Effects of periodic modulation on the Landau-Zener transition

We study the quantum tunnelling of a two-level crossing system which extends the standard Landau-Zener model with applying a periodic modulation on its energy sweep. By directly integrating the time evolution operator we obtain the analytic expressions of tunnelling probability in the cases of high and low modulation frequency limit as well as in weak inter-level coupling limit. Our formula clarify the conditions for resonance occurrence, with the help of it we can readily manipulate the system in a desired way, say, to enhance or suppress the tunnelling probability effectively through adjusting the modulation properly.