features observable in this plot. The first is the presence of equally-spaced n-photon transitions as a function of flux detuning, symmetrically located about the qubit step at $\delta f_{\textrm{dc}}=0$. As one might expect, the onset of the higher-photon transitions requires larger driving amplitudes. Remarkably, we observe up to 50-photon transitions in this scan. The second main feature is that for each n-photon resonance, the spectroscopy appears and disappears as a function of amplitude, which sweeps the interference phase $\Delta \theta_{12}$. For example, 14 oscillation lobes are visible for the 1-photon transition. A vertical slice of the spectroscopy (dashed white line in Fig. \[fig:3\]A) is plotted in Fig. \[fig:3\]C; at this particular amplitude, one can see the enhancement and suppression of the spectroscopy peaks as was observed in Fig. \[fig:1\]E.

The Mach-Zehnder interference for an n-photon transition yields a modified amplitude-dependent matrix element [@Oliver05a; @Nakamura01; @Tannoudji92a] $$\Delta_n = \Delta J_n(\lambda)$$ where $J_n(\lambda)$ is the nth-order Bessel-function, and its argument $\lambda = A / h \nu$ is the dimensionless driving amplitude. Intuitively, the Bessel-function dependence arises, because the qubit is driven harmonically through energy levels that are linear in flux detuning and, as a result, the interference phase accumulates with a harmonic 