42-EFT.WP.Heat.Decoherence v1.0 | Chapter 8: Thermal Coupling & Coherence Engineering — Shielding / Filtering / Cooling / Phononic Crystals

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Chapter 8: Thermal Coupling & Coherence Engineering — Shielding / Filtering / Cooling / Phononic Crystals

I. Objectives & Applicability

• Within a unified noise → spectrum → response → rate framework, establish minimal usable schemes for thermal coupling and coherence engineering: passive/active shielding, spectral filtering (electromagnetic/magnetic/mechanical/thermal domains), multi-stage cooling & thermal anchoring, and phononic-crystal/bandgap structures. Targets: increase T_1/T_2, reduce Γ_1/Γ_φ, and suppress frequency drift and hot-carrier injection.
• All formulas/symbols/definitions are in English and wrapped in backticks; SI units; ω/f and PSD conventions follow Chapter 2; spectrum→rate mapping per Chapters 4/5; heat transport per Chapter 6.

II. Minimal Statements & Principles (S80-*)

• S80-1 (Effective spectrum suppression)
  After control/filtering the effective spectrum is
  S_eff(ω) = |F(ω)|^2 S_{AA}(ω), with decoherence kernel
  χ(t) = (1/π) ∫_0^∞ S_eff(ω) (sin^2(ω t/2)/ω^2) dω.
• S80-2 (Multi-stage thermal chain & occupation)
  Node i balance in a cascaded thermal chain:
  Σ_j G_{ij}(T_i - T_j) + P_i^{diss} - P_i^{cool} = 0; oscillator occupation n̄(ω,T) = 1/(e^{ħω/k_B T}-1).
• S80-3 (Bandgap suppression)
  Phononic crystals with Bloch dispersion ω(k) yield a gap Ω_g = [ω_1, ω_2]; in-gap attenuation with
  L_att(ω) = 1/(2 Im k(ω)), hence J_{ph}(ω) → J_{ph}(ω) e^{-L/L_att(ω)}.
• S80-4 (Radiative & conductive loading)
  Radiative heat P_rad = ε σ A (T_h^4 - T_c^4); interface resistance ΔT = R_K q_n; effective conductance G = κ A / L.
• S80-5 (Filter networks & participation)
  For dielectric loss:
  1/Q_{die}(ω) = Σ_i p_i(ω) tanδ_i(ω); filter transfer H(ω) acts on input-coupled noise S_{in}(ω) yielding S_A(ω) = |H(ω)|^2 S_{in}(ω).
• S80-6 (Cooling limits)
  At operating point:
  P_{cool}(T_{stage}) = Σ G_{link}(T_{node}-T_{stage}) + P_{rad} + P_{diss}, defining achievable T_{node} and margin.

III. Engineering Modules & Methods (Shielding / Filtering / Cooling / Phononic Crystals)

• Shielding (Magnetic/EM/Vibration/Thermal):
  μ-metal/superconducting shields (magnetic), Faraday cages (EM); suspension/damping (vibration); multi-layer radiation shields (ε, A, spacing) and Cu braids/pure Cu/superconducting leads (conduction/thermal leak control).
• Filtering (frequency/space domains):
  Cryogenic RC/LC/TEE, Eccosorb/coax attenuators; mechanical acoustic low-pass (mass–spring chains) and phononic-crystal bandgaps; thermal R_th–C_th networks that smooth pulsed heat loads.
• Cooling (passive/active/cascaded):
  Cold-finger/thermal-anchoring layout, isothermal blocks, TIMs/solders/coatings; He-3/He-4/DR cascade budgeting; thermal-leak tables and clamps for cables/coax/fibers.
• Phononic crystals/bandgaps:
  1D periodic beams, 2D honeycomb/lattices, 3D metamaterials; target gap to cover ω_0 and harmonics; design L_att(ω) and duty cycle to meet attenuation-length constraints.

IV. Metrology Chain & Data Contract (Required Fields)

unit_system: "SI"

targets:

T2_min: "<s>", Q_min: "<->", nbar_max: "<->", T_node_max: "<K>"

noise_in:

SA_in: "<model|table>" # input-coupled noise spectrum S_in(ω)

H_chain: ["<H1(ω)>","<H2(ω)>","..."] # cascaded filters/attenuators

thermal_chain:

links: [{name:"Cu_braid", G:"<W/K>"}, {name:"SS_coax", G:"<W/K>"}]

rad: {epsilon:"<->", area:"<m^2>"}

stages: [{name:"4K", Pcool:"<W>"}, {name:"100mK", Pcool:"<mW>"}]

phononic:

dispersion: "<ω(k) or band diagram>", gap: {w1:"<rad/s>", w2:"<rad/s>"}, L: "<m>"

dielectrics:

p: [{region:"...", value:"<->"}], tan_delta: [{mat:"...", value:"<->"}]

control:

sequence: "Ramsey|Echo|CPMG|XY8|Uhrig|custom", timing: "<{t_k}>"

outputs:

Seff: "S_eff(ω)", rates: ["Gamma1","Gamma_phi","T1","T2","Q"], thermal: ["T_node","q_links","P_margin"]

uncertainty:

Σ_y: "<blocks>", priors: {R_K:"...", kappa:"...", epsilon:"...", gap:"..."}

references: ["Heat.Decoherence v1.0:Ch.4 S40-*","Ch.5 S50-*","Ch.6 S60-*","Ch.7 S70-*"]

V. Design & Optimization Workflows (M8-*)

• M8-1 (Backsolve from spectral → rate targets)
  With {T2_min, Q_min}, invert S80-1/5 to obtain allowed S_AA envelope and target |H(ω)|^2 roll-off (dB/dec).
• M8-2 (Thermal-chain balance & margin)
  Build the multi-stage chain → compute T_node, q_links, and the cooling margin P_margin = P_cool - P_load → iterate cable/anchor layout and materials.
• M8-3 (Bandgap synthesis)
  From desired Ω_g, length L, and fabrication constraints, optimize geometry (period, aperture, duty) to meet L_att(ω_0).
• M8-4 (Multi-objective optimization)
  Minimize
  Φ = w_1 · ∫ W(ω) S_eff(ω) dω + w_2 · T_node + w_3 · cost + w_4 · mass,
  subject to P_margin ≥ 0, L_att(ω_0) ≥ L_req, Q ≥ Q_min.
• M8-5 (Validation & regression)
  Validate Seff/Γ with time-domain (Ramsey/echo) and frequency-domain (PSD/CPSD) data; validate T_node with thermal imaging/drift; update versions.

VI. Implementation Bindings & Interfaces (I80-*)

• I80-1 synthesize_filter(chain, targets) -> {H(ω), Seff(ω), specs}
• I80-2 thermal_balance(links, stages, rad, diss) -> {T_node, q_links, P_margin}
• I80-3 design_phononic_gap(dispersion, gap, L, constraints) -> {geom*, L_att(ω), fab_notes}
• I80-4 map_to_rates(Seff, J, T, control) -> {Gamma1, Gamma_phi, T1, T2, Q}
• I80-5 multiobjective_opt(config, weights) -> {design*, Pareto_front}
• I80-6 validate_thermal_and_coherence(meas, model) -> {pass@coverage, residuals, update}

Error codes: E/INPUT (missing), E/UNIT (units), E/NUMERIC (non-convergence/energy imbalance), E/FAB (fabrication constraint), E/IDENTIFIABILITY (ill-conditioned).

VII. Quality Gates (This Chapter)

• Q1 Convention consistency: ω/f, one-/two-sided PSD, Seff and H(ω) units/normalization consistent with Chs. 2/5; pass check_dim.
• Q2 Energy & thermal balance: thermal_balance satisfies conservation; P_margin ≥ 0; radiative/conductive/interface loads accounted.
• Q3 Bandgap & attenuation: phononic Ω_g and L_att(ω) reconcile measurement/simulation; bandwidth/tolerance met.
• Q4 Rates & figures: {Γ_1, Γ_φ, T1, T2, Q} meet targets within tolerance; otherwise output minimal modification set (filter stages, materials, geometry, sequence).
• Q5 Traceability & repro: design cards include materials, surface treatment, geometry & fabrication tolerances, clamping/anchoring, and a repro_bundle.

VIII. Cross-References & Anchors (This Chapter)

• Cross-refs (fixed style): Ch. 4 (open systems, filter functions), Ch. 5 (FDT & spectra), Ch. 6 (heat & chains), Ch. 7 (platform channels); closed-loop evaluation with Ch. 9 (metrology & inference).
• Anchors: Minimal S80-1—S80-6; Workflows M8-1—M8-5; Interfaces I80-1—I80-6.

IX. Summary
This chapter integrates shielding/filtering/cooling/phononic-crystal measures into a unified spectrum–thermal–rate pathway, providing an executable route from target setting through structural synthesis and thermal balance to rates/coherence attainment, with data contracts, interfaces, and quality gates for cross-platform/cross-temperature comparability and auditability.