  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 treats these frameworks independently, lacking a unified
perspective. Our work addresses this gap by clarifying the role and status of
these approaches using a minimal single-level quantum dot model in a
two-terminal setup under voltage and temperature biases. We derive analytical
expressions for particle and energy currents and their fluctuations in both
steady-state and transient regimes. 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.
