Non-Markovian Quantum Stochastic Models

We give a derivation of the non-Markovian quantum state diffusion equation of Diosi and Strunz starting from a model of a quantum mechanical system coupled to a bosonic bath. We show that the complex trajectories arise 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 space is the one-particle Hilbert space of the bath quanta. We exploit this to derive the unravelling of the open quantum system dynamics and show equivalence to the equation of Diosi 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 Diosi-Strunz equation. Finally, we discuss the physical interpretation of the complex trajectories and show that they are intrinsically unobservable.