# Measuring Fluorescence to Track a Quantum Emitter's State: A Theory Review

## TL;DR

This paper reviews the theoretical framework for continuous fluorescence measurement of a qubit, illustrating how different measurement types relate to quantum trajectories and connecting theory with recent experimental results.

## Contribution

It provides a unified, Bayesian-based theoretical overview of fluorescence monitoring methods, including photodetection and homodyne detection, and explores advanced topics like the arrow of time in quantum measurement.

## Key findings

- Demonstrates the equivalence of Kraus operator framework and stochastic master equations.
- Illustrates methods for quantum trajectory analysis in fluorescence monitoring.
- Connects theoretical models with recent experimental observations.

## Abstract

We review the continuous monitoring of a qubit through its spontaneous emission, at an introductory level. Contemporary experiments have been able to collect the fluorescence of an artificial atom in a cavity and transmission line, and then make measurements of that emission to obtain diffusive quantum trajectories in the qubit's state. We give a straightforward theoretical overview of such scenarios, using a framework based on Kraus operators derived from a Bayesian update concept; we apply this flexible framework across common types of measurements including photodetection, homodyne, and heterodyne monitoring, and illustrate its equivalence to the stochastic master equation formalism throughout. Special emphasis is given to homodyne (phase-sensitive) monitoring of fluorescence. The examples we develop are used to illustrate basic methods in quantum trajectories, but also to introduce some more advanced topics of contemporary interest, including the arrow of time in quantum measurement, and trajectories following optimal measurement records derived from a variational principle. The derivations we perform lead directly from the development of a simple model to an understanding of recent experimental results.