wrapped in backticks; SI units; ω/f and one-/two-sided PSD conventions follow Chapter 2.

II. Minimal Statements & Principles (S50-*)

1. S50-1 (Quantum FDT)
   S_{xx}(ω) = 2 coth( ħω / (2 k_B T) ) · Im χ_{xx}(ω), with passivity Im χ_{xx}(ω) ≥ 0.
2. S50-2 (Classical limit of FDT)
   For k_B T ≫ ħω, coth(ħω/2k_B T) ≈ 2k_B T / (ħω), hence S_{xx}(ω) ≈ 4 k_B T · Im χ_{xx}(ω) / (ħω).
3. S50-3 (Johnson–Nyquist)
   Voltage noise of resistor R: S_VV(f) = 4 k_B T R (one-sided f, unit V^2/Hz); equivalent current noise S_II = 4 k_B T / R.
4. S50-4 (Cross-spectrum & coherence)
   γ_{xy}^2(ω) = |S_{xy}(ω)|^2 / ( S_{xx}(ω) S_{yy}(ω) ), with 0 ≤ γ_{xy}^2 ≤ 1.
5. S50-5 (1/f & TLS spectra)
   • 1/f: S(ω) = A^2 / |ω|^γ (0 < γ ≈ 1), low-frequency dominated pure dephasing Γ_φ = (1/2) S_{AA}(0).
   • TLS: S_{TLS}(ω) = Σ_i (4 τ_i)/(1 + ω^2 τ_i^2) · w_i, or with P(τ)∝1/τ continuum approaching 1/f.
6. S50-6 (Filter-function mapping)
   Under control sequences, effective spectrum S_eff(ω) = S_{AA}(ω) |F(ω)|^2; TLS-type qubit dephasing χ(t) = (1/π) ∫_0^∞ dω S_eff(ω) · (sin^2(ω t/2)/ω^2).
7. S50-7 (Parseval check)
   Var[x(t)] = (1/2π) ∫_{-∞}^{∞} S_{xx}(ω) dω (two-sided) for time/frequency energy consistency.

III. Models & Derivations (Johnson / 1/f / TLS / Response)

• Johnson–Nyquist: from conductance Y(ω)=1/Z(ω) and FDT, S_{II}(ω) = 2 ħω coth(ħω/2k_B T) Re{Y(ω)}; classical limit recovers 4 k_B T Re{Y}.
• 1/f model: broad relaxation spectrum P(τ)∝1/τ or structural defects; with finite observation time T_obs, low-frequency cutoff ω_L ≈ 2π/T_obs.
• TLS noise: sum of Lorentzian components; slopes of S(ω,T) vs. ω/T separate TLS from Johnson channels.
• Response function: obtain χ_{xx}(ω) from microscopic models or fits; passivity/causality enforce Im χ ≥ 0 and Kramers–Kronig relations.

IV. Metrology Chain & Data Contract (Required Fields)

unit_system: "SI"

spectral_conventions:

fourier: {forward: "∫ f(t) e^{-iωt} dt", inverse: "(1/2π) ∫ F(ω) e^{iωt} dω"}

psd: {type: "one-sided|two-sided", freq: "ω|f"}

fdt:

classical: "S_VV = 4 k_B T R"

quantum: "S_xx(ω) = 2 coth(ħω/2k_B T) · Im χ_xx(ω)"

models:

one_over_f: {A: "<unit>", gamma: "<~1>", omega_L: "<rad/s>", omega_H: "<rad/s>"}

tls: {weights: "<w_i>", taus: "<τ_i>", or_dist: "P(τ)∝1/τ", omega_c: "<rad/s>"}

estimation:

method: "Welch|Multitaper|Bayes(AR|GP)"

params: {fs:"<Hz>", nfft:n, window:"hann|dpss", overlap:"<0..1>", dpss_NW:"<...>"}

outputs:

psd: "S_xx(ω)", cpsd: "S_xy(ω)", coherence: "γ_xy^2(ω)", ci: "<method>"

uncertainty:

edof: "<effective dof>", ci_level: 0.95

references: ["Heat.Decoherence v1.0:Ch.2 S20-*","Ch.4 S40-*","Ch.7 S70-*"]

V. Spectral-Estimation Algorithms (M5-*)

1. M5-1 (Welch)
   Segment + window → overlap → average; record effective DOF edof ≈ 2K (K independent averages); correct window normalization and ENBW.
2. M5-2 (Multitaper)
   Use DPSS (NW parameter) to build orthogonal tapers; estimate Ŝ(ω) = Σ_l w_l |F_l(ω)|^2; Thomson estimator balances leakage/variance.
3. M5-3 (Bayesian PSD)
   • AR/ARMA priors with MAP/MCMC;
   • GP on log-PSD: log S(ω) ~ GP(m,k), adaptive smoothing with credible intervals.
4. M5-4 (Cross-spectra/coherence)
   Compute S_{xy}, γ_{xy}^2; correct for leakage and mis-registration via phase alignment or FRF deconvolution.
5. M5-5 (CIs & bias correction)
   χ^2 approximations or (block) bootstrap; enforce Parseval consistency and unit checks.

VI. Implementation Bindings & Interfaces (I50-*)

• I50-1 estimate_psd(signal, fs, method, params) -> {S_xx(ω), edof, ci}
• I50-2 estimate_cpsd(x, y, fs, method, params) -> {S_xy(ω), coherence}
• I50-3 fit_noise_models(S_xx, family) -> {A, gamma, w_tls, τ_tls, omega_c, ci}
• I50-4 fdt_map_to_chi(S_xx, T) -> {Im χ(ω), Re χ(ω) via K-K}
• I50-5 build_S_eff(S_AA, F) -> {S_eff(ω), χ(t)}

Error codes: E/INPUT (missing), E/UNIT (unit mismatch), E/NUMERIC (non-positive/divergent spectrum), E/MODEL (fit failure).

VII. Quality Gates (This Chapter)

• Q1 Convention consistency: PSD one-/two-sided and ω/f selections are consistent and versioned in the dataset.
• Q2 Units/dimensions: pass check_dim; explicit rad^2/Hz for phase PSD, V^2/Hz for voltage, A^2/Hz for current.
• Q3 Leakage & bias: record window normalization, ENBW, and frequency resolution; correct for leakage and scalloping.
• Q4 Stationarity & aliasing: detect non-stationarity/drift; enforce anti-aliasing and bandwidth control; detrend/demean if needed.
• Q5 Parseval & cross-validation: time/frequency energy match; validate 1/f/TLS fits & intervals on independent data or across temperatures/bands.

VIII. Cross-References & Anchors (This Chapter)

• Cross-refs (fixed style): Ch. 2 (conventions), Ch. 4 (open quantum systems), Ch. 7 (platform-specific channels), Ch. 10 (robust strategies—DD & filter functions).
• Anchors: Minimal S50-1—S50-7; Workflows M5-1—M5-5; Interfaces I50-1—I50-5.

IX. Summary
With FDT as the bridge, this chapter unifies Johnson, 1/f, and TLS noise models with spectral-estimation methods, forming a closed loop from measurement → spectrum → response/channels → decoherence. Standardized data contracts, quality gates, and interfaces enable reliable S_{xx}(ω) and S_eff(ω) across platforms/temperatures, providing traceable inputs for Chapters 4 and 7.