dot, Fig. \[fig:2\]A) and driven longitudinally from that point by making $\varepsilon(t)$ time dependent. We distinguish here between “weak driving,” in which the resulting excursion is not large enough to reach the avoided crossing, and “strong driving” for which the avoided crossing is traversed. Under strong-driving conditions, the asymptotic probability $P_{\textrm{D-LZS}}$ of a transition between diabatic states, $$P_{\textrm{D-LZS}} \equiv 1 - P_{\textrm{E-LZS}} = 1 - \exp{
        \frac{- \pi \Delta^2}{2 \hbar \zeta_i}},
 \label{Eq:P_LZS}$$ is governed by the relative-energy sweep rate $\zeta_i$ $$\zeta_i = \frac{d\varepsilon(t)}{dt} |_{t=t_i}$$ evaluated at the time $t=t_i$ at which the system is swept through the avoided crossing. In the original LZS formulation, the system was driven with a fixed sweep rate, whereas we will later consider a harmonically driven system. We also note that we have elected to monitor the probability $P_{\textrm{D-LZS}}$ of a transition between the diabatic states $| L \rangle$ and $| R \rangle$, because our readout detector is sensitive to changes in diabatic state. This can be written with no loss of generality in terms of the probability $P_{\textrm{E-LZS}}$ of a transition between the system eigenstates.

There are two strong-driving limits characterized by the relative sizes of the sweep velocity $\zeta$ and the avoided crossing 