Driving not so forbidden state transitions in a frequency-tunable transmon

In a frequency-tunable transmon, transitions between the |0> and |2> states are nominally forbidden by selection rules, yet an experiment (Sank et al., PRL 117) observed that it is possible to drive these transitions. In this talk we explain that observation. We show that the apparent selection rule violation is a direct consequence of parametric modulation of the Josephson energy of the compound junction of the device. From a theoretical analysis of the dc SQUID, we derive a drive term that explains the violation. We validate our theory using experimentally measured Rabi oscillations and Ramsey interferometry on the |0> --> |2> transition. Surprisingly, the transition occurs for a transmon driven through a capacitor, indicating a modest degree of stray inductive coupling between the XY drive line and the compound junction of the transmon. These results identify a leakage channel and should inform efforts to integrate cryogenic control systems with arrays of frequency-tunable transmons.