Stochastic analysis and simulation of spin star systems

We discuss two methods of an exact stochastic representation of the non-Markovian quantum dynamics of open systems. The first method employs a pair of stochastic product vectors in the total system's state space, while the second method uses a pair of state vectors in the open system's state space and a random operator acting on the state space of the environment. Both techniques lead to an exact solution of the von Neumann equation for the density matrix of the total system. Employing a spin star model describing a central spin coupled to bath of surrounding spins, we perform Monte Carlo simulations for both variants of the stochastic dynamics. In addition, we derive analytical expression for the expectation values of the stochastic dynamics to obtain the exact solution for the density matrix of the central spin.