must also play a role in this problem, since the qubit state continues to evolve for the remainder of the driving period. It is physically meaningful to consider the total phase $\theta$ accumulated over a single period: $$\theta = \frac1{\hbar} \oint \epsilon(t)dt=2\pi\epsilon_0/\hbar\omega,$$ which is [*independent*]{} of the driving amplitude. Over many periods of the driving field, the cascaded pairs of LZS transitions (cascaded interferometers) will constructively interfere provided $\theta = 2\pi n$. One can view this as a “time-domain” formulation for the familiar n-photon resonance condition $$\varepsilon_{0,n} = n h \nu$$ where n is the number of photons involved in the transition. It is only when the product $n h \nu$ equals the energy separation $\varepsilon_{0,n}$ that the cascaded LZS transitions lead to a net buildup of state population and, as a function of $\Delta \theta_{12}$, the observed interference fringes. These oscillations are related to photoassisted transport [@Tien63a; @Kouwenhoven94; @NakamuraTsai99] and Rabi oscillations [@Nakamura01; @Saito06a] in the multiphoton regime.

Mach-Zehnder-type interference in the discrete-resonance limit driven towards saturation is shown in Fig. \[fig:3\]A for driving frequency $\nu=270$ MHz. This frequency is larger than the resonance linewidth (Fig. \[fig:3\]B), and so individual n-photon resonances can be resolved. There are two main 