inhomogeneous transverse decay time $T_2^* \sim 10 \, \textrm{ns}$. The experiments were performed in a dilution refrigerator at a base temperature of 20 mK. The device was magnetically shielded, and all electrical lines were carefully filtered and attenuated to reduce noise (see Ref.  for details).

Landau-Zener-Stückelberg Transitions {#sec:LZS}
====================================

Landau-Zener-Stückelberg (LZS) transitions occur when a quantum system is driven through an energy-level avoided crossing [@Landau32a; @Zener32a; @Stueckelberg32a]. The resulting quantum dynamics of the LZS mechanism  in driven systems [@Grifoni98a] have been developed [@Kayanuma93a; @Kayanuma94a] within a two-level coherent scattering formalism [@Kayanuma97a; @Kayanuma00a; @Shytov03; @Ashab07a] with potential application to quantum information science and technology [@Shytov03; @Ashab07a; @Wubs06a; @KSaito06a; @KSaito07a].

![image](Fig2_Oliver_final.png){width="5in"}

We model a LZS transition at a single crossing using the two-level Hamiltonian $$\mathcal{H} =
    -\frac{1}{2} \left(
    \begin{matrix}
        \varepsilon(t) & \Delta \\
        \Delta & -\varepsilon(t)
    \end{matrix}
    \right),
 \label{Eq:Hamiltonian}$$ in which $\varepsilon$ is the energy difference between diabatic-state energy levels (dashed lines in Fig. \[fig:2\]a), and $\Delta$ is the size of the avoided crossing and corresponds to the coupling strength between diabatic states $| \textrm{L} \rangle$ and $| \textrm{R} \rangle$.

The system is prepared in state $| R \rangle$ at a static value $\varepsilon_0 \gg \Delta$ far from the avoided crossing (blue 