Reproducing kernel Hilbert space approach to non-Markovian quantum stochastic models

We give a derivation of the non-Markovian quantum state diffusion equation of Diósi and Strunz starting from a model of a quantum mechanical system coupled to a bosonic bath. We show that the complex trajectories arises as a consequence of using the Bargmann-Segal (complex wave) representation of the bath. In particular, we construct a reproducing kernel Hilbert space for the bath auto-correlation and realize the space of complex trajectories as a Hilbert subspace. The reproducing kernel naturally arises from a feature space where the underlying feature map is the free dynamical evolution in the one-particle Hilbert space of the bath quanta. We exploit this to derive the unraveling of the open quantum system dynamics and show equivalence to the equation of Diósi and Strunz. We also give an explicit expression for the reduced dynamics of a two-level system coupled to the bath via a Jaynes-Cummings interaction and show that this does indeed correspond to an exact solution of the Diósi-Strunz equation. Finally, we discuss the physical interpretation of the complex trajectories and show that they are intrinsically unobservable.