Geometric phases in superconducting qubits beyond the   two-level-approximation
  Geometric phases, which accompany the evolution of a quantum system and depend only on its trajectory in state space, are commonly studied in two-level systems. Here, however, we study the adiabatic geometric phase in a weakly anharmonic and strongly driven multi-level system, realised as a superconducting transmon-type circuit. We measure the contribution of the second excited state to the two-level geometric phase and find good agreement with theory treating higher energy levels perturbatively. By changing the evolution time, we confirm the independence of the geometric phase of time and explore the validity of the adiabatic approximation at the transition to the non-adiabatic regime. 
S. Berger, M. Pechal, S. Pugnetti, A. A. Abdumalikov Jr, L. Steffen,   A. Fedorov, A. Wallraff, S. Filipp
quant-ph cond-mat.supr-con