GENERALIZED DEMKOV MODEL - GEOMETRICAL ADIABATIC PHASE-FACTORS

We consider a two-state model which extends the exponential time-depen dent Demkov model of non-adiabatic coupling to complex energies and in teractions. Using the method of steepest descent, exact and practicabl e asymptotic expansions of the Hankel functions H(upsilon-j)(1)(lambda ), H(upsilon-j)(2)(lambda) are obtained when \upsilon(j)\ and \lambda\ are large and of the same order. The semiclassical scattering two cha nnel matrix within the strong-coupling Demkov model and embracing non- adiabatic parameters is derived. Closed form expressions for the trans ition probabilities, wich incorporate the effects of the classical tur ning point, are also derived. The matrix elements are shown to reduce correctly in known limiting cases. These include the case of exact sym metric resonance, the Landau-Zener and Callaway-Bartling model and the Stueckelberg noncrossing formula. In the case of a Hamiltonian matrix which is complex Hermitian on the real t axis, Berry has shown that t he phenomenon of anholonomy arises as the physical manifestation of a non-integrable real geometric phase in adiabatic parallel transport ro und a closed circuit in real space. More recently Berry has reinterpre ted this theory as arising from transport around a circuit in complex time. The geometric amplitude factor, which is simply the anholonomic phase corresponding to the connection on a complex fibre bundle, occur s naturally in the Demkov model which is an example of a closed system in parameter space driven by a complex Hermitian Hamiltonian. It is s hown that the Berry phase phenomenon also involves contributions from both real and complex circuit adiabatic phases.