Exact Non-Markovian Master Equation for Quantum Dot Decoherence via Influence Functional — E8 Intelligence Research

FINDING: Exact master equation for non-Markovian decoherence in quantum dots, derived via Feynman-Vernon influence functional, captures memory effects beyond Lindblad form. MATH: - Exact master equation: \( \dot{\rho}(t) = -i[H_S(t),\rho] + \sum_{ij} \Delta_{ij}(t)[a_j^\dagger \rho a_i - \frac{1}{2}\{a_i a_j^\dagger, \rho\}] + \sum_{ij} \Upsilon_{ij}(t)[a_j^\dagger \rho a_i^\dagger - \frac{1}{2}\{a_i^\dagger a_j^\dagger, \rho\}] + \text{h.c.} \) — time-dependent coefficients \(\Delta_{ij}(t), \Upsilon_{ij}(t)\) encode memory kernel. - Spectral density (Ohmic): \( J(\omega) = \kappa \omega^s e^{-\omega/\omega_c} \), with \(s=1\) for Ohmic, \(s<1\) sub-Ohmic, \(s>1\) super-Ohmic. - Memory kernel: \( \mathcal{K}(t-t') = \int_0^\infty d\omega J(\omega) \sin[\omega(t-t')] \) — non-Markovian when \( \mathcal{K}(t-t') \) decays slowly relative to system timescale. - Decoherence rate: \( \Gamma(t) = \int_0^t dt' \int_0^{t'} dt'' \mathcal{K}(t''-t') \) — for Ohmic bath at zero tempera Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com