N-boson time-dependent problem: A reformulation with stochastic wave functions

We present a numerically tractable method to solve exactly the evolution of a N boson system with binary interactions. The density operator of the system $\rho$ is obtained as the stochastic average of particular operators $|{\Psi }_1〉〈{\Psi }_2|$ of the system. The states $|{\Psi }_{1,2}〉$ are either Fock states $|N:{\phi }_{1,2}〉$ or coherent states $|\mathrm{coh}:{\phi }_{1,2}〉$ with each particle in the state ${\phi }_{1,2}\mathrm{}\left(x\right).$ We determine the conditions on the evolution of ${\phi }_{1,2},$ which involves a stochastic element, under which we recover the exact evolution of $\rho .$ We discuss various possible implementations of these conditions. The well known positive P representation arises as a particular case of the coherent state ansatz. We treat numerically two examples: a two-mode system and a one-dimensional harmonically confined gas. These examples, together with an analytical estimate of the noise, show that the Fock state ansatz is the most promising one in terms of precision and stability of the numerical solution.