Geometric Amplification in Asymmetric Logarithmic Superpositions: A Theoretical Model of Phase Drift with Extensions to Open Quantum Systems

We derive a universal geometric bias in the phase accumulation of finite asymmetric exponential superpositions \( F(τ) = ∑_k c_k e(α_k + iβ_k)τ \) (\( x > 1 \), \( τ = ln x \)), when growth rates \( α_k \) are positively correlated with frequencies \( β_k \). Under quasi-uniform \( β_k \) distribution (mean density \( ≈ 1 \)), the unwrapped phase \( Θ(τ) = ∫_0^τ (F'(u)/F(u))\,du \) exhibits an asymptotic linear drift \( (π + 2δπ)τ + o(τ) \), where \( δ > 0 \) measures the correlation strength. The amplification factor of 2 originates from quadratic intensity weighting in \( |F(τ)|^2 \). Rigorous error bounds of \( O((ln τ)^2) \) are provided, together with a direct analogy to correlated pure dephasing in Lindblad dynamics. The framework extends naturally to bosonic quantum error correction, particularly cat qubits subject to occupation-correlated photon loss and weak Kerr nonlinearity, predicting a doubled logical phase drift \( Δφ(t) ≈ 2κ n t \) in the large-amplitude regime. Numerical simulations in two-mode and multi-mode cases confirm the bias and its robustness. These results highlight a fundamental geometric effect in intensity-weighted oscillatory sums and correlated dissipative channels, with implications for coherence optimization in next-generation bosonic processors.