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Transport in open quantum systems can be explored through various theoretical frameworks, including the quantum master equation, scattering matrix, and Heisenberg equation of motion. The choice of framework depends on factors such as the presence of interactions, the coupling strength between the system and environment, and whether the focus is on steady-state or transient regimes. Existing literature mainly treats these frameworks independently. Our work establishes connections between them by clarifying the role and status of these approaches using two paradigmatic models for single and multipartite quantum systems in a two-terminal setup under voltage and temperature biases. We derive analytical solutions in both steady-state and transient regimes for the populations, currents, and current correlation functions. Exact results from the Heisenberg equation are shown to align with scattering matrix and master equation approaches within their respective validity regimes. Crucially, we establish a protocol for the weak-coupling limit, bridging the applicability of master equations at weak-coupling with Heisenberg or scattering matrix approaches at arbitrary coupling strength. Published by the American Physical Society 2024
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