Exact solutions to Landau–Zener problems by evolution operator method

Various analytic approaches have been developed to solve the famous Landau–Zener (LZ) problem. Here, we introduce a time-evolution operator method to investigate such a problem by numerically solving the induced algebraic equations (by Runge–Kutta method). Based on these calculations, transitions between two levels driven by various time-dependent external fields can be simulated in detail, typically near the so-called avoided crossing points. Far from these points, our results reduce to the original Landau–Zener transition (LZT).