On the semiclassical approach to cold atomic collisions

Abstract

We extend the semiclassical description of two-state atomic collisions to low energies for which the impact parameter treatment fails. The problem reduces to solving a system of first-order differential equations with coefficients whose semiclassical asymptotes experience the Stokes phenomenon in the complex coordinate plane. Primitive semiclassical and uniform Airy approximations are discussed. The theory of collisions between two atomic systems goes back to the early days of quantum mechanics (Mott 1931; Teller 1937; Landau & Teller 1936) and the basic 1930s models of state interaction are detailed in the recent review of Nikitin (1999). Typically, the colliding atoms undergo electronic transitions and one needs to solve a number (in the simplest case, two) of coupled radial Schrodinger equations. It has been noted since the early thirties that the relative motion of the heavy nuclei can be described classically (Mott 1931; Rosen & Zener 1932). Semiclassically, Stueckelberg (1932) first suggested the analytical continuation of the JWKB (Jeffreys-Wentzel- Kramers-Brillouin) wave functions into the complex plane of the internuclear sepa- ration and a proper handling of the Stokes phenomenon. His solution of the avoided crossing time-independent problem (Stueckelberg 1932) shows why a description of the interference in terms of adiabatic quasiclassical phases fails if the phase difference accumulated during the adiabatic motion of the two atoms between the centre of the coupling region and the turning points is small. The vexatious problem of the choice of branch cuts and the determination of Stokes constants via the comparison equa- tion method, were largely resolved by Crothers (1971). Equally well, it was shown by Coveney et al . (1985) that the classically forbidden region is less amenable to generalized phase-integral analysis. The classical approximation of the nuclear motion leads to the impact parameter treatment (Gaunt 1927; Landau 1932; Zener 1932). The impact parameter approx- imation (IPA) offers a simple physical picture in which fast electronic transitions occur as the slow nuclei follow a classical trajectory. By its nature, the IPA neglects