Sideband transitions in a two-mode Josephson circuit driven beyond the rotating wave approximation

Driving quantum systems periodically in time plays an essential role in the coherent control of quantum states. A good approximation for weak and nearly resonance driving fields, experiments often require large detuning and strong driving fields, for which the RWA may not hold. In this work, we experimentally, numerically, and analytically explore strongly driven two-mode Josephson circuits in the regime of strong driving and large detuning. Specifically, we investigate beam-splitter and two-mode squeezing interaction between the two modes induced by driving two-photon sideband transition. Using numerical simulations, we observe that the RWA is unable to correctly capture the amplitude of the sideband transition rates, which we verify using an analytical model based on perturbative corrections. Interestingly, we find that the breakdown of the RWA in the regime studied does not lead to qualitatively different dynamics, but gives the same results as the RWA theory at higher drive strengths, enhancing the coupling rates compared to what one would predict. Our work provides insight into the behavior of time-periodically driven systems beyond the RWA, and provides a robust theoretical framework for including these in the calculation and calibration of quantum protocols in circuit quantum electrodynamics.