42-EFT.WP.Heat.Decoherence v1.0 | Chapter 13: Case Library — SC / NV / QD / Optomechanics / Acoustics / 2D

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Chapter 13: Case Library — SC / NV / QD / Optomechanics / Acoustics / 2D

I. Objectives & Scope

• Provide cross-platform end-to-end reference implementations (data → cleaning/conventions → observation construction → rate mapping/estimation → evaluation → design recommendations) covering SC superconducting, NV centers, semiconductor quantum dots, optomechanical systems, acoustic resonators, and 2D materials.
• All formulas/symbols/definitions are in English with backticks; SI units; ω/f and PSD conventions per Ch. 2; spectrum–response–rate mapping per Chs. 4/5; heat transport & BCs per Ch. 6; platform channels per Ch. 7; engineering measures per Ch. 8; metrology & inversion per Ch. 9.

II. Common Data Contract & Pipeline (Required)

• S130-1 (Case dataset card)
• unit_system: "SI"
• case_id: "<SC|NV|QD|Optomech|Acoustic|2D>"
• scene_tag: "<mK-cryostat|room-temp|UHV|...>"
• device:
• omega0: "<rad/s>"
• bias: {lambda: "<Φ|Q|I_c|B|Vg|strain>", d_omega_d_lambda: "<rad/s/unit>"}
• baths:
• J_omega: {family: "ohmic|piezo|custom", params: {eta:"<...>", s:"<...>", omega_c:"<rad/s>"}}
• S_in: {family: "Johnson|one_over_f|TLS|white|custom", params:{...}, type:"one-sided|two-sided", freq:"ω|f"}
• thermal:
• kappa_T: "<W/m/K>", R_K: "<K m^2/W>", sources: {Q:"<W/m^3>"}, T0: "<K>"
• protocols:
• type: ["Ramsey","Echo","CPMG","T1","NoiseTherm"]
• timing: {t: "<s>", N: "<int>", phases: "<{φ_k}>"}
• readout:
• snr: "<->", enbw: "<Hz>", tau_int: "<s>", model: "Gaussian|Poisson|custom"
• uncertainty:
• Σ_y: "<block-diagonal or sparse>", priors: {eta:"...", s:"...", omega_c:"...", tan_delta:"..."}
• references: ["Heat.Decoherence v1.0:Ch.2–Ch.12 anchors"]
• S130-2 (Cleaning & conventions)
  Standardize Fourier/FDT/PSD conventions; if reusing arrival-time chain, record convention/delta_form/gamma(ell)/d_ell; phase unwrapping and ENBW checks are written into Σ_y.
• S130-3 (Observations & rate mapping)
  Build h(·), F(ω) and S_{ΔωΔω}(ω) per Chs. 4/5; map J(ω) → {Γ_1, Γ_φ} and produce T_1/T_2/Q.
• S130-4 (Fusion & inversion)
  Apply Ch. 9 MLE/Bayesian inversion, merging multiple protocols (Ramsey/Echo/CPMG/T1) with thermal-chain/platform priors.
• S130-5 (Evaluation & design)
  Output CRLB/posterior CI, NEES/NIS, pass@coverage, and design recommendations for filtering/cooling/bandgaps per Ch. 8.
• M13-1 (Card → cleaning)
  Import card → validate conventions/units → phase unwrapping/leakage correction → initial thermal-chain estimate → update Σ_y.
• M13-2 (Cleaning → observations/matrices)
  Build y, H and S_{ΔωΔω}(ω); form J, B, Σ_y (Ch. 10).
• M13-3 (Observations → estimation)
  MLE/Bayes to obtain {T_1, T_2, Γ_1, Γ_φ, θ} with CI; identify non-identifiable directions.
• M13-4 (Estimation → evaluation/design)
  Generate spectral/rate response surfaces and thermal margins; propose minimal modification set and engineering quotas.

III. Case A: SC — Superconducting Circuits

• S130-6 (Channels)
  1/f flux/charge, dielectric TLS, quasiparticles, radiative leakage.
• S130-7 (Mapping)
  Γ_φ^{(λ)} = (1/2)(∂ω_0/∂λ)^2 S_{λλ}(0); Γ_1 = ω_0/(2Q_{die}) + Γ_1^{(qp)} + Γ_1^{(rad)}.
• S130-8 (Protocols)
  T1 + Ramsey/Echo/CPMG; record cryogenic thermal chain and readout-line attenuation/filter chain.
• Outputs: {T1,T2,Γ1,Γφ}, S_eff(ω), attainment vs Q_min; advice: tighter p_i control, increase P_margin, quotas for coax/attenuators and shielding.

IV. Case B: NV — Diamond NV Centers

• S130-9 (Channels)
  Nuclear Overhauser, surface spins/electric noise, phonon-induced T_1.
• S130-10 (Mapping)
  Γ_φ^{(hf)} = (γ_e)^2 S_{B_zB_z}(0)/2; Γ_1^{(ph)} ∝ J_ph(ω_0) coth(ħω_0/2k_B T).
• S130-11 (Protocols)
  Ramsey/Echo/XY8; noise thermometry and stress/temperature sweeps to separate TLS/Johnson.
• Outputs: {T1,T2} and recommendations on surface treatment, microwave filtering, and bandgaps (phononic membranes).

V. Case C: Quantum Dots (Semiconductor)

• S130-12 (Channels)
  Charge-trap 1/f, e–ph (deformation/piezo), interface roughness.
• S130-13 (Mapping)
  Γ_1^{(ph)}(ω_0) ∝ J_ph(ω_0); Γ_φ^{(Q)} = (1/2)(∂ω_0/∂Q)^2 S_{QQ}(0).
• S130-14 (Protocols)
  Gate scans + Ramsey/Echo/CPMG; use CPMG to upsample low-frequency window.
• Outputs: S_{QQ}(ω) shape, T_1/T_2, dielectric/stress-engineering guidance.

VI. Case D: Optomechanical Systems

• S130-15 (Channels)
  Mechanical internal loss (Akhiezer/thermoelastic/clamping), radiation-pressure backaction, absorption heating.
• S130-16 (Mapping)
  Γ_{th} ≈ Γ_m n̄(ω_m,T); suppress via |F(ω)|^2 acting on S_{FF}(ω).
• S130-17 (Protocols)
  Ring-up/pump with weak probe, thermal imaging/drift, PSD/CPSD.
• Outputs: Q, Γ_m(T), and design advice for cooling/cavity coupling (g_0, κ) and phononic bandgaps.

VII. Case E: Acoustic Resonators (MEMS/SAW/BAW)

• S130-18 (Channels)
  Akhiezer, Landau–Rumer, thermoelastic, surface/anchor losses.
• S130-19 (Mapping)
  Q^{-1} = Q_{AKE}^{-1} + Q_{LR}^{-1} + Q_{TED}^{-1} + Q_{surf}^{-1}; Γ_1 = ω_0/Q.
• S130-20 (Protocols)
  Loop readout + Ramsey/Echo-equivalent sequences; thermal-chain/support modeling.
• Outputs: κ_eff, R_K with structure/material/interface recommendations.

VIII. Case F: 2D Materials (Graphene / MoS₂ / 2D SC)

• S130-21 (Channels)
  Interface/edge TLS, stress-induced 1/f, low-D phonon scattering, 2D-SC quasiparticle/radiation channels.
• S130-22 (Mapping)
  Size/layer dependence for J_ph(ω) and R_K; Γ_φ dominated by S_{λλ}(0).
• S130-23 (Protocols)
  Raman/thermal imaging + Ramsey/Echo; thin-film bandgap/support optimization.
• Outputs: T_1/T_2/Q, S_eff(ω), and layer/interface engineering advice.

IX. Unified Outputs & Report Template

• S130-24 (Report schema)
• case_report:
• summary: {...}
• estimates: {T1:"...", T2:"...", Gamma1:"...", Gamma_phi:"...", Q:"..."}
• spectra: {S_dw:"...", S_eff:"..."}
• thermal: {T_node:"...", q_links:"...", P_margin:"..."}
• channels: {contribs: {TLS:"...", Johnson:"...", phonon:"...", qp:"...", rad:"..."}, ranking:"..."}
• design: {mods: ["filter","cooling","phononic","materials","bias"], quotas: {...}}
• consistency: {NEES:"...", NIS:"...", coverage:"..."}
• reproducibility: {scripts:"...", env:"...", anchors:["S130-*","M13-*","I13-*"], hash:"..."}
• M13-5 (Report generation)
  Aggregate estimates & statistics, channel decomposition & priorities, design quotas, and regression versions.

X. Implementation Bindings & Interfaces (I13-*)

• I13-1 load_case_card(path) -> {data, meta}
• I13-2 clean_and_standardize(data, contracts) -> {clean, Σ_y, notes}
• I13-3 build_observables(clean, models) -> {y, H, S_dw(ω), J(ω)}
• I13-4 estimate_params(y, H, priors) -> {T1, T2, Gamma1, Gamma_phi, θ, cov}
• I13-5 decompose_and_rank(channels, rates) -> {contribs, ranking}
• I13-6 recommend_design(metrics, constraints) -> {mods, quotas}
• I13-7 emit_case_report(artifacts) -> {report.yaml, repro_bundle}

Error codes: E/INPUT (missing), E/UNIT (units), E/NUMERIC (non-convergence/non-positive spectrum), E/IDENTIFIABILITY (ill-conditioned), E/CONSISTENCY (cross-protocol inconsistency).

XI. Quality Gates (This Chapter)

• Q1 Conventions & units: match Chs. 2/5; pass check_dim.
• Q2 Consistency/coverage: NEES/NIS and pass@coverage near nominal; cross-protocol (Ramsey/Echo/CPMG/T1) agreement.
• Q3 Energy & thermal chain: reconcile with Ch. 6 thermal chain, P_margin ≥ 0; thermal imaging/drift consistent.
• Q4 Identifiability: Fisher conditioning within threshold; provide sampling/timing/bandwidth improvements if insufficient.
• Q5 Traceability: materials/geometry/conventions/filtering/cooling/bandgaps and version hashes traceable in the dataset; complete repro_bundle.

XII. Cross-References & Anchors (This Chapter)

• Cross-refs (fixed style): Chs. 2 (conventions), 4 (open systems), 5 (FDT/spectra), 6 (heat), 7 (platform channels), 8 (coherence engineering), 9 (metrology & inversion), 11 (simulation stack).
• Anchors: Minimal S130-1—S130-24; Workflows M13-1—M13-5; Interfaces I13-1—I13-7.

XIII. Summary
This chapter delivers SC / NV / QD / Optomech / Acoustic / 2D reproducible case patterns and report templates, spanning a unified metrology & inversion chain across spectrum–thermal–rate–protocol. It outputs channel decomposition and engineering quotas, with quality gates and repro_bundle ensuring cross-platform/temperature consistency, traceability, and auditability.