Fractional harmonic instabilities in a quantum driven non-linear oscillator*

The action of drives on non-linear modes to engineer parametric processes is ubiquitous in circuit QED. In this work, we provide a map in parameter space of the instabilities that are created when the ratio of the drive and the mode transition frequencies coincide with rational numbers. With careful engineering of system parameters and knowledge of such a map, once could avoid these instabilities, particularly at large drive-strengths. Our findings are supported by the bifurcation map of the classical analogue of our system where under certain drive conditions there exist a manifold of degenerate steady states that would lead to the quantum heating shown in the first part of the talk. One could also harness this robust degeneracy and exploit it to generate cat-like states which could be used to store and manipulate quantum information. Preliminary experimental results will be shown.