band occurs at $I_b = 17.624\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$, for which ${\ensuremath{\omega_{0 1}}} / 2 \pi = 6.4\ {\ensuremath{{\ensuremath{\mathrm{GHz}}}}}$. This again indicates an ac Stark shift of this transition, which we denote by $\Delta {\ensuremath{\omega_{0 1}}} \equiv {\ensuremath{\omega_{r\!f}}}- {\ensuremath{\omega_{0 1}}} \approx 2 \pi \times 100\ {\ensuremath{{\ensuremath{\mathrm{MHz}}}}}$. In addition, the higher levels have suppressed the oscillation frequency below the bare Rabi frequency of ${\ensuremath{\Omega_{0 1}}} / 2 \pi = 620\ {\ensuremath{{\ensuremath{\mathrm{MHz}}}}}$ \[calculated with [Eq. [(\[eqf\])]{}]{}\].

![\[F032206MNstats\] (Color online) The (a) on-resonance Rabi oscillation frequencies [$\Omega^{min}_{R,0 1}$]{} and [$\Omega^{min}_{R,0 2}$]{} and (b) resonance frequency shifts $\Delta {\ensuremath{\omega_{0 1}}} = {\ensuremath{\omega_{r\!f}}}- {\ensuremath{\omega_{0 1}}}$ and $\Delta {\ensuremath{\omega_{0 2}}} = 2 {\ensuremath{\omega_{r\!f}}}- {\ensuremath{\omega_{0 2}}}$ are plotted as a function of the microwave current, for data taken at $110\ {\ensuremath{{\ensuremath{\mathrm{mK}}}}}$ with a microwave drive of frequency ${\ensuremath{\omega_{r\!f}}}/ 2 \pi = 6.5\ {\ensuremath{{\ensuremath{\mathrm{GHz}}}}}$ and powers ${\ensuremath{P_S}}= -23, -20, -17, -15, -10\ {\ensuremath{{\ensuremath{\mathrm{dBm}}}}}$. Values extracted from data for the [$0 \rightarrow 1$]{} ([$0 \rightarrow 2$]{}) transition are plotted as open circles (filled squares), while five-level rotating wave solutions for a junction with $I_{01} = 17.736\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$ and $C_1 = 4.49\ {\ensuremath{{\ensuremath{\mathrm{pF}}}}}$ are shown as solid (dashed) lines. In (a), the dotted line is from a simulation of a 