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Localisation d'Anderson avec des atomes froids : dynamique dans le désordre et perspectives avec des modèles chaotiques

Tony Prat

Published Sep 25, 2017• 0 citations
Publisher

Abstract

This thesis theoretically investigates several effects related to Anderson localization, focusing on the context of disordered and chaotic cold-atomic systems. In cold-atomic systems, optical speckle patterns are often used to create the disorder. The resulting potentials felt by the atoms differ from Gaussian random potentials, commonly assumed in the description of condensed-matter systems. In the first part of the thesis, we discuss their specificities, with an emphasis on the spectral properties. Atom-optics experiments offer interesting possibilities, such as the possibility to directly probe the atoms inside the disordered potential. In view of these possibilities, we consider in the second part of the thesis the spreading of matter wave packets initially launched with a non-zero velocity. We find that after an initial ballistic motion, the packet center-of-mass experiences a retroreflection and slowly returns to its initial position, mimicking a boomerang. Atom-atom interactions are then introduced in a third part. We consider dilute condensed bosonic gases, and treat the interactions at the mean-field (Gross-Pitaevskii) level. Various situations are studied numerically, in particular the quantum boomerang scenario, and the dynamical spreading both in momentum and energy of matter waves prepared as plane waves. In the last part, we show that chaotic models offer interesting prospects for the study of Anderson localization. On the one hand, we present strong evidences in favor of a spinless kicked rotor in the sympletic ensemble. On the other hand, a second look at a commonly studied quasi-periodically modulated kicked rotor reveals intriguing results.

DOI
10.70675/d41abb3dz5bb4z424cza4fbz977b8d6518d2

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Anderson localization with cold atoms : dynamics in disorder and prospects from chaos

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Localisation d'Anderson avec des atomes froids : dynamique dans le désordre et perspectives avec des modèles chaotiques Dans cette thèse, nous étudions théoriquement plusieurs effets liés à la localisation d'Anderson, dans le contexte des atomes froids. Dans les systèmes d'atomes froids, le désordre est généralement créé à l'aide d'une figure de tavelure optique. Dans la première partie de la thèse, nous discutons des spécificités de ces potentiels optiques, et nous nous intéressons en particulier aux propriétés spectrales. Les expériences usant de l'interaction lumière-matière offrent d'intéressantes possibilités. Dans ce cadre, nous considérons dans une deuxième partie de la thèse l'étalement d'un paquet d'ondes atomique, initialement lancé avec une vitesse non nulle dans un potentiel désordonné. Nous trouvons qu'après un mouvement balistique, le centre de masse du paquet subit une rétro-réflection et retourne lentement à sa position initialle, se comportant comme un boomerang. Nous introduisons ensuite les interactions inter-atomiques dans une troisième partie. Nous considèrons des gaz dilués de bosons condensés, et traitons les interactions au niveau champ moyen. Plusieurs situations sont étudiées numériquement, en particulier le boomerang quantique, et l'étalement dynamique -- à la fois en impulsion et en énergie -- d'ondes de matière préparées en ondes planes. Dans la dernière partie de la thèse, nous montrons que des modèles chaotiques offrent des perspectives intéressantes pour l'étude de la localisation d'Anderson. D'une part, nous présentons des éléments probants en faveur d'un kick rotor sans spin dans l'ensemble symplectique. D'autre part, le réexamen de modèles communément étudiés de kick rotors quasi-périodiques révèle des résultats intrigants.

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Localization and spreading of matter waves in disordered potentials

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In this thesis we address relevant problems of the physics of quantum disordered systems from a numerical and theoretical point of view, with specific attention to the connection of our findings with ultracold atomic gases experiments. We concentrate on two main issues: the interplay between localization and interaction in disordered systems and the problem of localization in correlated random potentials. The first problem is investigated considering the expansion of a weakly interacting Bose gas in a bichromtic optical lattice. We observe that interaction has a destructive effect on the disorder-induced localization and leads to a subdiffusive expansion of the atomic gas. By comparing three characteristic energy scales of the system one can identify three different spreading regimes: weak chaos, strong chaos and self-trapping. The spreading behaviour in these regimes is predicted theoretically and verified numerically. We also interpreted existing experimental data on the basis of our findings and showed that there is a qualitative agreement between our numerical simulations and experiments. The second problem is investigated proposing a new model of correlated disorder that can be implemented experimentally using ultracold dipolar gases. We show that this model is characterized by the presence of both short and long range correlations. We study the localization properties of the model and highlight the role played by short and long range correlations in the determination of those properties. In particular we show that when short-range correlations are dominant, extended states can appear in the spectrum. The effect of long-range correlations is instead to restore localization over the whole spectrum and lead to counterintuitive behaviours of the localization length. More precisely, depending on the localization regime they can enhance or reduce the localization length at the centre of the band.

•Jan 1, 2013•0 citations
This thesis theoretically investigates several effects related to Anderson localization, focusing on the context of disordered and chaotic cold-atomic systems. In cold-atomic systems, optical speckle patterns are often used to create the disorder. The resulting potentials felt by the atoms differ from Gaussian random potentials, commonly assumed in the description of condensed-matter systems. In the first part of the thesis, we discuss their specificities, with an emphasis on the spectral properties. Atom-optics experiments offer interesting possibilities, such as the possibility to directly probe the atoms inside the disordered potential. In view of these possibilities, we consider in the second part of the thesis the spreading of matter wave packets initially launched with a non-zero velocity. We find that after an initial ballistic motion, the packet center-of-mass experiences a retroreflection and slowly returns to its initial position, mimicking a boomerang. Atom-atom interactions are then introduced in a third part. We consider dilute condensed bosonic gases, and treat the interactions at the mean-field (Gross-Pitaevskii) level. Various situations are studied numerically, in particular the quantum boomerang scenario, and the dynamical spreading both in momentum and energy of matter waves prepared as plane waves. In the last part, we show that chaotic models offer interesting prospects for the study of Anderson localization. On the one hand, we present strong evidences in favor of a spinless kicked rotor in the sympletic ensemble. On the other hand, a second look at a commonly studied quasi-periodically modulated kicked rotor reveals intriguing results.
Localisation d'Anderson avec des atomes froids : dynamique dans le désordre et perspectives avec des modèles chaotiques Dans cette thèse, nous étudions théoriquement plusieurs effets liés à la localisation d'Anderson, dans le contexte des atomes froids. Dans les systèmes d'atomes froids, le désordre est généralement créé à l'aide d'une figure de tavelure optique. Dans la première partie de la thèse, nous discutons des spécificités de ces potentiels optiques, et nous nous intéressons en particulier aux propriétés spectrales. Les expériences usant de l'interaction lumière-matière offrent d'intéressantes possibilités. Dans ce cadre, nous considérons dans une deuxième partie de la thèse l'étalement d'un paquet d'ondes atomique, initialement lancé avec une vitesse non nulle dans un potentiel désordonné. Nous trouvons qu'après un mouvement balistique, le centre de masse du paquet subit une rétro-réflection et retourne lentement à sa position initialle, se comportant comme un boomerang. Nous introduisons ensuite les interactions inter-atomiques dans une troisième partie. Nous considèrons des gaz dilués de bosons condensés, et traitons les interactions au niveau champ moyen. Plusieurs situations sont étudiées numériquement, en particulier le boomerang quantique, et l'étalement dynamique -- à la fois en impulsion et en énergie -- d'ondes de matière préparées en ondes planes. Dans la dernière partie de la thèse, nous montrons que des modèles chaotiques offrent des perspectives intéressantes pour l'étude de la localisation d'Anderson. D'une part, nous présentons des éléments probants en faveur d'un kick rotor sans spin dans l'ensemble symplectique. D'autre part, le réexamen de modèles communément étudiés de kick rotors quasi-périodiques révèle des résultats intrigants.
Static and dynamical properties of atomic Bose-Einstein condensate in a random potential. The presence of disorder in systems with long-range coherence leads to fascinating effects such as the localization of waves and the appearance of novels quantum phases. While it is often difficult to isolate the effects of disorder in experiments, atomic Bose-Einstein condensates can be manipulated in a controlled random environment. In this report, several aspects of atomic Bose-Einstein condensates in a random potential induced by laser speckles are discussed. After presenting the realization and the characterization of the speckle potential we first study some properties of the trapped, disordered condensates, in particular the enhancement of large density modulations during time-of-flight. We show that this effect is not related to initial phase fluctuations but originates from small initial density modulations in the trap. Secondly, we report on collective modes of a disordered Bose-Einstein condensate. The absence of frequency shifts and the damping of the dipole mode in the presence of disorder are discussed. Then we present measurements of the velocity of sound in a disordered condensate and we show the absence of shift compared to the non-disordered case. At last, we report on the suppression of the 1D expansion of an interacting matter-wave in the presence of disorder in two different regimes. In a regime of strong disorder a detailed experimental analysis supports the proposed scenario of trapping where reflexions on a single modulation of the random potential and interactions play a crucial role. A different regime of weak disorder is then investigated theoretically and we show that Anderson localization can happen in this case under the conditions we establish. This scheme has been realized recently in experiments, leading to the observation of the Anderson localization of a matter-wave.
Topic of the thesis is a theoretical description of the ultracold atomic gases in one- and two-dimensional optical lattices in the presence of the disorder leading to the Anderson localization. The disorder is created by interaction of the main fraction of atoms with the second immobilized fraction distributed randomly over the lattice. In low-dimensional systems there is no transition from the Anderson localized to the conducting phase, although in the presence of correlations a discrete set of extended states can exist. The first part of the thesis is devoted to properties of such states. In the finite size lattices, the presence of those states results in the appearance of `windows of transport' -- energy ranges, in which the localization length is longer than the system size.The analytical method of determining the extended states energies for correlations of generalized NNN-mers type is presented, along with a proof that indeed those states are extended in the infinite system. Subsequently, the way of experimental creation of this type of correlations is proposed, as well as the technique of generation of the disorder in the tunneling amplitudes, which significantly enhances tunability of the proposed energy filters. The second part of the thesis describes the method which allows to simulate a specific type of the disorder: random magnetic field. Systems of such a class, created in a two dimensional lattice using simultaneous fast periodic modulation of the lattice height and interactions with immobilized species, may allow in future to investigate the range of topics from the condensed matter physics, for example fractional quantum Hall effect at half-filling. In the thesis, the most interesting features observed upon investigation of such systems are presented. Especially, the anomalously low localization length for correlated disorder is explained.
This manuscript presents a selection of works on the coherent evolution of atomic matter waves in random potentials. In this context, a number of interference, « mesoscopic » phenomena surviving a disorder average are discussed, in particular in the regime where Anderson localization shows up: coherent backscattering, coherent forward scattering, mesoscopic echo and quantum boomerang effect. The interplay between weak atomic interactions and Anderson localization is also touched upon for spreading wave packets, a scenario frequently considered in experiments. The last chapter, finally, gives an overview of recent works by the author on multiple scattering of light in media of dimension « (2+1) ». These peculiar systems, originally introduced to demonstrate Anderson localization of light in 2D, have sparked a growing interest in the recent years, for light propagation is governed by an effective Schrödinger equation and thus mimics the behavior of a matter wave. We here show that when the vector character of light is properly taken into account, these systems in fact display a physics much richer than the one pertained to this sole analogy.
We revisit the problem of Anderson localization in a trapped Bose–Einstein condensate in 1D and 3D in a disordered potential, applying Quantum Monte Carlo technique because the disorder cannot be treated accurately in a perturbative way as even a small amount of disorder can produce dramatic changes in the physical properties of the system under investigation. Till date no unambiguous evidence of localization has been observed for matter waves in 3D. Matter waves made up of cold atoms are good candidates for such investigations. Simulations are performed for Rb gas in continuous space using canonical ensemble in the case of random and quasi-periodic potentials. To realize random and quasiperiodic potentials numerically we use speckle and bichromatic potentials, respectively. Owing to the high degree of control over the system parameters we specifically study the interplay of disorder and interaction in the system. A dilute Bose gas placed in a random environment falls into a fragmented localized state and the ergodicity (the repetitiveness of the wave function) is lost. An arbitrary Interaction can slowly overcome the effect of disorder and restore the ergodicity again. We observe that as the interaction strength increases, the wave functions become more and more delocalized. Since vanishing of Lyapunov exponent is only a necessary but not a sufficient condition for delocalization for probing the localization we calculate the mean square displacements as an alternative measure of localization. The path integral Monte Carlo technique in this paper numerically establishes the existing predictions of the scaling theory so far and paves a clear path for the further investigation of scaling theory to calculate more complicated properties like ‘critical exponents’ etc. in disordered quantum gases.
In this thesis we address relevant problems of the physics of quantum disordered systems from a numerical and theoretical point of view, with specific attention to the connection of our findings with ultracold atomic gases experiments. We concentrate on two main issues: the interplay between localization and interaction in disordered systems and the problem of localization in correlated random potentials. The first problem is investigated considering the expansion of a weakly interacting Bose gas in a bichromtic optical lattice. We observe that interaction has a destructive effect on the disorder-induced localization and leads to a subdiffusive expansion of the atomic gas. By comparing three characteristic energy scales of the system one can identify three different spreading regimes: weak chaos, strong chaos and self-trapping. The spreading behaviour in these regimes is predicted theoretically and verified numerically. We also interpreted existing experimental data on the basis of our findings and showed that there is a qualitative agreement between our numerical simulations and experiments. The second problem is investigated proposing a new model of correlated disorder that can be implemented experimentally using ultracold dipolar gases. We show that this model is characterized by the presence of both short and long range correlations. We study the localization properties of the model and highlight the role played by short and long range correlations in the determination of those properties. In particular we show that when short-range correlations are dominant, extended states can appear in the spectrum. The effect of long-range correlations is instead to restore localization over the whole spectrum and lead to counterintuitive behaviours of the localization length. More precisely, depending on the localization regime they can enhance or reduce the localization length at the centre of the band.