Exactly soluble two-state quantum models with linear couplings
Boyan T. Torosov, Nikolay V. Vitanov · Journal of Physics A Mathematical and Theoretical · 2008
A class of exact analytic solutions of the time-dependent Schrödinger equation is presented for a two-state quantum system coherently driven by a nonresonant external field. The coupling is a linear function of time with a finite duration and the detuning is constant. Four special models are considered in detail, namely the shark, double-shark, tent and zigzag models. The exact solution is derived by rotation of the Landau–Zener propagator at an angle of π/4 and is expressed in terms of Weber's parabolic cylinder function. Approximations for the transition probabilities are derived for all four models by using the asymptotics of the Weber function; these approximations demonstrate various effects of physical interest for each model.