Landau-Zener transition

A Landau-Zener transition occurs when a changing parameter sweeps two coupled quantum states through an avoided crossing. In a two-state description,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 $\exp [-2\pi |\Delta {|}^2/(\hslash |\alpha |)]$. 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.