  We provide a systematic approach to compute different kinds of
non-equilibrium Green's functions for open quantum systems which are
essentially two-point correlation functions in time. We reveal that the
definition of Green's functions based on the Born approximation does not
provide the correct results in the leading order of the system-bath coupling.
We next provide a systematic correction term in Green's functions by going
beyond the Born approximation and incorporating a finite correlation between
the system and the bath. We primarily focus on two paradigmatic models of open
quantum systems, namely, the dissipative Caldeira-Leggett model and the
dissipative spin-boson model. We show that the inclusion of such a correction
correctly reproduces the Kadanoff-Baym type equation for the so-called lesser
or greater components of the Green's functions and provides the correct
long-time result up to the first non-zero order of the system-bath coupling,
satisfying the detailed balance condition in thermal equilibrium. We further
extend our study to a system coupled to multiple reservoirs simultaneously that
are maintained at different temperatures and obtain expressions for
non-equilibrium steady-state energy current, once again correct up to the first
non-zero order of the system-bath coupling.
