$\Delta$. In the adiabatic limit (Fig. \[fig:2\]A), the sweep velocity is small ($h \zeta \ll \Delta^2$) and the probability of a transition from state $| R \rangle$ to state $| L \rangle$ approaches unity, $P_{\textrm{D-LZS}}\rightarrow 1$. In this case, the system dynamics are slow enough that the system adiabatically follows the ground eigenstate through the avoided crossing. In the nonadiabatic limit (Fig. \[fig:2\]B), the sweep velocity is large ($h \zeta \gg \Delta^2$) and the probability of a transition approaches zero $P_{\textrm{D-LZS}}\rightarrow 0$. In this case, the dynamics are too fast for the system to follow; the system remains in diabatic state $| R \rangle$ and, thereby, effectively jumps the energy gap at the avoided crossing.

More generally, a superposition state $\alpha | L \rangle + \beta | R \rangle$ results from an excursion through the avoided crossing, as illustrated in Fig. \[fig:2\]C. Following an idea discussed by Shytov [*et al.*]{} [@Shytov03], the LZS transition acts as a beamsplitter for the atomic state. The amplitudes $\alpha$ and $\beta$ are determined by a unitary transformation $U_{\textrm{D-LZS}}$, effectively a $2 \times 2$ “beamsplitter matrix” comprising complex reflection $r$ and transmission $t$ coefficients related to the adiabaticity parameter [@H_Nakamura01a] $\Delta^2 / \hbar \zeta$ present 