HEAVY-PARTICLE COLLISIONS AND QUANTUM OPTICS - THE PARABOLIC NONCROSSING MODEL

The problem of deriving analytic formulas for transition probabilities in two-level systems is studied. The two-level systems are described by a pair of first-order differential equations coupled by a time-depe ndent potential. One such model is given by da(m)/dt=-i beta f(t)a(n)e ((-1)ni alpha t) (m,n=1,2; m not equal n), which describes certain typ es of ion-atom collisions and some quantum-optics two-level problems. It will be shown that the correct approach in solving the coupled equa tions is to adopt a Zwaan-Stueckelberg phase-integral analysis of the four-transition-point problem based on the parabolic noncrossing model of Crothers [J. Phys. B 9, 635 (1976)]. Alternatively, one may obtain an approximation by employing adiabatic perturbation theory, but such an approach can at best provide only weak-coupling solutions and can never guarantee unitarity in the probability amplitudes. The advantage of the phase-integral method is that it produces a strong-coupling ap proximation by embracing the appropriate asymptotic expansions for cyl inder functions of large order and argument [D. S. F. Crothers, J. Phy s. A 5, 1680 (1972)] and it also ensures analyticity, unitarity, and s ymmetry.