  We employ phase-sensitive amplification to perform homodyne detection of the
resonance fluorescence from a driven superconducting artificial atom.
Entanglement between the emitter and its fluorescence allows us to track the
individual quantum state trajectories of the emitter conditioned on the
outcomes of the field measurements. We analyze the ensemble properties of these
trajectories by considering trajectories that connect specific initial and
final states. By applying the stochastic path integral formalism, we calculate
equations-of-motion for the most likely path between two quantum states and
compare these predicted paths to experimental data. Drawing on the mathematical
similarity between the action formalism of the most likely quantum paths and
ray optics we study the emergence of caustics in quantum trajectories---places
where multiple extrema in the stochastic action occur. We observe such multiple
most likely paths in experimental data and find these paths to be in reasonable
quantitative agreement with theoretical calculations.
