School on Anderson Localization Abstracts

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Anderson localization: Introduction and known results — Dominique Delande

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  • Abstract: Delande will present a general overview of Anderson localization, that is localization in a quantum (or more generally wave) system due to interefences, with emphasis on cold atomic gases. He will introduce the basic notions such as mean free path, diffusion equation, weak and strong localization, as well as discuss the presently available theoretical approaches and why this is a difficult problem in dimension larger than 1.

The Landscape theory of localization — Marcel Filoche

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  • Abstract: Standing waves in disordered or complex systems can be subject to a strange and intriguing phenomenon which has puzzled the physics and mathematical communities for more than 60 years, namely wave localization. This phenomenon consists of a concentration (or a focusing) of the wave energy in a very restricted sub-region of the entire domain. It has been evidenced experimentally in mechanics, in acoustics, and in quantum physics. Determining the conditions for the onset of localization, depending on the disorder amplitude, the energy, or the wave type, is the aim of many theoretical studies. We will present a theory that tends to unify various aspects of wave localization within a single mathematical framework. To that end, we will introduce the notion of "localization landscape", solution to a Dirichlet problem associated to the wave equation, and an "effective localization potential", reciprocal of the landscape. This potential allows us to predict the localization region, the energies of the localized modes, the density of states, and the long range decay of the wave functions. Through a Weyl-Wigner approach in phase space, this theory give also access to the computation of spectral functions, both in the classical and quantum regimes. We will present experimental and numerical examples of this theory, as well as theoretical perspectives in cold atom systems.

Anderson localization of light by cold atoms — Sergei Skipetrov

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  • Abstract: Large ensembles of cold, immobile atoms might serve as ideal model systems to study Anderson localization of light. In contrast to dielectric systems, atomic clouds allow for a fine experimental control and admit a precise theoretical description. In this talk, we will review the recent theoretical developments aimed at paving a path towards an experimental realization of Anderson localization of light in three-dimensional (3D) atomic systems. Because atoms are the simplest point scatterers, considering light scattering on them is a good opportunity to illustrate several basic concepts and methods of the localization theory (Thouless conductance and Ioffe-Regel criterion, the mobility edge and the critical behavior, the scaling and self-consistent theories, the finite-size scaling procedure, etc.) on a concrete example. We will show that, surprisingly, Anderson localization of light by atoms is impossible in 3D if no special efforts are made to suppress near-field interactions between atoms. The necessary suppression may be achieved by subjecting the atoms to a constant external magnetic field or by arranging them in a crystalline structure in which a small amount of disorder is introduced. However, even when spatially localized states appear in a system of immobile atoms, their experimental observation remains challenging because of the residual thermal motion of atoms. 

Quantum fluids of light in quasi-periodic potential — Jacqueline Bloch

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  • Abstract: Propagation and localization of light and matter waves dramatically depend on the landscape they explore. In periodic media, plane waves propagate ballistically wih a modified group velocity. On the contrary, in disordered media, waves can be scattered and localize so that propagation is inhibited. In between these extreme situations, the case of quasi-periodic media is fascinating. They are neither periodic, nor disordered but present long range correlations and give rise to exotic localization and propagation properties. Two paradigmatic examples of quasi-crystals are the Aubry-André and Fibonacci models. In the former one, depending on a characteristic parameter, eigenmodes are either localized or delocalized, with a region in between where they become critically localized. For the Fibonacci model the situation is very different since the eigenmodes are critically localized, whatever the value of the Fibonacci potential contrast. In the present talk, I will describe a recent work we have developed in collaboration with the Oded Zilberberg. We explored how wave localization evolves when continuously interpolating between these two paradigmatic models that is when continuously deforming an Aubry-André potential into a Fibonacci one. We discovered that the spatial fractality of the Fibonacci critical eigenmodes does not appear abruptly, but progressively develops through a cascade of delocalization localization transitions.  In parallel to these theoretical investigations, we have realized experimentally these exotic quasicrystals by laterally sculpting one-dimensional semiconductor microcavities. Optical spectroscopy experiments revealed the predicted delocalization localization transition and part of the localization diagram. These results open very interesting perspectives with the exploration of such quasi-periodic photonic structures in presence of non-linearities.

Observation of two-dimensional Anderson localisation of ultracold atoms — Maarten Hoogerland

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  • Abstract: We report on our recent experimental observation of two-dimensional Anderson localisation of ultra cold atoms. We use a two-dimensional trap consisting of a single “pancake” of a pair of interfering red-detuned laser beams, and a “starry sky” potential landscape produced by a spatial light modulator. We will discuss the results of the experiment, as well as GPE simulations and insights obtained from localisation landscape theory.

3D Anderson localization with ultracold atoms — Vincent Josse

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  • Abstract: The study of disordered systems with ultracold atoms has attracted a lot of attention over past decade, in particular to investigate the Anderson transition that occurs in three-dimensional systems between localized and diffusive states. However significant discrepancies have been reported between experiments and numerics about the precise location of the mobility edge (energy of the transition), rendering new investigations desirable. In this seminar, Josse will present recent progress along that line, with the measurement of the spectral functions in laser speckle disordered potential. The method relies on the use of a state-dependent disorder and the controlled spectroscopic transfer of atoms to create well-defined energy states. By scanning the energy across the mobility edge, this method opens new prospects to study the 3D Anderson transition.

Engineering Hamiltonians and symmetries in the Quantum Kicked Rotor — Jean-Claude Garreau

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  • Abstract: Symmetries are fundamental for our understanding of physics. This is particularly true for disordered systems and even more for quantum chaotic systems, classified according to Random Matrix ensembles constrained by symmetry. In this talk I will show how the flexibility of the cold-atom Quantum Kicked Rotor, which belongs both to quantum-chaotic and to quantum-disordered systems allows us to experimentally construct Hamiltonians whose parity and time-reversal symmetries can be easily manipulated. These Hamiltonian are used to study weak-localization properties like Coherent Back and Forward Scattering and to measure the famous b(G) scaling function of the Anderson localization.
  • In collaboration with: Clément Hainaut, Isam Manai, Jean-François Clément Pascal Szriftgiser, Gabriel Lemarié, Nicolas Cherroret, Dominique Delande & Radu Chicireanu

Localization of ultracold atoms in optical quasicrystals — Ulrich Schneider

  • Abstract: Quasicrystals are a novel form of condensed matter that is not periodic, but nonetheless long-range ordered. Similar to periodic crystals, quasicrystals give rise to diffraction patterns consisting of sharp Bragg peaks—but with rotational symmetries forbidden for periodic structures.  Many foundational concepts of periodic systems such as Blochwaves or Brillouin zones are not applicable to quasicrystals, thereby giving rise to new physics. Examples include Anderson localization, phasonic degrees of freedom, fractal band structures, and a direct link to higher dimensions via cut-and-project techniques. In this talk, Schneider will present his experimental realization of an eightfold symmetric optical quasicrystal for ultracold atoms. This will be followed by an experimental characterisation of the localization transition in these potentials and I will close by discussing the prospect for observing the 2D Bose glass and Many-body localization in these potentials.

Localization and interactions — Thierry Giamarchi

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  • Abstract: Disorder leads for quantum systems to the celebrated Anderson localization. This phenomenon occurs already for single particles or waves.  One interesting question, pertinent for the understanding of several realistic experimental systems, is to understand what happens when several particles are present and interact with one another. Indeed in that case the energy of a single particle is not conserved, and thus one can expect the interactions to affect drastically the phenomenon of localization. On the other hand if the particles are localized when non-interacting instead of being in a plane-wave state, one can expect also the interactions to be drastically affected as well. This interplay between the effects of disorder and the effects of interactions is thus particularly rich and also particularly difficult to tackle, since the size of the Hilbert space to consider grows exponentially fast with the number of particles. I will review in this lecture the main points and challenges of such problem, starting with equilibrium situation and discussing the consequences for transport in such disordered interacting many-body systems (localization of interacting particles). I will also if time permits discuss the question of what happens for isolated systems and whether such systems can reach equilibrium at all (the so-called many-body localization problem).

Weakly interacting disordered Bose gases out of equilibrium — Nicolas Cherroret

Exploring many-body localization in two dimensions via quantum-gas microscopy — Antonio Rubio Abadal

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  • Abstract: Ultracold atoms in optical lattices have become a versatile platform for the study of out-of-equilibrium quantum dynamics. In this talk, I will describe recent experiments performed in a bosonic “quantum-gas microscope”. The resolution of the imaging system allows us to generate site-resolved programmable disorder potentials1, and hence realize the disordered Bose-Hubbard model,  as well as to prepare density patterns at the single-atom level. Using these tools, we prepare initial states at high energy density, and track their dynamics under the presence of interactions, tunneling and disorder. For sufficiently strong disorder, the initially prepared pattern survives for hundreds of tunneling times, which is a signature of many-body localization. Additionally, I will describe an experiment2, in which we explored the effects of coupling an initially localized system to a quantum bath, i.e. a thermalizing quantum system made only of few degrees of freedom. By preparing a fraction of the atoms in a disorder-insensitive state, we produce a hybrid system of clean and dirty particles, and observe that, while a sufficiently large number of clean particles delocalizes the system, localization can survive for long times when the fractions of the clean component is small.
  • [1] J.-y. Choi et al., Science 352, 1547-1552 (2016); [2] A. Rubio-Abadal et al., Phys. Rev. X 9, 041014 (2019).

Many-body localization in a simple quantum spin chain problem — Nicolas Laforencie

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  • Absract: In this talk, I will focus on the paradigmatic example of a quantum spin chain in a random magnetic field at infinite temperature. This (apparently simple) problem provides a fine example for MBL physics, accessible to exact numerical calculations. Thus, I will try to address some important aspects, such as the existence of a mobility edge; the slow dynamics; multifractal properties ; the universality class of the MBL transition