  In this book chapter we analyze the high excitation nonlinear response of the
Jaynes-Cummings model in quantum optics when the qubit and cavity are strongly
coupled. We focus on the parameter ranges appropriate for transmon qubits in
the circuit quantum electrodynamics architecture, where the system behaves
essentially as a nonlinear quantum oscillator and we analyze the quantum and
semi-classical dynamics. One of the central motivations is that under strong
excitation tones, the nonlinear response can lead to qubit quantum state
discrimination and we present initial results for the cases when the qubit and
cavity are on resonance or far off-resonance (dispersive).
