Publication : t98/025

Etats propres de systemes classiquement chaotiques dans l'espace des phases

Nonnenmacher S. (CEA, DSM, SPhT (Service de Physique Théorique), F-91191 Gif-sur-Yvette, FRANCE)
The aim of this work is to study classically chaotic quantum systems. We restrict ourselves to one-dimensional dynamics, and pay a particular attention to eigenstates, using both analytical and numerical methods. Quantum states are represented using phase space probability densities (Husimi densities), so that they can be easily compared to classically invariant measures, in the semiclassical limit. On the other hand, a quantum state can be built directly from the knoowledge of its constellation, i.e. the set of zeros of its Husimi density. We first study an integrable Hamiltonian system with a fixed unstable point. A precise description of Husimi densities of eigenstates near the critical energu is provided by uniform WKB approximations. While densities concentrate exponentially around the separatrix, zeros are stributed along classically defined (anti-Stokes) lines. We then study area-preserving maps on the torus, in particular Arnold's ``cat'' maps and the baker's map, which are both proven to be fully chaotic and for which we know a consistent quantization procedure. Due to arithmetical properties of cat maps, we can build families of very ergodic eigenstates, for which the constellations form crystals on the torus. More generally, we show that eigenstates of these quantum maps have, {\it on average}, similar properties to Gaussian random states: their Husimi densities and constellations are grossly equidistributed over the whole torus in the semiclassical limit, and their fluctuations around the ergodic measure are universal. On the other hand, we argue that the specic features of an {\it individual} eigenstate (e.g. a {\it scar} above a periodic point) can be robustly extracted from the first few Fourier coefficients of its constellation.
Année de publication : 1998
Soutenance de thèse : Université Paris XI ; 1998-01-30
Lien :
Keywords : Application quantique chaotique du tore, représentation de Bargmann, analyse semi-classique, approximation WKB uniforme, fonction thęta aléatoire, cicatrice d'orbite périodique
Langue : Français
NB : CCSD:identifiant tel-00000855
Editeurs : Voros A.

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