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Diagrammatic approach for analytical non-Markovian time-evolution: Fermi's two atom problem and causality in waveguide quantum electrodynamics

Non-Markovian time-evolution of quantum systems is a challenging problem, often mitigated by employing numerical methods or making simplifying assumptions. In this work, we address this problem in waveguide QED by developing a diagrammatic approach, which performs fully analytical non-Markovian time evolution of single-photon states. By revisiting Fermi's two atom problem, we tackle the impeding question of whether rotating-wave approximation violates causality in single-photon waveguide QED. Afterward, we introduce and prove the \emph{no upper half-plane poles (no-UHP) theorem}, which connects the poles of scattering parameters to the causality principle. Finally, we visualize the time-delayed coherent quantum feedback mediated by the field, discuss the Markovian limit for microscopically separated qubits where short-distance causality violations occur and the emergence of collective decay rates in this limit. Our diagrammatic approach is the first method to perform exact and analytical non-Markovian time evolution of multi-emitter systems in waveguide QED.