Observation of the Bloch-Siegert shift in a driven quantum-to-classical   transition
  We show that the counter-rotating terms of the dispersive qubit-cavity Rabi model can produce relatively large and nonmonotonic Bloch-Siegert shifts in the cavity frequency as the system is driven through a quantum-to-classical transition. Using a weak microwave probe tone, we demonstrate experimentally this effect by monitoring the resonance frequency of a microwave cavity coupled to a transmon and driven by a microwave field with varying power. In the weakly driven regime (quantum phase), the Bloch-Siegert shift appears as a small constant frequency shift, while for strong drive (classical phase) it presents an oscillatory behaviour as a function of the number of photons in the cavity. The experimental results are in agreement with numerical simulations based on the quasienergy spectrum. 
I. Pietik\"ainen, S. Danilin, K. S. Kumar, A. Veps\"al\"ainen, D. S.   Golubev, J. Tuorila, and G. S. Paraoanu
cond-mat.mes-hall quant-ph