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: fcpp_final.tex — LaTeX source (12 pages, 7 figures, 1 table) fcpp_final.pdf — Compiled PDF cover_letter.tex / .pdf — Cover letter for Physical Review E Figures (300 DPI PNG): Fig. 1: fig1_slow_mode_limit.png — Slow-mode limitation & core FCPP demo Fig. 2: fig2_biorthogonal_accuracy.png — Biorthogonal accuracy (3×3) Fig. 3: fig3_jordan_transition.png — Jordan crossover & Lambert W Fig. 4: fcpp_unified_floor.png — Transient re-excitation & unified predictor (main result) Fig. 5: fcpp_zhang2025.png — Trapped-ion (Zhang et al. parameters) Fig. 6: fcpp_2qubit.png — Two-qubit 16×16 Liouvillian Fig. 7: fig4_regime_map.png — Regime map Numerical Code (Python 3.12+, numpy/scipy/matplotlib): fcpp_nonnormal.py — 2×2 & 3×3 non-normal systems fcpp_jordan.py — Jordan block & Lambert W verification fcpp_lindbladian.py — Single-qubit Lindbladian fcpp_zhang_fix.py — Trapped-ion (Zhang et al. 2025) fcpp_floor.py — Unified predictor with cieff fcpp_2qubit.py — Two-qubit interacting system fcpp_torino_v2.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)