  Assuming that the effect of the residual interaction beyond mean-field is
weak and has a short memory time, two approximate treatments of correlation in
fermionic systems by means of Markovian quantum jump are presented. A
simplified scenario for the introduction of fluctuations beyond mean-field is
first presented. In this theory, part of the quantum correlations between the
residual interaction and the one-body density matrix are neglected and jumps
occur between many-body densities formed of pairs of states $D=| \Phi_a > <
\Phi_b |/< \Phi_b . |\Phi_a >$ where $| \Phi_a >$ and $| \Phi_b >$ are
antisymmetrized products of single-particle states. The underlying Stochastic
Mean-Field (SMF) theory is discussed and applied to the monopole vibration of a
spherical $^{40}$Ca nucleus under the influence of a statistical ensemble of
two-body contact interaction. This framework is however too simplistic to
account for both fluctuation and dissipation. In the second part of this work,
an alternative quantum jump method is obtained without making the approximation
on quantum correlations. Restricting to two particles-two holes residual
interaction, the evolution of the one-body density matrix of a correlated
system is transformed into a Lindblad equation. The associated dissipative
dynamics can be simulated by quantum jumps between densities written as $D = |
\Phi >< \Phi |$ where $| \Phi >$ is a normalized Slater determinant. The
associated stochastic Schroedinger equation for single-particle wave-functions
is given.
