  We formulate a model for the steady state response of a nonlinear quantum
oscillator structure, such as those used in a variety of superconducting qubit
experiments, when excited by a steady, but not necessarily small, ac tone. We
show that this model can be derived directly from a circuit description of some
recent qubit experiments in which the state of the qubit is read out directly,
without a SQUID magnetometer. The excitation profile has a rich structure
depending on the detuning of the tone from the small-signal resonant frequency,
on the degree of damping, and on the excitation amplitude. We explore two
regions in detail: First, at high damping there is a trough in the excitation
response as a function of detuning, near where the classical Duffing
bifurcation occurs. This trough has been understood as a classical interference
between two metastable responses with opposite phase. We use Wigner function
studies to show that while this picture is roughly correct, there are also more
quantum mechanical aspects to this feature. Second, at low damping we study the
emergence of sharp, discrete spectral features from a continuum response. We
show that these the structures, associated with discrete transitions between
different excited-state eigenstates of the oscillator, provide an interesting
example of a quantum Fano resonance. The trough in the Fano response evolves
continuously from the "classical" trough at high damping.
