  We develop a general theory for multiphoton qubit-resonator interactions
enhanced by a qubit drive. The interactions generate qubit-conditional
operations in the resonator when the driving is near $n$-photon
cross-resonance, namely, the qubit drive is $n$-times the resonator frequency.
We pay special attention to the strong driving regime, where the interactions
are conditioned on the qubit dressed states. We consider the specific case
where $n=2$, which results in qubit-conditional squeezing (QCS). We propose to
use the QCS protocol for amplifying resonator displacements and their
superpositions. We find the QCS protocol to generate a superposition of
orthogonally squeezed states following a properly chosen qubit measurement. We
outline quantum information processing applications for these states, including
encoding a qubit in a resonator and performing a quantum non-demolition
measurement of the qubit inferred from the resonator's second statistical
moment. Next, we employ a two-tone drive to engineer an effective $n$-photon
Rabi Hamiltonian in any desired coupling regime. In other words, the effective
coupling strengths can be tuned over a wide range, thus allowing for the
realization of new regimes that have so far been inaccessible. Finally, we
propose a multiphoton circuit QED implementation based on a transmon qubit
coupled to a resonator via an asymmetric SQUID. We provide realistic parameter
estimates for the two-photon operation regime that can host the aforementioned
two-photon protocols. We use numerical simulations to show that even in the
presence of spurious terms and decoherence, our analytical predictions are
robust.
