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Géométrie et adiabaticité des systèmes photodynamiques quantiques

Abstract : The numerical treatment of the interaction of an electromagnetic field with an atom or a molecule presents some severe computational problems. The description of the photodynamical system by traditional methods requires the use of a very large set of basis funtions; this in turn leads to the need for a large computer memory store and to long computational times. In our modelling we use only a relatively small vector space, the active space, and adopt an approach which relies heavily on adiabatic time evolutions, since the parameters describing the system are taken to change slowly. Adiabatic theorems then lead to the result that the system remains confined to a spectral subspace associated with an isolated group of eigenvalues. In mathematical terms the wave function can be described by means of a horizontal lift in the principal bundle associated with the Berry phase. Since this phase does not commute with the dynamical phase we propose a description based on the use of a composite bundle, which can simultaneously describe both geometric and dynamical phases. Our geometrical viewpoint leads to a new way of modelling a photodynamical system; the concept of the virtual magnetic monopole is found to be useful in the construction of the appropriate computational tools to treat dynamical systems. We investigate our geometrical approach in the context of the time-dependent wave operator formalism, in which the effective Hamiltonian plays a major role. We establish the value of our approach in that context by deriving a wave operator adiabatic theorem. We present numerical applications to simple two and three level model atoms and to the H2+ molecular ion.
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https://tel.archives-ouvertes.fr/tel-00011145
Contributor : David Viennot <>
Submitted on : Monday, December 5, 2005 - 9:59:15 AM
Last modification on : Thursday, April 8, 2021 - 3:38:54 AM
Long-term archiving on: : Friday, April 2, 2010 - 11:21:08 PM

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  • HAL Id : tel-00011145, version 1

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David Viennot. Géométrie et adiabaticité des systèmes photodynamiques quantiques. Physique mathématique [math-ph]. Université de Franche-Comté, 2005. Français. ⟨tel-00011145⟩

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