  Many superconducting qubit systems use the dispersive interaction between the
qubit and a coupled harmonic resonator to perform quantum state measurement.
Previous works have found that such measurements can induce state transitions
in the qubit if the number of photons in the resonator is too high. We
investigate these transitions and find that they can push the qubit out of the
two-level subspace, and that they show resonant behavior as a function of
photon number. We develop a theory for these observations based on level
crossings within the Jaynes-Cummings ladder, with transitions mediated by terms
in the Hamiltonian that are typically ignored by the rotating wave
approximation. We find that the most important of these terms comes from an
unexpected broken symmetry in the qubit potential. We confirm the theory by
measuring the photon occupation of the resonator when transitions occur while
varying the detuning between the qubit and resonator.
