  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.
