Driven multiphoton qubit-resonator interactions

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, i.e., when the qubit drive is $n$ times the resonator frequency. We pay special attention to the strong driving regime, where the resulting effective interactions are conditioned on the qubit dressed states. Next, we investigate the use of a two-tone drive to engineer an effective $n$-photon Rabi Hamiltonian with widely tunable effective system parameters, which could enable the realization of new regimes that have so far been inaccessible. Then, we discuss applications for the specific case where $n=2$, which results in qubit-conditional squeezing (QCS). We show that the QCS protocol can be used 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 via the superposition of orthogonally squeezed states. We show how the QCS operation can be used to realize a controlled-squeeze gate and its use in bosonic phase estimation. The QCS protocol can also be utilized to achieve faster unitary operator synthesis on the joint qubit-resonator Hilbert space. 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.