Do diffusive non-Markovian stochastic Schr\"odinger equations (SSEs) for open quantum systems have a physical interpretation? In a recent paper [Phys. Rev. A 66, 012108 (2002)] we investigated this question using the orthodox interpretation of quantum mechanics. We found that the solution of a non-Markovian SSE represents the state the system would be in at that time if a measurement was performed on the environment at that time, and yielded a particular result. However, the linking of solutions at different times to make a trajectory is, we concluded, a fiction. In this paper we investigate this question using the modal (hidden variable) interpretation of quantum mechanics. We find that the noise function
z(t) appearing in the non-Markovian SSE can be interpreted as a hidden variable for the environment. That is, some chosen property (beable) of the environment has a definite value
z(t) even in the absence of measurement on the environment. The non-Markovian SSE gives the evolution of the state of the system ``conditioned'' on this environment hidden variable. We present the theory for diffusive non-Markovian SSEs that have as their Markovian limit SSEs corresponding to homodyne and heterodyne detection, as well as one which has no Markovian limit.
We present a stochastic projection formalism for the description of quantum dynamics in bosonic or spin environments. The Schrödinger equation in the coherent state representation with respect to the environmental degrees of freedom can be reformulated by employing the Feshbach partitioning technique for open quantum systems based on the introduction of suitable non-Hermitian projection operators. In this picture the reduced state of the system can be obtained as a stochastic average over pure state trajectories, for any temperature of the bath. The corresponding non-Markovian stochastic Schrödinger equations include a memory integral over the past states. In the case of harmonic environments and linear coupling the approach gives a new form of the established non-Markovian quantum state diffusion stochastic Schrödinger equation without functional derivatives. Utilizing spin coherent states, the evolution equation for spin environments resembles the bosonic case with, however, a non-Gaussian average for the reduced density operator.