  Despite the high measurement fidelity that can now be reached, the dispersive
qubit readout of circuit quantum electrodynamics is plagued by a loss of its
quantum nondemolition character and a decrease in fidelity with increased
measurement strength. In this work we elucidate the nature of this dynamical
process, which we refer to as transmon ionization. We develop a comprehensive
framework which provides a unified physical picture of the origin of transmon
ionization. This framework consists of three complementary levels of
descriptions: a fully quantized transmon-resonator model, a semiclassical model
where the resonator is treated as a classical drive on the transmon, and a
fully classical model. Crucially, all three approaches preserve the full cosine
potential of the transmon, and lead to similar predictions. This framework
identifies the multiphoton resonances responsible for transmon ionization. It
also allows us to efficiently compute numerical estimates of the photon number
threshold for ionization, which are in remarkable agreement with recent
experimental results. The set of tools developed within this work are both
conceptually and computationally simple, and we expect them to become an
integral part of the theoretical support of all circuit QED experiments.
