  We present a formalism that enables the analytic calculation of the
interaction of a spin-half particle with a polychromatic electromagnetic field.
This powerful new approach provides a clear physical picture even for cases
with highly degenerate energy levels, which are complicated to interpret in the
standard dressed-atom picture. Typically semi-classical methods are used for
such problems (leading to equations that are solved by Floquet theory). Our
formalism is derived from quantum electrodynamics and thus is more widely
applicable. In particular it makes accessible the intermediate regime between
quantum and semi-classical dynamics. We give examples of the application to
multi-frequency multi-photon processes in strong fields by deriving the
Hamiltonians of such systems, and also to the dynamics of weak fields at long
times for which semi-classical methods are insufficient.
