42-EFT.WP.Heat.Decoherence v1.0 | Chapter 9: Metrology Chain & Estimation — T1/T2/Ramsey/Echo/Noise Thermometry

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Chapter 9: Metrology Chain & Estimation — T1/T2/Ramsey/Echo/Noise Thermometry

I. Objectives & Applicability

• Establish a minimal usable framework for the decoherence metrology chain and parameter estimation, covering experimental protocols for relaxation/coherence times T_1/T_2 (Ramsey/echo/multipulse), observation→parameter mappings, Fisher/CRLB and Bayesian inversion, and noise thermometry based on measured spectra.
• All formulas/symbols/definitions are in English and wrapped in backticks; SI units; ω/f and PSD conventions per Chapter 2; spectrum→rate mapping per Chs. 4/5; heat transport per Ch. 6; platform channels per Ch. 7.

II. Minimal Statements & Principles (S90-*)

• S90-1 (T1 relaxation)
  For a TLS, excited-state population P_e(t) = P_e(0) e^{-t/T_1}, T_1 = 1/Γ_1, with
  Γ_1(ω_0,T) = (π/2) |A_{eg}|^2 J(ω_0) coth(ħω_0/2k_B T).
• S90-2 (T2 coherence)
  1/T_2 = Γ_φ + (1/2) Γ_1; for quasi-static noise variance σ_φ^2, short-time C(t) ≈ e^{-(t/T_φ)^2}, T_φ = 1/σ_φ.
• S90-3 (Ramsey decay)
  Envelope C_R(t) = exp{ - χ_R(t) }, χ_R(t) = (1/π) ∫_0^∞ dω S_{ΔωΔω}(ω) (sin(ω t/2)/ω)^2.
• S90-4 (Echo / CPMG)
  Echo filter |F_E(ω,t)|^2 gives
  C_E(t) = exp{ - (1/π) ∫_0^∞ dω S_{ΔωΔω}(ω) |F_E(ω,t)|^2 / ω^2 }; with N pulses, CPMG shifts sensitivity to ~ N/t.
• S90-5 (Noise thermometry via FDT)
  From measured S_{xx}(ω) infer T:
  S_{xx}(ω) = 2 coth(ħω/2k_B T) · Im χ_{xx}(ω); if χ is known, invert for T or use the limit k_B T ≫ ħω.
• S90-6 (Measurement matrix & CRLB)
  With y = h(θ) + ε, ε ~ 𝒩(0, Σ_y), Fisher
  F(θ) = (∂h/∂θ)^T Σ_y^{-1} (∂h/∂θ), and cov(θ̂) ⪰ F^{-1} (CRLB).
• S90-7 (Frequency drift & sensitivity)
  If Δω = (∂ω_0/∂λ) δλ, then S_{ΔωΔω}(0) = (∂ω_0/∂λ)^2 S_{λλ}(0) mapping device-bias noise to Γ_φ.
• S90-8 (Readout chain & noise floor)
  Readout SNR via ENBW and integration time τ_int folds into Σ_y, with σ_y^2 ∝ ENBW/τ_int.

III. Observation Models & Protocols (Ramsey / Echo / Multipulse / Noise Thermometry)

1. Ramsey: P_e(t_i) = 1/2 [ 1 + V_R e^{-χ_R(t_i)} cos(Δ t_i + φ) ]; fit T_2^* or spectral params {A, γ, …}.
2. Hahn Echo: P_e(t_i) = 1/2 [ 1 + V_E e^{-χ_E(t_i)} ]; suppresses low-ω drift, probes midband spectral weight.
3. Multipulse (CPMG/XY/UDD): choose pulse count N and total time t to place passbands; effectively sample S_{ΔωΔω}(ω_k) at ω_k ≈ kπ N/t.
4. Relaxation (T1): single-exponential or multi-process (quasiparticle + dielectric + radiation); co-fit with Ch. 7 channel decomposition.
5. Noise thermometry:
   • High-ω: measure S_{xx}(ω) and Im χ(ω), invert T via S90-5;
   • Low-ω: use Johnson–Nyquist S_VV = 4 k_B T R or resonator occupancy n̄(ω_0,T).

IV. Metrology Chain & Data Contract (Required Fields)

unit_system: "SI"

protocol:

type: "Ramsey|Echo|CPMG|XY|T1|NoiseTherm"

timing: {t: "<s>", N: "<pulses>", phases: "<{φ_k}>"}

readout:

snr: "<->", enbw: "<Hz>", tau_int: "<s>", model: "Gaussian|Poisson|custom"

models:

rates: {Gamma1: "<model|table>", Gamma_phi: "<model|table>"}

spectrum: {S_dw: "<model|table>", chi: "<Im χ(ω) or transfer>"}

priors:

theta: {A:"<...>", gamma:"<...>", omega_c:"<rad/s>", T:"<K>", …}

observables:

y: "<P_e(t) or signal>", times: "<{t_i}>", covariance: "Σ_y"

design:

target_ci: {T1:"<%>", T2:"<%>", T:"<K>"}, budget: {time:"<s>", repeats:n}

outputs:

estimates: ["T1","T2","Gamma1","Gamma_phi","T"], spectra: ["S_dw(ω)","Seff(ω)"], ci: "<95%>"

references: ["Heat.Decoherence v1.0:Ch.2 S20-*","Ch.4 S40-*","Ch.5 S50-*","Ch.7 S70-*"]

V. Algorithmic Workflows (M9-*)

• M9-1 (Observation→model assembly)
  Build h(θ) and F(ω) per protocol; construct Σ_y.
• M9-2 (Initialization & linearization)
  Seed {T_1, T_2^*, A, γ} from priors/quick fits; compute J = ∂h/∂θ and Fisher F.
• M9-3 (MLE / LS)
  Solve θ̂ = argmin (y - h(θ))^T Σ_y^{-1} (y - h(θ)); report cov(θ̂) ≈ F^{-1}.
• M9-4 (Bayesian inversion)
  With p(θ) and likelihood p(y|θ), run HMC/NUTS or VI; output posterior means/intervals and logZ.
• M9-5 (Spectral reconstruction & thermometry)
  Reconstruct S_{ΔωΔω}(ω) and Seff(ω) from θ̂ or posterior draws; solve T̂ for noise thermometry with uncertainty.
• M9-6 (Experiment-design iteration)
  Optimize {t_i, N, repeats} and readout allocation via Fisher or D/A/E-opt criteria to minimize parameter variances.
• M9-7 (Cross-validation & coverage)
  Validate with independent data/alternate sequences; NEES/NIS and pass@coverage near nominal.

VI. Implementation Bindings & Interfaces (I90-*)

• I90-1 assemble_protocol(protocol, design) -> {h(θ), F(ω), Σ_y}
• I90-2 estimate_T1T2(y, times, model, priors) -> {T1, T2, Gamma1, Gamma_phi, cov}
• I90-3 infer_spectrum(y, protocol, priors) -> {S_dw(ω), Seff(ω), post}
• I90-4 noise_thermometry(S_xx, chi, band) -> {T_hat, ci}
• I90-5 design_experiment(Fisher, budget, objective) -> {times*, N*, repeats*, expected_ci}
• I90-6 validate_coverage(estimates, cov, metrics) -> {NEES, NIS, pass@coverage}

Error codes: E/INPUT (missing), E/UNIT (units), E/NUMERIC (non-convergence), E/IDENTIFIABILITY (ill-conditioned), E/MODEL (model mismatch).

VII. Quality Gates (This Chapter)

• Q1 Convention consistency: ω/f and PSD one-/two-sided choices match Chs. 2/5; filter normalization correct.
• Q2 Units/dimensions: unit columns complete for P_e(t), S_{xx}(ω), χ(ω), and Σ_y; pass check_dim.
• Q3 Identifiability: Fisher conditioning within bounds; if large cond(F), add time points/pulses or priors.
• Q4 Coverage & consistency: NEES/NIS and pass@coverage near nominal; Ramsey/Echo/CPMG agreement.
• Q5 Traceability: protocol, timing, readout model, priors, and version hashes recorded; emit repro_bundle for replay.

VIII. Cross-References & Anchors (This Chapter)

• Cross-refs (fixed style): Ch. 2 (metrology & conventions), Ch. 4 (open QS & filters), Ch. 5 (FDT & spectral estimation), Ch. 7 (platform channels), Ch. 11 (simulation stack & SBC).
• Anchors: Minimal S90-1—S90-8; Workflows M9-1—M9-7; Interfaces I90-1—I90-6.

IX. Summary
This chapter establishes a closed metrology loop from protocol design → observation modeling → Fisher/Bayesian estimation → noise thermometry → experiment-design iteration, yielding traceable estimates of T_1/T_2/Γ_1/Γ_φ/T and S_{ΔωΔω}(ω), with standardized data contracts, quality gates, and interfaces for cross-platform, cross-temperature, and cross-protocol consistency and auditability.