  Qubit measurement and control in circuit QED rely on microwave drives, with
higher drive amplitudes ideally leading to faster processes. However,
degradation in qubit coherence time and readout fidelity has been observed even
under moderate drive amplitudes corresponding to few photons populating the
measurement resonator. Here, we numerically explore the dynamics of a driven
transmon-resonator system under strong and nearly resonant measurement drives,
and find clear signatures of transmon ionization where the qubit escapes out of
its cosine potential. Using a semiclassical model, we interpret this ionization
as resulting from resonances occurring at specific resonator photon
populations. We find that the photon populations at which these spurious
transitions occur are strongly parameter dependent and that they can occur at
low resonator photon population, something which may explain the experimentally
observed degradation in measurement fidelity.
