  A vast collection of light-matter interactions are described by the
single-frequency Rabi model. However, the physical world is polychromatic, and
until now there is no general method to find analytic solutions to the
multi-frequency Rabi model. We present the first general method to solve the
Rabi model with N frequency modes. Analytic expressions are derived for a
spin-half particle in a multi-frequency coherent field, and accurately describe
the time evolution even when the interactions are strong. The dynamics are
solved in an extended dressed atom picture whereby the spin is progressively
dressed by each frequency mode. For weak fields, the closed form expressions
for the time evolution of atoms or their analogues take a particularly simple
form. These results analytically solve problems in a wide range of fields
including quantum information processing, coherent control and resonant
nonlinear optics.
