Finite-speed effects in non adiabatic transitions

The Landau-Zener formula works because it is based on a universal quantum normal form of non-adiabatic transitions (also known as mode conversion). However, the normal form is exact only in the non-physical limit of infinitely slow approach to the conversion point or infinite bandwidth. When the Hamiltonian depends smoothly on time, Colin de Verdiere has shown that non-adiabatic transition amplitudes are determined entirely by its local structure near the transition point. Using quantum normal form theory, we calculate the leading finite-speed correction to the Landau-Zener formula in terms of the transition point Taylor expansion of the Hamiltonian.