  Many-body quantum systems in nonequilibrium remain one of the frontiers of
many-body physics. While there has been significant advances in describing the
short-time evolution of these systems using a variety of different numerical
algorithms, it has been quite difficult to evolve a system from an equilibrium
state prior to the application of a driving field, to the long-time steady (or
periodically oscillating) state. These dynamics are complex: the retarded
quantities tend to approach their long-time limit much faster than the lesser
(or greater) quantities. Recent work on strongly correlated electrons in DC
electric fields illustrated that the system may evolve through successive
quasi-thermal states obeying an effective fluctuation-dissipation theorem in
time. We demonstrate an extrapolation scheme that uses the short-time transient
calculation to obtain the retarded quantities and to extract how the
lesser/greater quantities vary with time and then extend the numerical
solutions all the way to the steady state, with minimal additional
computational cost. Our approach focuses on extrapolating the electronic
self-energy and then employing that to determine the Green's function and
various experimentally relevant expectation values.
