Observation of the Bloch-Siegert shift in the driven quantum-to-classical transition of a dispersive transmon qubit

We demonstrate the Bloch-Siegert effect in a dispersively coupled qubit-cavity system which is driven through a quantum-to-classical transition. The observed dispersive shift of the resonance frequency displays strongly non-monotonic dependence on the number of cavity photons and escapes the scope of the analytic results obtained with either a simple rotating-wave approximation, or with a more refined counter-rotating hybridized rotating wave approach. We measured the transition energy with a weak resonant probe, and the obtained data is in a good agreement with our numerical simulations of the quasienergy spectrum.