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Application of strong-coupling asymptotic expansions to the complex Nikitin model of atomic collisions

We consider a phase integral interpretation of the two-state time-dependent exponential model of Nikitin. We generalize to complex energies and complex interactions following previous workers (Krstić & Janev 1988). Applying the method of steepest descent to the Whittaker functions for which all three parameters are large and complex we obtain the adiabatic semi-classical scattering matrix of non-adiabatic transitions. The matrix elements which are expressed in terms of phase integrals taken over physical potentials, are shown to reduce correctly in known limiting cases. These include the Demkov, Rosen-Zener and parabolic models and also the case of symmetric and non-symmetric resonance. It has been previously shown by Coveney et al. that the elastic Stueckelberg non-adiabatic phase may be interpreted as arising from adiabatic parallel transport of the state vector round a closed contour or circuit in the complex time (t) plane, even though the hamiltonian is real hermitian on the real t-axis. For a hamiltonian matrix which is complex hermitian on the real t-axis, it has been shown by Berry that anholonomy results in a non-integrable real geometric phase in adiabatic parallel transport around a circuit in real space. It is shown that the Berry phase interpretation also applies to the generalized Whittaker model and all of the above mentioned models, and that it involves contributions from both real and complex closed-circuit adiabatic phases.