# Effet tunnel dans les systèmes complexes

Abstract : The present work is developed within the general framework of the description of the tunneling effect in the semiclassical limit $\hbar \rightarrow 0$. We introduce a new method for the direct computation of the tunneling splittings. We get a trace formula involving the evolution operator continued in the complex plane using a complex time $T$. The next step is to obtain semiclassical expansion of these traces. Within the framework of one dimensionnal integrable systems, we show the key role of a complex time. When performing semiclassical calculations, an appropriate complex-time paths provide an efficient criterion in order to select the dominant complex trajectories involved in the traces. We will show that our approach includes instanton techniques in the limit of a purely imaginary time and describes dynamical tunneling and resonant tunneling for which a complete $\textsc{Wick}$ is not sufficient. We will show also how our method works for the decay rates. Finally, as a perspective, we will study resonant tunneling from integrable models which exhibit prominent islands surrounded by chains of tori. From these models, we will try to apply the theory of resonant assisted tunneling to integrable systems.
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https://tel.archives-ouvertes.fr/tel-00599680
Contributor : Jérémy Le Deunff <>
Submitted on : Friday, June 10, 2011 - 3:42:30 PM
Last modification on : Thursday, March 5, 2020 - 5:33:19 PM
Document(s) archivé(s) le : Friday, November 9, 2012 - 3:01:00 PM

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• HAL Id : tel-00599680, version 1

### Citation

Jérémy Le Deunff. Effet tunnel dans les systèmes complexes. Physique mathématique [math-ph]. Université François Rabelais - Tours, 2011. Français. ⟨tel-00599680⟩

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