Avoided crossings and Landau-Zener transitions

In any problem involving avoided crossings, there are two modes of variation. A physical system will evolve adiabatically, i.e. it will remain on one or other branch if the intensity A is varied slowly. If. however, the variation is rapid then diabatic or Landau-Zener transitions may occur [508]. The probability of a transition from one branch to the other at the avoided crossing is given by 

Floquet theory provides a natural means of studying the dynamics of the response of an atomic system to a pulsed laser field. Indeed, it yields a general framwork for discussing the adiabatic motion of any periodically driven quantum system. If we define a characteristic time St = 6A/(dA/dt) which is the time taken to sweep through the avoided crossing, then one can distinguish between three regimes  

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