Stability of the electron cyclotron resonance

We consider the magnetic AC Stark effect for the quantum dynamics of a single particle in the plane under the influence of an oscillating homogeneous electric and a constant perpendicular magnetic field. We prove that the electron cyclotron resonance is insensitive to impurity potentials.Comment: version to appear in Comm. Math. Phy

Spectral Stability of Unitary Network Models

We review various unitary network models used in quantum computing, spectral analysis or condensed matter physics and establish relationships between them. We show that symmetric one dimensional quantum walks are universal, as are CMV matrices. We prove spectral stability and propagation properties for general asymptotically uniform models by means of unitary Mourre theory

Localization Properties of the Chalker-Coddington Model

The Chalker Coddington quantum network percolation model is numerically pertinent to the understanding of the delocalization transition of the quantum Hall effect. We study the model restricted to a cylinder of perimeter 2M. We prove firstly that the Lyapunov exponents are simple and in particular that the localization length is finite; secondly that this implies spectral localization. Thirdly we prove a Thouless formula and compute the mean Lyapunov exponent which is independent of M.Comment: 29 pages, 1 figure. New section added in which simplicity of the Lyapunov spectrum and finiteness of the localization length are proven. To appear in Annales Henri Poincar

The GRAVITY+ Project: Towards All-sky, Faint-Science, High-Contrast Near-Infrared Interferometry at the VLTI

The GRAVITY instrument has been revolutionary for near-infrared interferometry by pushing sensitivity and precision to previously unknown limits. With the upgrade of GRAVITY and the Very Large Telescope Interferometer (VLTI) in GRAVITY+, these limits will be pushed even further, with vastly improved sky coverage, as well as faint-science and high-contrast capabilities. This upgrade includes the implementation of wide-field off-axis fringe-tracking, new adaptive optics systems on all Unit Telescopes, and laser guide stars in an upgraded facility. GRAVITY+ will open up the sky to the measurement of black hole masses across cosmic time in hundreds of active galactic nuclei, use the faint stars in the Galactic centre to probe General Relativity, and enable the characterisation of dozens of young exoplanets to study their formation, bearing the promise of another scientific revolution to come at the VLTI.Comment: Published in the ESO Messenge

Étude de la stabilité de systèmes dynamiques quantiques

Jury : Yves COLIN DE VERDIÈRE (Université de Grenoble I), Président ; Joachim ASCH (CPT-Marseille, Université de Toulon et du Var), Co-Directeur ; Monique COMBESCURE (CNRS, IPNL, Université de Lyon I), rapporteur ; Stephan DE BIÈVRE (Université des Sciences et Techniques, Lille I), rapporteur ; Pierre DUCLOS (CPT-Marseille, Université de Toulon et du Var) ; Alain JOYE (Université de Grenoble I), Co-Directeur.The dynamics of a periodic time-dependent quantum system may be described by means of a Floquet operator on a suitable Hilbert space. The spectral properties of this operator give some information on the long time behaviour of the system. Two models are considered here. First, we investigate the spectral properties of the Floquet operators of some stationary quantum systems with a discrete and simple spectrum, periodically perturbed by a rank one kick. Our first result is the following: if the perturbation is suitably chosen and if the eigenvalues of the stationary system are given by a polynomial with some arithmetic condition on its coefficients, the spectrum of the Floquet operator is purely singular continuous. Then, we prove this is still true for some rank one perturbation, if the eigenvalues grow suitably fast and if the period belongs to a set of full Lebesgue measure. These results complete a previous study of Combescure. The spectral properties of a family of Floquet operators with a matrix representation displaying a band structure are also analyzed. Such operators appear in the study of some electronic conduction models. Although they depend on a larger number of parameters, we prove their spectral properties depend in some limit, on the choice of two infinite sequences of phases. We prove therefore that the spectrum remains purely singular if the phases are determined by some ergodic processes. However, the spectrum is absolutely continuous and may possess a finite number of isolated simple eigenvalues if the phases are built according to a periodic procedure.La dynamique d'un système quantique gouverné par un hamiltonien dépendant du temps de manière périodique peut être décrite à l'aide d'un opérateur de Floquet sur un espace de Hilbert convenable. La nature spectrale de cet opérateur donne des informations sur le comportement temporel asymptotique du système concerné. Deux modèles sont étudiés dans cette perspective. La première analyse que nous proposons, prolonge et complète les travaux de Combescure sur la dynamique de systèmes stationnaires à spectre discret, simple, frappés périodiquement par une perturbation de rang un. Un premier résultat est d'abord obtenu lorsque les valeurs propres du système stationnaire sont données par un polynôme vérifiant certaines conditions arithmétiques et lorsque la perturbation est convenablement choisie : le spectre de l'opérateur de Floquet est alors singulier continu. Nous montrons ensuite que sous certaines hypothèses sur la croissance de ces valeurs propres, ce spectre reste singulier continu pour presque toute période au sens de la mesure de Lebesgue et tout choix convenable de la perturbation de rang un. Une stratégie d'analyse spectrale différente est ensuite mise en place pour une classe d'opérateurs de Floquet intervenant dans un modèle de conduction électronique et ayant une représentation matricielle multi-diagonale. Bien que ces opérateurs soient bâtis autour d'un nombre de paramètres plus importants, nous montrons que dans une certaine limite motivée par des considérations physiques, l'étude spectrale est seulement gouvernée par deux suites de phases. Lorsque ces phases sont engendrées par certains processus ergodiques, nous montrons que le spectre de l'opérateur de Floquet est singulier. Lorsqu'elles sont données par une construction périodique, le spectre présente une portion absolument continue ainsi qu'un nombre fini de valeurs propres isolées et de multiplicité un

Singular continuous Floquet operator for systems with increasing gaps

AbstractConsider the Floquet operator of a time-independent quantum system, periodically perturbed by a rank one kick, acting on a separable Hilbert space: e−iH0Te−iκT|φ〉〈φ|, where T and κ are the period and the coupling constant, respectively. Assume the spectrum of the self-adjoint operator H0 is pure point, simple, bounded from below and the gaps between the eigenvalues (λn) grow like λn+1−λn∼Cnd with d⩾2. Under some hypotheses on the arithmetical nature of the eigenvalues and the vector φ, cyclic for H0, we prove the Floquet operator of the perturbed system has purely singular continuous spectrum