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Transport in open quantum systems can be explored through various theoreticalframeworks, including the quantum master equation, scattering matrix, andHeisenberg equation of motion. The choice of framework depends on factors suchas the presence of interactions, the coupling strength between the system andenvironment, and whether the focus is on steady-state or transient regimes.Existing literature treats these frameworks independently, lacking a unifiedperspective. Our work addresses this gap by clarifying the role and status ofthese approaches using a minimal single-level quantum dot model in atwo-terminal setup under voltage and temperature biases. We derive analyticalexpressions for particle and energy currents and their fluctuations in bothsteady-state and transient regimes. Exact results from the Heisenberg equationare shown to align with scattering matrix and master equation approaches withintheir respective validity regimes. Crucially, we establish a protocol for theweak-coupling limit, bridging the applicability of master equations atweak-coupling with Heisenberg or scattering matrix approaches at arbitrarycoupling strength.

Exact finite-time correlation functions for multi-terminal setups: Connecting theoretical frameworks for quantum transport and thermodynamics