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.
Furthermore, these transitions 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
which are typically ignored by the rotating wave approximation. We confirm the
theory by measuring the photon occupation of the resonator when transitions occur
while varying the detuning between the qubit and resonator.