Partial Landau-Zener transitions and applications to qubit shuttling

The transition dynamics of two-state systems with time-dependent energy levels, first considered by Landau, Zener, Majorana, and St\"uckelberg, is one of the basic models in quantum physics and has been used to describe various physical systems. We propose here a generalization of the Landau-Zener (LZ) problem characterized by distinct paths of the instantaneous eigenstates as the system evolves in time while keeping the instantaneous eigenenergies exactly as in the standard LZ model. We show that these paths play an essential role in the transition probability $P$ between the two states, and can lead to a substantial reduction of $P$, being possible even to achieve $P=0$ in an instructive extreme case, and also to large $P$ even in the absence of any anticrossing point. The partial LZ model can describe valley transition dynamics during charge and spin shuttling in semiconductor quantum dots.