Quantum jump trajectories, hybrid systems, non-Hermitian evolutions, quantum/classical walks

TL;DR

This paper develops a general framework for quantum jump trajectories and hybrid systems, unifying various quantum evolution models and providing tools to analyze jump probabilities and waiting times.

Contribution

It introduces the notions of 'typical trajectory' and 'exclusive probability densities' to construct solutions of stochastic master equations and describe jump statistics.

Findings

Unified description of quantum jump processes and hybrid systems.

Method to recursively construct solutions of non-linear stochastic master equations.

Characterization of jump waiting times and probability distributions.

Abstract

Quantum stochastic master equations of jump type are formulated in a general way and connections with quantum/classical hybrid systems and quantum filtering theory are discussed. By introducing the notion of ``typical trajectory", we show how to recursively construct the solution of the non-linear stochastic master equation (the conditional state). Moreover, by the notion of ``exclusive probability densities" we can describe all the probabilities related to the jumps, in particular, the waiting times of the jumps and their probability distributions. This general formulation and the idea of hybrid system allow to unify and generalize different fields: evolutions under non-Hermitian Hamiltonians, unitary dynamics interspersed by quantum channels at random times, quantum renewal processes, continuous time open quantum walks, Lindblad rate equation, ...