42-EFT.WP.Heat.Decoherence v1.0 | Chapter 4: Open Quantum Systems — Lindblad / Redfield / Influence Functional

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Chapter 4: Open Quantum Systems — Lindblad / Redfield / Influence Functional

I. Objectives & Applicability

• Establish a minimal usable description for open quantum systems: starting from H = H_S + H_B + H_{SB}, provide the Markov and non-Markov master equations (Lindblad/GKSL, Redfield/TCL) and the path-integral influence functional (Feynman–Vernon/Keldysh) in a unified convention, and map environmental spectral density J(ω) and response χ(ω) to decoherence rates Γ_1, Γ_φ and dynamical observables.
• All formulas/symbols/definitions are in English and wrapped in backticks; SI units; ω/f and PSD conventions follow Chapter 2.

II. Minimal Statements & Principles (S40-*)

• S40-1 (GKSL / Lindblad canonical form)
  \dot ρ = -i[H_S, ρ] + Σ_k 𝒟[L_k]ρ, with 𝒟[L]ρ = LρL^\dagger - (1/2){L^\dagger L, ρ}. Enforce trace preservation and complete positivity (CP).
• S40-2 (Born–Markov–Secular approximations)
  Under weak coupling and short bath memory, jump operators L_k and rates γ_k(ω) are set by J(ω) and n̄(ω,T); KMS guarantees equilibrium ρ_β ∝ e^{-βH_S}.
• S40-3 (Bloch–Redfield tensor)
  \dot ρ_{mn} = -i ω_{mn} ρ_{mn} + Σ_{pq} R_{mn,pq} ρ_{pq}, where R comes from Fourier transforms of bath correlators C_{ab}(t) = ⟨B_a(t)B_b(0)⟩; without RWA it may be non-CP.
• S40-4 (TCL / TCL2 master equation)
  \dot ρ(t) = 𝒦(t) ρ(t), with a time-local expansion 𝒦 = 𝒦_1 + 𝒦_2 + …; second order TCL2 uses only two-point correlators.
• S40-5 (Influence functional / Feynman–Vernon)
  Open-path propagator 𝒥[x,x'] = exp{ -Φ[x,x'] }, with noise kernel ν(t) and dissipation kernel η(t); linked to J(ω), χ(ω), S_{xx}(ω) via FDT.
• S40-6 (Quantum Brownian Motion / Caldeira–Leggett)
  H_{SB} = x Σ_k c_k q_k; for Ohmic J(ω) at high T, obtain quantum Langevin and QBM master equation with diffusion and friction coefficients.
• S40-7 (Rate mapping & control filtering)
  Two-level rates: Γ_1(ω_0) ∝ J(ω_0) coth(ħω_0/2k_B T); pure dephasing Γ_φ = (1/2) S_{AA}(0). Under control, effective spectrum S_eff(ω) = S_{AA}(ω) |F(ω)|^2.

III. Models & Derivations (Approximation Domains)

• Weak-coupling Markov (Lindblad): valid for ||H_{SB}|| τ_B ≪ 1 and τ_S ≫ τ_B; with RWA/diagonalization jumps, non-negative rates respect detailed balance.
• Non-Markov (Redfield/TCL): non-negligible bath correlation time τ_B; Redfield allows memory kernels but may violate CP; TCL2 remains time-local and captures short-memory effects.
• Influence functional: for strong coupling or non-Gaussian spectra use numerical path integrals (QUAPI/ISPI) or Keldysh Dyson equations; obtain ν(t), η(t) from J(ω) integrals.
• Control & filter functions: transform by control H_c(t); the filter F(ω) and S_{AA}(ω) convolution yields residual decoherence; used for dynamical decoupling and noise ID.

IV. Metrology Chain & Data Contract (Required Fields)

unit_system: "SI"

oQS_model:

type: "GKSL|Redfield|TCL2|IF|QBM"

h_sys: "<H_S in chosen basis>"

coupling: {ops: ["A1","A2",...], map_to: "<charge|flux|spin|x|p>"}

bath:

J_omega: {family: "ohmic|sub|super|custom", params: {eta:"<…>", s:"<…>", omega_c:"<rad/s>"}}

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

approximations:

born_markov: true|false

secular_rwa: true|false

cp_enforced: true|false

control:

frame: "lab|rotating(ω0)"

pulses: {sequence: "CPMG|Uhrig|XY8|custom", timing: "<{t_k}>", phase: "<{φ_k}>"}

outputs:

rates: ["Gamma1","Gamma_phi"]

observables: ["⟨σ_z⟩(t)","⟨x⟩(t)","S_eff(ω)"]

uncertainty:

Σ_y: "<meas covariance>", priors: {eta:"...", s:"...", omega_c:"..."}

references: ["Heat.Decoherence v1.0:Ch.2 S20-*","Ch.3 S30-*","Ch.5 S50-*"]

V. Algorithmic Workflows (M4-*)

• M4-1 (Rate generation)
  Given J(ω), n̄(ω,T), A_eg, ω_0 → compute Γ_↓, Γ_↑, Γ_1, Γ_φ via S40-4/7.
• M4-2 (Model selection)
  Choose GKSL / Redfield / TCL2 / IF by τ_B/τ_S and coupling strength; project Redfield to CP if required.
• M4-3 (Time-domain simulation)
  Diagonalize H_S or integrate on the Lie algebra: ρ(t+Δt) = e^{𝒦Δt} ρ(t); or splitting/Krylov/PCG solvers.
• M4-4 (Control filtering)
  Build F(ω) from the sequence, compute S_eff(ω) and residual decoherence; optimize pulse timing to minimize J = ∫ W(ω) S_eff(ω) dω.
• M4-5 (Parameter inference)
  From T_1/T_2/Ramsey/echo/noise spectrum infer {η,s,ω_c} and {Γ_1, Γ_φ} by Fisher/Bayesian methods; output confidence and identifiability.

VI. Implementation Bindings & Interfaces (I40-*)

• I40-1 compute_lindblad_rates(H_S, L_ops, J, T) -> {Γ_↓, Γ_↑, Γ_1, Γ_φ}
• I40-2 redfield_tensor(H_S, couplings, C_t) -> {R, CP_flag}
• I40-3 tcl2_generator(H_S, couplings, C_t) -> {K(t)}
• I40-4 influence_functional(J, T) -> {ν(t), η(t)}
• I40-5 simulate_oQS(H_S, model, params, control) -> {ρ(t), observables}
• I40-6 filter_function(sequence, timing) -> {F(ω)}
• I40-7 dd_optimize(targets, constraints) -> {seq*, S_eff(ω), Γ_res}
• I40-8 infer_oQS_params(data, priors, model) -> {θ̂, cov, logZ}

Error codes: E/INPUT (missing), E/UNIT (units), E/NUMERIC (divergence/non-positive spectrum), E/CP (non-CP), E/IDENTIFIABILITY (ill-conditioned).

VII. Quality Gates (This Chapter)

• Q1 Trace & CP: GKSL ensures CP and Tr ρ = 1; if Redfield/TCL produce negatives, project or tighten approximations.
• Q2 Energy balance & drift: with no drive and thermal bath, ρ(t→∞)→ρ_β; otherwise check KMS and non-negative rates.
• Q3 Numerical stability: control generator spectral radius and steps; use implicit/exponential integrators for stiffness; ensure kernels are PSD in path integrals.
• Q4 Identifiability & bandwidth: control cond(F) = J_θ^T Σ^{-1} J_θ; expand measurement band/temperature or add priors if needed.
• Q5 Traceability & repro: dataset records J(ω), χ(ω), control sequences, and approximation switches (born_markov, secular_rwa, cp_enforced); emit a repro_bundle.

VIII. Cross-References & Anchors (This Chapter)

• Cross-refs (fixed style): Ch. 2 (metrology & conventions), Ch. 3 (baths & spectra), Ch. 5 (FDT & spectral estimation), Ch. 7 (platform channels), Ch. 9 (metrology & inference), Ch. 11 (simulation stack).
• Anchors: Minimal S40-1—S40-7; Workflows M4-1—M4-5; Interfaces I40-1—I40-8.

IX. Summary
Using GKSL/Redfield/TCL/influence functional as the backbone, this chapter maps J(ω), χ(ω) and temperature T to observable dynamics and decoherence rates, and provides executable flows for control filtering and parameter inference. Unified data contracts and interfaces ensure cross-platform comparability, auditability, and engineering utility.