SEMINAIRE MERCREDI 16 AVRIL 2008
14 heures
Salle Séminaire 5
Centre de Physique Théorique
Marseille-Luminy

Olivier Sapin
Laboratoire de Physique, Université de Bourgogne, Dijon

Titre: Exposant de Lyapunov quantique et relations d'Anosov pour
l'oscillateur paramétrique quantique quasi-périodique

Abstract: Anosov properties and Lyapunov exponents are well-established
characterization of classical dynamics and it is natural to search for
similar concepts applicable to quantum dynamics. Several definitions
have been given in the literature. Due to the fact that there are few
examples on which these definitions can be explicitly tested in
detail, their range of applicability is not well established. We
introduce here a definition of upper Lyapunov exponent for quantum
systems in the Heisenberg representation, and apply it to parametric
quantum oscillators. We provide a simple proof that the upper quantum
Lyapunov exponent ranges from zero to a positive value, as the
parameters range from the classical system's region of stability to
the instability region. Moreover, we generalize the definition of
quantum Anosov properties defined by Emch, Narnhofer, Sewell and
Thirring to the case of quantum systems driven by a classical flow. We
show that this definition can be interpreted as regular Anosov
properties in an enlarged Hilbert space, in the framework of a
generalized Floquet theory. This extension allows us to describe the
hyperbolicity properties of almost-periodic quantum parametric
oscillators. As second example, we show that the configurational
quantum cat system satisfies quantum Anosov properties.