Reproducing Kernel Hilbert Space Approach to Non-Markovian Quantum Stochastic Models

TL;DR

This paper derives a non-Markovian quantum state diffusion equation using a reproducing kernel Hilbert space framework, providing explicit solutions and physical insights into complex trajectories in open quantum systems.

Contribution

It introduces a novel Hilbert space approach to derive and interpret non-Markovian quantum stochastic models, connecting complex trajectories to the bath's feature space.

Findings

Derived the non-Markovian quantum state diffusion equation from a bath model.

Constructed a reproducing kernel Hilbert space for bath auto-correlation.

Provided an explicit exact solution for a two-level system with Jaynes-Cummings interaction.

Abstract

We give a derivation of the non-Markovian quantum state diffusion equation of Di{\'o}si 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 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 Di{\'o}si 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 Di{\'o}si-Strunz equation. Finally, we discuss the physical interpretation of the complex trajectories and show that they are intrinsically unobservable.