= Landau-Zener transition
{c}
A Landau-Zener transition occurs when a changing parameter sweeps two coupled <quantum states> through an <avoided crossing>. In a two-state description,
$$
H(t)=\begin{pmatrix}\alpha t/2&\Delta\\\Delta^*&-\alpha t/2\end{pmatrix},
$$
the uncoupled <energy levels> cross at $t=0$, but the coupled <energy eigenvalues> have minimum separation $2|\Delta|$. The <adiabatic theorem> predicts following an instantaneous <energy eigenstate> during a sufficiently slow sweep; a fast sweep can retain the original uncoupled-state character and thus change instantaneous <energy eigenstate>.
For an ideal linear sweep extending from $t=-\infty$ to $+\infty$, the probability of remaining in the initial uncoupled state is $\exp[-2\pi|\Delta|^2/(\hbar|\alpha|)]$. The exponent compares the square of the coupling with the rate of change of the uncoupled <energy> difference. This model helps estimate when a molecular passage through an <avoided crossing> or a controlled <qubit> sweep ceases to obey the <adiabatic theorem>.
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