Tracking an open quantum system using a finite state machine: stability analysis

TL;DR

This paper investigates the stability of finite state machine-based monitoring schemes for tracking open quantum systems, demonstrating their stability analytically and numerically, especially in the context of qubit resonance fluorescence.

Contribution

It provides the first analytical proof of stability for finite state machine monitoring schemes in open quantum systems and explores their behavior through examples.

Findings

Finite state machine schemes are always stable for cyclic jumps in qubits.

Analytical and numerical evidence supports stability in resonance fluorescence.

Different monitoring schemes exhibit diverse trajectory behaviors.

Abstract

A finite-dimensional Markovian open quantum system will undergo quantum jumps between pure states, if we can monitor the bath to which it is coupled with sufficient precision. In general these jumps, plus the between-jump evolution, create a trajectory which passes through infinitely many different pure states, even for ergodic systems. However, as shown recently by us [Phys. Rev. Lett. \textbf{106}, 020406 (2011)], it is possible to construct {\em adaptive} monitorings which restrict the system to jumping between a finite number of states. That is, it is possible to track the system using a {\em finite state machine} as the apparatus. In this paper we consider the question of the stability of these monitoring schemes. Restricting to cyclic jumps for a qubit, we give a strong analytical argument that these schemes are always stable, and supporting analytical and numerical evidence for the example of resonance fluorescence. This example also enables us to explore a range of behaviors in the evolution of individual trajectories, for several different monitoring schemes.