  We discuss how coherent driving of a two-level quantum system can be used to
induce a complex phase on the ground state and we discuss its geometric and
dynamic contributions. While the global phase of a wave function has no
physical significance, coherent dynamics in a two-level subspace provides
relative phases and is an essential building block for more advanced dynamics
in larger systems. In this regard, we note that one must be careful with
intuitive accounts of the phase dynamics as it depends on the interaction
picture applied. To mitigate ambiguities in practical analyses, we suggest to
complement the Bloch sphere picture with the path taken by the ground state
amplitude in the complex plane, and we show how the two-level pure state
dynamics can serve as a starting point for the study of the dynamics explored
in three-level lambda systems, four-level tripod systems, and open quantum
systems.
