{\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$; thus a second band appears there, corresponding to two-photon [$0 \rightarrow 2$]{} Rabi oscillations. Similarly, three-photon [$0 \rightarrow 3$]{} oscillations are visible near $17.572\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$ where ${\ensuremath{\omega_{0 3}}} / 6 \pi = 6.5\ {\ensuremath{{\ensuremath{\mathrm{GHz}}}}}$. Finally, large escape rates occur near $17.549\ {\ensuremath{\mu {\ensuremath{\mathrm{A}}}}}$ corresponding to a four-photon [$0 \rightarrow 4$]{} transition, although no oscillations are apparent in Fig. \[F010206H1\](a). Note that the effective junction parameters $I_{01}$ and $C_1$ (given in the caption) used to predict [$\omega_{n m}$]{} and other level properties are slightly different than those for Figs. \[FEnergyGamma\]–\[F031806Dfosc\], as the data sets were taken two months apart.

The rotating wave solution provides a simple way to predict the oscillation frequencies. Calculations of [$\Omega_{R,0 n}$]{} using [Eq. [(\[eqd\])]{}]{} for a seven-level system are shown as dashed curves in Fig. \[F010206H1\](b) for the four lowest multiphoton transitions ($n = 1, 2, 3, 4$). The microwave amplitude ${\ensuremath{I_{r\!f}}}= 24.4\ {\ensuremath{{\ensuremath{\mathrm{nA}}}}}$ is the only free parameter in the calculation and the rotating wave solution using this value reproduces the oscillation frequencies of the different processes well, even at large detuning.

Figure \[F010206H1\](b) shows that the minimum oscillation frequency ${\ensuremath{\Omega^{min}_{R,0 1}}} / 2 \pi = 540\ {\ensuremath{{\ensuremath{\mathrm{MHz}}}}}$ of the first (experimental) 