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 \( F(\tau) = \sum_k c_k e^{(\alpha_k + i\beta_k)\tau} \) (\( x > 1 \), \( \tau = \ln x \)), when growth rates \( \alpha_k \) are positively correlated with frequencies \( \beta_k \). Under quasi-uniform \( \beta_k \) distribution (mean density \( \approx 1 \)), the unwrapped phase \( \Theta(\tau) = \int_0^\tau \Im(F'(u)/F(u))\,du \) exhibits an asymptotic linear drift \( (\pi + 2\delta\pi)\tau + o(\tau) \), where \( \delta > 0 \) measures the correlation strength. The amplification factor of 2 originates from quadratic intensity weighting in \( |F(\tau)|^2 \).

Rigorous error bounds of \( O((\ln \tau)^2) \) 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 \( \Delta\phi(t) \approx 2\kappa \langle n \rangle t \) 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.