Non-Markovian Extension

Non-Markovian Quantum Processes with Memory Kernels A Rigorous(wip) Extension of the Lindblad Formalism via the Generalized Nakajima-Zwanzig Equation Description This repository contains the complete theoretical foundations of a (futur) rigorous extension to the Lindblad master equation for describing non-Markovian quantum processes. The work introduces a general mathematical framework based on memory kernels $K(t,s)$ to capture the effects of a structured environment with finite memory, going beyond the Markovian approximation. It is accompanied by an interactive, web-based version of this work, featuring a fully functional simulator, dynamic graphs, a 3D Bloch sphere visualization, and unit tests, available at: ((...)) Core Contributions 1. Axiomatic Definition of $\mathcal{K}_{CP}$ A rigorous definition of the space of memory kernels that preserve the fundamental properties of quantum states: complete positivity and trace conservation. Necessary and sufficient conditions for a kernel to be admissible are established and proven. Axiom A1 (Hermiticity): $K(t,s) = K^*(s,t)$ Axiom A2 ($L^2$-integrability): $\int_0^\infty\!\!\int_0^\infty |K(t,s)|^2\,ds\,dt < \infty$ Axiom A3 (Complete Positivity): $\sigma_{\min}(\hat{K}(\omega)) \geq 0,\ \forall \omega$ 2. Characterization of the Instantaneous Generator $L(t)$ A formal link between the memory kernel $K(t,s)$ and the instantaneous generator $L(t)$ is established, with explicit formulas for different kernel types: $$L(t) = \mathcal{L} + \int_0^t K(\tau)\,d\tau - K(0)$$ 3. Exactly Solvable Examples Detailed analysis of four canonical memory kernels: Exponential (Ornstein-Uhlenbeck process): $\rho_{11}(t) = \rho_{11}(\infty) + [\rho_{11}(0) - \rho_{11}(\infty)]e^{-\gamma t}$ Oscillatory (Cavity QED, damped Rabi oscillations): $K(t,s) = \Omega_R^2\cos[\Omega_R(t-s)]e^{-\kappa|t-s|}$ Compact Support (finite memory) Power-Law ($1/f$ noise, non-ergodicity): $K(t) \sim 1/t^{1-\alpha}$ 4. Quantitative Measures of Non-Markovianity Definition of three new, operationally significant measures to quantify the degree of memory in a quantum process: Measure Definition Interpretation $\mathcal{M}_L$ $\frac{1}{T}\int_0^T |\dot{L}(t)|\,dt$ Based on the temporal variation of $L(t)$. $\mathcal{M}_L = 0$ iff Markovian. $\mathcal{M}_K$ $\inf_{\{K_i\}} \| \mathcal{K} - \sum_i K_i \otimes K_i^* \|$ Distance to a simple Kraus representation. $\Deg(K)$ $\lim_{t\to\infty} \frac{\ln|K(t)|}{\ln(1/t)}$ Asymptotic exponent classifying the memory length (0 for Markovian, 1 for strongly non-Markovian). 5. Comprehensive Validation Comparison with experimental data from NMR, superconducting cavities, and the FMO complex. Proof-of-concept numerical simulations using an adaptive step-size Runge-Kutta 4 (RK45) integrator, integrated into the interactive version. Interactive Simulator (Online Version) The interactive version of this work, located at ((...)), includes a powerful falsification toolbox: Real-time RK45 Simulator with 5 selectable kernel types. Dynamic 2D plots for population $\rho_{11}(t)$, information flow $\sigma(t)$, and comparison with Lindblad. 3D Bloch sphere visualization of the qubit state. Integrated unit tests for trace conservation and complete positivity. Metrics display ($\mathcal{M}_L$, $\mathcal{M}_K$, $\Deg(K)$, $\tau_{\text{mem}}$, RK4 error). Comparison with experimental datasets (NMR, FMO, etc.). Export options for PDF, LaTeX, Markdown, and JSON data. Repository Contents CORE-IMMUABLE - extension-non-markovienne.md (8.6 MB): The comprehensive, publication-ready manuscript in Markdown format. It contains all theoretical derivations, proofs, and references. This is the permanent, citable record of the work.Online interactive version Citation If you use this work, please cite it as: DESVAUX, G. J. Y. (2023). Non-Markovian Extension. Zenodo. https://doi.org/10.5281/zenodo.19102132 Keywords Open quantum systems, Non-Markovianity, Memory kernels, Lindblad equation, Nakajima-Zwanzig equation, Complete positivity, Kraus representation, Quantum simulator, Runge-Kutta, $1/f$ noise Additional Information Languages: The core manuscript is in English. Titles are also available in French ("Extension non-markovienne") and Chinese ("非马尔可夫扩展").