  Using circuit QED, we consider the measurement of a superconducting transmon
qubit via a coupled microwave resonator. For ideally dispersive coupling,
ringing up the resonator produces coherent states with frequencies matched to
transmon energy states. Realistic coupling is not ideally dispersive, however,
so transmon-resonator energy levels hybridize into joint eigenstate ladders of
the Jaynes-Cummings type. Previous work has shown that ringing up the resonator
approximately respects this ladder structure to produce a coherent state in the
eigenbasis (a dressed coherent state). We numerically investigate the validity
of this coherent state approximation to find two primary deviations. First,
resonator ring-up leaks small stray populations into eigenstate ladders
corresponding to different transmon states. Second, within an eigenstate ladder
the transmon nonlinearity shears the coherent state as it evolves. We then show
that the next natural approximation for this sheared state in the eigenbasis is
a dressed squeezed state, and derive simple evolution equations for such states
using a hybrid phase-Fock-space description.
