system (dashed line). This effect is a hallmark of a multilevel system and has been previously observed in a similar phase qubit.[@Claudon04a; @Claudon07b] There are two distinct phenomena that affect [$0 \rightarrow 1$]{} Rabi oscillations in such a device.[@Goorden03a; @Meier05a; @Strauch06a; @Shevchenko07a] To describe these, we must first define resonance as occurring when the Rabi frequency [$\Omega_{R,0 1}$]{} is at a minimum, as the detuning between the microwave drive frequency [$\omega_{r\!f}$]{} and level spacing [$\omega_{0 1}$]{} is varied (for a fixed drive power). In a two-level system, this happens for ${\ensuremath{\omega_{0 1}}} = {\ensuremath{\omega_{r\!f}}}$. For our phase qubit, resonance occurs when ${\ensuremath{\omega_{0 1}}} < {\ensuremath{\omega_{r\!f}}}$ due to the decreasing level spacings with increasing state [$\left| n \right>$]{} \[see Fig. \[FEnergyGamma\](a)\]. This shift in resonance is an analog of the ac Stark effect. Resonance shifts occur under strong driving in other superconducting qubits as well.[@Schuster05a; @Dutton06a]

However, this shift does not explain the data shown in Fig. \[F031806Dfosc\], which were taken at a fixed bias current (and thus off resonance at high power). The higher levels also affect the frequency of the Rabi oscillations. We will refer to the minimum value of [$\Omega_{R,n m}$]{} as the on-resonance Rabi frequency [$\Omega^{min}_{R,n m}$]{}. 