Measuring Fluorescence to Track a Quantum Emitter's State: A Theory Review. (arXiv:1908.04720v1 [quant-ph])

We review and expand on the continuous monitoring of a qubit through its
spontaneous emission. 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 overview of such scenarios, using
a simple and flexible theoretical framework across common types of
measurements, such as photodetection, homodyne, and heterodyne monitoring.
Specifically, we adopt an approach using Kraus operators derived from a
Bayesian--update concept, and apply it across a wider range of measurements
than it has been previously used for in the literature; we demonstrate the
equivalence between this approach and other common ones, such as a stochastic
master equation, for every type of measurement we consider. Our emphasis is on
simple examples explained through a single theoretical framework, rather than
on developing wholly new results; our aim is to offer a pedagogical review of
this research area. Special emphasis is given to homodyne (phase--sensitive)
monitoring of fluorescence, the time symmetry of the resulting dynamical
equations, and the optimal paths connecting different states under such

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