  Electron dynamics in a two-sites Hubbard model is studied using the
nonequilibrium Green's function approach. The study is motivated by the
empirical observation that a full solution of the integro-differential
Kadanoff-Baym equation (KBE) is more stable and often accompanied by artificial
damping [Marc Puig von Friesen, C. Verdozzi, and C.-O. Almbladh (2009)] than
its time-linear reformulations relying on the generalized Kadanoff-Baym ansatz
(GKBA). Additionally, for conserving theories, numerical simulations suggest
that KBE produces natural occupations bounded by one and zero in agreement with
the Pauli exclusion principle, whereas, in some regimes, GKBA-based theories
violate this principle. As the first step for understanding these issues, the
electron dynamics arising in the adiabatic switching scenario is studied.
Many-body approximations are classified according to the channel of the
Bethe-Salpeter equation in which electronic correlations are explicitly
treated. They give rise to the so-called second Born, $T$-matrix and $GW$
approximations. In each of these cases, the model is reduced to a system of
ordinary differential equations, which resemble equations of motion for a
driven harmonic oscillator with time-dependent frequencies. A more complete
treatment of electronic correlations is achieved by combining different
correlation channels, with parquet theory serving as a starting point.
