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An Isaac Newton Institute Workshop

The Theory of Highly Oscillatory Problems

Beyond the adiabatic approximation: exponentially small coupling terms

28th March 2007

Author: Stefan Teufel (University of Tuebingen)


For multi-level time-dependent quantum systems one can construct superadiabatic representations in which the coupling between separated levels is exponentially small in the adiabatic limit. We explicitly determine the asymptotic behavior of the exponentially small coupling term for generic two-state systems with real-symmetric Hamiltonian. The superadiabatic coupling term takes a universal form and depends only on the location and the strength of the complex singularities of the adiabatic coupling function. First order perturbation theory for the Hamiltonian in the superadiabatic representation then allows to describe the time-development of exponentially small adiabatic transitions and thus to rigorously confirm Michael Berry’s predictions on the universal form of adiabatic transition histories (Joint work with V. Betz from Warwick).

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