Equivalence of stochastic formulations and master equations for open systems

Several stochastic formulations have been developed to investigate the quantum dynamics of open systems. We show the equivalence between the Liouville equation based on the stochastic decoupling of the system-bath interaction and the non-Markovian quantum state diffusion approach. The procedure we use provides not only a means of calculating the unknown functional derivative in the latter, but also a possibility of developing more efficient theoretical schemes. Further, we demonstrate that the stochastic decoupling approach can feasibly be used to derive master equations for the spontaneously decaying multistate systems, which are difficult to obtain by other available methods.