  A superconducting qubit coupled to a read-out resonator is currently the
building block of multiple quantum computing as well as quantum optics
experiments. A typical qubit-resonator system is coupled in the dispersive
regime, where the detuning between qubit and resonator is much greater than the
coupling between them. In this work, we fabricated and measured a
superconducting transmon-resonator system in the non-dispersive regime. The
dressed states formed by the mixing of the bare qubit and resonator states can
be further mixed by applying a drive on the qubit, leading to the formation of
polariton states. We report experimental studies of transitions between
polariton states at varying driving powers and frequencies and show how the
non-dispersive coupling of the higher levels of the qubit-resonator system
modifies the polariton eigenstates and the corresponding transition
frequencies. We also report close agreement with numerical results obtained
from a driven Jaynes-Cummings Model beyond the dispersive regime.
