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Time Development of Exponentially Small Non-Adiabatic Transitions

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Abstract

Optimal truncations of asymptotic expansions are known to yield approximations to adiabatic quantum evolutions that are accurate up to exponentially small errors. In this paper, we rigorously determine the leading order non–adiabatic corrections to these approximations for a particular family of two–level analytic Hamiltonian functions. Our results capture the time development of the exponentially small transition that takes place between optimal states by means of a particular switching function. Our results confirm the physics predictions of Sir Michael Berry in the sense that the switching function for this family of Hamiltonians has the form that he argues is universal.

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Acknowledgements.

George Hagedorn wishes to thank the Institut Fourier of the Université de Grenoble I for its kind hospitality and support. Alain Joye wishes to thank Virgina Tech for its kind hospitality and the NSF for travel support. We also wish to thank Ovidiu Costin for many useful discussions about this problem.

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Authors and Affiliations

  1. Department of Mathematics and Center for Statistical Mechanics and Mathematical Physics, Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061-0123, U.S.A

    George A. Hagedorn

  2. Institut Fourier, Unité Mixte de Recherche CNRS-UJF 5582, Université de Grenoble I, BP 74, 38402, Saint Martin d’Hères Cedex, France

    Alain Joye

Authors
  1. George A. Hagedorn
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  2. Alain Joye
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Additional information

Communicated by B. Simon

Partially supported by National Science Foundation Grants DMS–0071692 and DMS–0303586.

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Hagedorn, G., Joye, A. Time Development of Exponentially Small Non-Adiabatic Transitions. Commun. Math. Phys. 250, 393–413 (2004). https://doi.org/10.1007/s00220-004-1124-5

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