concept is generally applicable to other solid-state artificial atoms [@vanderWiel03a; @Hanson07a] and generalized spin systems [@Tannoudji92a] (e.g., molecular magnets [@Friedman96a; @Thomas96a; @Wernsdorfer99a], natural atoms [@Baruch92a; @Yoakum92a], and molecules [@Mark07a; @Mark07b; @Lang08a]) that exhibit avoided level crossings, and it is extensible to multiple energy levels as we demonstrate in Section \[fig:5\].

Mach-Zehnder-type Interferometry {#sec:MZ}
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The structure of the n-photon spectroscopy peaks seen in Fig. \[fig:1\]E consists of regularly-spaced resonances positioned according to the condition $\textrm{n}\times$1.2 GHz being resonant with the energy level separation. Notably, however, for this particular value of driving amplitude, the n=1, 3, and 6 photon peaks are missing. As we describe in this section, the presence and absence of these peaks arise from Mach-Zehnder-type quantum interference at a level crossing. The interference phase is tunable via the driving amplitude, leading to a “Bessel staircase” interference pattern in the observed spectroscopy. The interference oscillations are known as Stückelberg oscillations [@Stueckelberg32a], and they have been observed in both artificial [@Oliver05a; @Berns06a; @Sillanpaa06a; @Wilson07a] and natural [@Baruch92a; @Yoakum92a; @Mark07a] atomic systems.

![image](Fig3_Oliver_final_red_size.png){width="5in"}

In a conventional Mach-Zehnder interferometer, an optical signal is passed through two beamsplitters. The first beamsplitter coherently divides the signal into two output paths, which may have different 