Transiently Renormalized Modal Bottlenecks in Non-Normal Relaxational Dynamics

Transiently Renormalized Modal Bottlenecks in Non-Normal Relaxational Dynamics Author: Takayuki Takagi (高木高正) Affiliation: Independent Researcher, Higashimatsuyama, Saitama, JapanORCID: 0009-0003-5188-2314Date: March 30, 2026Status: Preprint — Submitted to Physical Review E Abstract We propose a unified convergence-time predictor for non-normal relaxational dynamics, based on transiently renormalized modal bottlenecks. In relaxation governed by a linear generator L, convergence time is commonly assumed to scale with distance to the target state. We show instead that the rate-limiting quantity is the modal bottleneck: the eigenmode whose biorthogonal coupling strength and decay rate produce the longest-lasting contribution. For non-normal generators, static modal overlaps drastically underestimate convergence times because transient dynamics re-excite slow modes even when the initial overlap vanishes. We resolve this by introducing effective coefficients cieff that capture the maximal transient repopulation of each mode. Key Results System Dim R² (distance) R² (standard) R² (unified) Key failure mode 2×2 upper-tri 4 0. 54 ~1. 00 ~1. 00 None 3×3 upper-tri 9 0. 47 ~1. 00 ~1. 00 None 1-qubit Lindblad 4 0. 71 0. 78 0. 94 Coherence degeneracy Trapped-ion (9×9) 9 0. 09 0. 88 0. 96 Transient re-excitation 2-qubit (16×16) 16 0. 33 0. 23 0. 84 Coherent beating The unified predictor improves R² from 0. 23 to 0. 84 on a 16×16 two-qubit Liouvillian where the standard predictor performs worse than distance. The trapped-ion system uses the experimental parameters of Zhang et al. (Nat. Commun. 16, 301, 2025). Package Contents Manuscript: fcppfinal. tex — LaTeX source (12 pages, 7 figures, 1 table) fcppfinal. pdf — Compiled PDF coverₗetter. tex /. pdf — Cover letter for Physical Review E Figures (300 DPI PNG): Fig. 1: fig1ₛlowₘodeₗimit. png — Slow-mode limitation & core FCPP demo Fig. 2: fig2biorthogonalₐccuracy. png — Biorthogonal accuracy (3×3) Fig. 3: fig3ⱼordanₜransition. png — Jordan crossover & Lambert W Fig. 4: fcppᵤnifiedfloor. png — Transient re-excitation & unified predictor (main result) Fig. 5: fcppᵦhang2025. png — Trapped-ion (Zhang et al. parameters) Fig. 6: fcpp₂qubit. png — Two-qubit 16×16 Liouvillian Fig. 7: fig4ᵣegimeₘap. png — Regime map Numerical Code (Python 3. 12+, numpy/scipy/matplotlib): fcppₙonnormal. py — 2×2 & 3×3 non-normal systems fcppⱼordan. py — Jordan block & Lambert W verification fcppₗindbladian. py — Single-qubit Lindbladian fcppᵦhangfix. py — Trapped-ion (Zhang et al. 2025) fcppfloor. py — Unified predictor with cieff fcpp₂qubit. py — Two-qubit interacting system fcppₜorinoᵥ2. py — IBM Torino GHZ decoherence (TSTT connection) Related Work TSTT Quantum Experiments V3. 2 (doi: 10. 5281/zenodo. 17589109): Structural coherence in GHZ states on IBM Quantum. The present work provides a dynamical (FCPP) interpretation of the structural coherence measured in that study. Zhang et al. (Nat. Commun. 16, 301, 2025): Experimental observation of quantum strong Mpemba effect. FCPP is tested against their experimental parameters. Main Theorem The convergence time is predicted by: Tε ≈ maxi (1/|Re (λi) |) · log (cieff · ‖vi‖ / ε) where cieff = maxt≥0 |⟨wi, exp (Lt) ·δx₀⟩| · exp (|Re (λi) |·t)