Semiclassical computation of non-adiabatic quantum dynamics

We develop efficient computational methods for oscillatory hyperbolic systems arising in non-adiabatic quantum dynamics involving band-crossing. Band-crossing is a quantum dynamical behavior that contributes to important physics and chemistry phenomena such as quantum tunneling, Berry connection, chemical reaction etc. In this talk, we will discuss our works in developing semiclassical methods for band-crossing in surface hopping. For such systems, which are hyperbolic but with oscillatory terms modeling non-adiabatic quantum transitions between different energy bands, we will introduce a nonlinear geometric optics method based "asymptotic-preserving" method that is accurate uniformly for all wave numbers, including the problem with random uncertain band gaps.