abstract
When the state of a quantum system is slowly swept through an avoided crossing, there is a finite probability of a non-adiabatic transition across the gap. The non-adiabatic transition can serve as a switching mechanism for a qubit whose levels are coupled by a chirped frequency control. However, limited bandwidth in the control of the qubit implies that the transition probability is not precisely described by the universal Landau-Zener formula. I will use the nonlinear quantum normal form theory developed by Colin de Verdiere to express the transition amplitudes in terms of the curvature of the control signal. It is a starting point for an asymptotic expansion of the scattering matrix analogous to the one used for reaction rates in quantum chemistry.