Quantum Monte-Carlo methods and exact treatment of the two-body problem with Hartree-Fock Bogoliubov states

In this article, we show that the exact two-body problem can be replaced by quantum jumps between densities written as $D=| \Psi_a \right> \left$ and $| \Psi_b \right>$ are vacuum for different quasi-particles operators. It is shown that the stochastic process can be written as a Stochastic Time-Dependent Hartree-Fock Bogoliubov theory (Stochastic TDHFB) for the generalized density ${\cal R}$ associated to $D$ where ${\cal R}^2 = {\cal R}$ along each stochastic trajectory