Wave transport in strongly scattering, disordered media is investigated experimentally using ultrasonic techniques. Several cases of anomalous wave transport (deviations from conventional diffusion) are studied through experiments on aluminum mesoglass samples, which were designed and created for this purpose. The anomalous wave behaviour is contrasted with conventional diffusion, observed at some frequencies via both transmission and backscattering measurements on wide, thick, slab-shaped samples. The coherent backscattering (CBS) effect is measured experimentally for strongly scattered acoustic waves in three dimensions (3D), and is compared for the first time with theory for diffusive elastic waves in 3D to give an estimate of the diffusion coefficient. At other frequencies, an Anderson localization regime is observed, and is studied in detail. The first experimental study of CBS for localized elastic waves in 3D is presented. By comparing both backscattering and transmission measurements with predictions from a ‘local’ self-consistent theory of localization, the first experimental observation of a complete Anderson mobility gap for elastic waves in 3D is reported. In this mobility gap, large contributions to backscattered intensity from recurrent scattering were observed, enabling the first experimental study of recurrent scattering on its own. The time-dependence of the recurrent scattering, R(t), is shown to agree with theoretical predictions in the diffuse and localized regimes. At the mobility edge, R(t) shows a surprisingly slow decay, prompting further theoretical work. Localization and criticality are also investigated via statistical measurements of ultrasound from cubic mesoglasses of different sizes. Finite-size scaling of multifractal quantities is observed in these cubic samples, and a preliminary fit with theory to determine critical parameters of the Anderson transition is demonstrated. Finally, a sample is which is a candidate to exhibit superdiffusion of ultrasound is studied via a range of experimental techniques, showing subtle deviations from diffusion and opening doors for the next steps in this study. / May 2016

Computational techniques were employed to investigate pathways that would improve the properties and characteristics of transition metal (i.e., ruthenium) catalysts, and to explore their mechanisms. The studied catalytic pathways are particularly relevant to catalytic hydroarylation of olefins. These processes involved the +2 to +3 oxidation of ruthenium and its effect on ruthenium-carbon bond strengths, carbon-hydrogen bond activation by 1,2-addition/reductive elimination pathways appropriate to catalytic hydrogen/deuterium exchange, and the possible intermediacy of highly coordinatively unsaturated (e.g., 14-electron) ruthenium complexes in catalysis. The calculations indicate a significant decrease in the Ru-CH3 homolytic bond dissociation enthalpy for the oxidation of TpRu(CO)(NCMe)(Me) to its RuIII cation through both reactant destabilization and product stabilization. This oxidation can thus lead to the olefin polymerization observed by Gunnoe and coworkers, since weak RuIII-C bonds would afford quick access to alkyl radical species. Calculations support the experimental proposal of a mechanism for catalytic hydrogen/deuterium exchange by a RuII-OH catalyst. Furthermore, calculational investigations reveal a probable pathway for the activation of C-H bonds that involves phosphine loss, 1,2-addition to the Ru-OH bond and then reversal of these steps with deuterium to incorporate it into the substrate. The presented results offer the indication for the net addition of aromatic C-H bonds across a RuII-OH bond in a process that although thermodynamically unfavorable is kinetically accessible. Calculations support experimental proposals as to the possibility of binding of weakly coordinating ligands such as dinitrogen, methylene chloride and fluorobenzene to the "14-electron" complex [(PCP)Ru(CO)]+ in preference to the formation of agostic Ru-H-C interactions. Reactions of [(PCP)Ru(CO)(1-ClCH2Cl)][BAr'4] with N2CHPh or phenylacetylene yielded conversions that are exothermic to both terminal carbenes and vinylidenes, respectively, and then bridging isomers of these by C-C bond formation resulting from insertion into the Ru-Cipso bond of the phenyl ring of PCP. The QM/MM and DFT calculations on full complexes [(PCP)(CO)Ru=(C)0,1=CHPh]+ and on small models [(PCP')(CO)Ru=(C)0,1=CH2]+, respectively, offered data supportive of the thermodynamic feasibility of the suggested experimental mechanisms and their proposed intermediates.

Quantum chaos has emerged in the half of the last century with the notorious problem of scattering of heavy nuclei. Since then, theoreticians have developed powerful techniques to approach disordered quantum systems. In the late 70's, Casati and Chirikov initiated a new field of research by studying the quantum counterpart of classical problems that are known to exhibit chaos. Among the several quantum-classical chaotic systems studied, the kicked rotor stimulated a lot of enthusiasm in the scientific community due to its equivalence to the Anderson tight binding model. This equivalence allows one to map the random Anderson model into a set of fully deterministic equations, making the theoretical analysis of Anderson localization considerably simpler. In the one-dimensional linear regime, it is known that Anderson localization always prevents the diffusion of the momentum. On the other hand, for higher dimensions it was demonstrated that for certain conditions of the disorder parameter, Anderson localized modes can be inhibited, allowing then a phase transition from localized (insulating) to delocalized (metallic) states. In this thesis we will numerically and theoretically investigate the properties of a multidimensional quantum kicked rotor in a nonlinear medium. The presence of nonlinearity is particularly interesting as it raises the possibility of having soliton waves as eigenfunctions of the systems. We keep the generality of our approach by using an adjustable diffusive nonlinearity, which can describe several physical phenomena. By means of Variational Calculus we develop a chaotic map which fully describes the soliton dynamics. The analysis of such a map shows a rich physical scenario that evidences the wave-particle behavior of a soliton. Through the nonlinearity, we trace a correspondence between quantum and classical mechanics, which has no equivalent in linearized systems. Matter waves experiments provide an ideal environment for studying Anderson localization, as the interactions in these systems can be easily controlled by Feshbach resonance techniques. In the end of this thesis, we propose an experimental realization of the kicked rotor in a dipolar Bose Einstein Condensate.

Complex systems are ensembles of interconnected elements where mutual interaction and an optimized amount of disorder produce advanced functionalities. These systems are abundant in our daily experience: typical examples are the brain, biological ecosystems, society, and finance. In the last century, researchers have investigated the fundamental properties of disordered systems, unveiling fascinating and counterintuitive dynamics. The main aim of this Dissertation is the study of a new platform of disorder-enhanced photonics systems, denoted as Complex Metamaterials. Due to its ultrafast time scale nanophotonics represents an ideal framework to investigate and harness complex dynamics. Starting from the theoretical modeling of disordered plasmonic systems, I discuss advanced real-life applications, including the generation of highly-resistant structural colors from porous metal surfaces and the realization of early-stage cancer detectors based on surface roughness and self-similarity. In addition to the effects of structural disorder on plasmonic systems I also investigate the complex emission dynamics from non-conventional nanolasers. Lasers represent the quintessential example of a complex photonic system due to the simultaneous presence of strong nonlinearities and multi-mode interactions. At the same time, the integration of nanolasers with silicon-based electronic circuitry represents one of the greatest technological challenges in the field of nanophotonics. By combining ab-initio simulations and analytical modeling, I characterize the nonlinear emission from three-dimensional plasmonic nanolasers known as SPASERs. My results show for the first time the occurrence of a spontaneous rotational emission in spherical SPASERs, which originates from the nonlinear interaction of several lasing modes. I further discuss how rotating nanolasers can be employed as a fundamental building block for integrated quantum simulators, random information sources, and brain-inspired photonics platforms. Leveraging the practical limitations of SPASERs, I also propose a novel concept of near-field nanolaser based on invisible anapole modes. Anapoles constitute a peculiar state of electromagnetic radiation with no far-field emission and they have been recently discovered in dielectric nanoparticles. By integrating anapole lasers in a silicon-compatible platform, I discuss several advanced applications such as spontaneously polarized nanolasers and ultrafast pulse generators on-chip.

Neste trabalho, realizamos o estudo da transição metal-isolante e da localização de Anderson na liga de Pb1-xEuxTe do tipo p para x variando de 0 até 0.1. As propriedades de transporte nessa liga (mobilidade, concentração de portadores e resistividade elétrica) foram obtidas utilizando o método de caracterização elétrica por efeito Hall entre as temperaturas de 300 K e 10 K. Nessa região de temperatura, foi possível observar uma transição metal-isolante para x > 0.05. Verificamos que a transição é do tipo Anderson e ocorre devido à desordem presente na liga. Para baixas temperaturas (T < 10 K) e em amostras com x > 0.01, verificamos a presença de magnetorresistência positiva e negativa aplicando campos magnéticos de até 11T. Nas amostras metálicas, a presença de magnetorresistência negativa é causada pelo efeito conhecido como localização de Anderson (efeito de interferência quântica construtiva entre as funções de onda) e a presença de magnetorresistência positiva é causada, principalmente, pelo acoplamento spin-órbita, e é chamada de antilocalização. Nas amostras isolantes, a magnetorresistência negativa é originada pelo efeito Zeeman enquanto que a magnetoresistência positiva é causada pela redução do comprimento de localização. Assim, os valores positivos e negativos da magnetoresistência têm origens diferentes dependendo do regime de condução (metálicoou isolante). Por esse motivo, o estudo dos resultados experimentais apresentados nesse trabalho foi dividido em duas partes: uma parte que trata as amostras metálicas (região de desordem fraca) e outra parte para as amostras isolantes (região de desordem forte). A partir dessa divisão, e utilizando os modelos teóricos disponíveis na literatura, foi possível fazer uma análise das medidas experimentais de magnetotransporte. Como resultado, identificamos os principais mecanismos de interação (espalhamento inelástico, efeito Zeeman, acoplamento spin-órbita, etc.) que interferem no transporte e nos efeitos de localização e antilocalização. / In this work, we investigated Anderson localization and the metal-insulator transition in p-type films of Pb1-xEuxTe for x varying from 0 up to 0.1. The transport properties of this alloy (mobility, carrier concentration and electrical resistivity) were obtained using the Hall method of electrical caracterization for temperatures ranging from 300 K down to 10 K. In this temperature range, it was possible to observe a metal-insulator transition for x > 0:05. The transition is of the Anderson type and is due to the disorder present in the alloy. For low temperatures (T < 10 K) and for samples with x > 0.01, we observed positive and negative magnetoresistance for magnetic fields up to 11 T. For metallic samples, the negative magnetoresistance originates from Anderson’s localization (constructive quantum interference effect between the wave functions) while positive magnetoresistence is caused, mainly, by the spin-orbit scattering, and it is called antilocalization. For insulating samples, negative magnetoresistance is originated from the Zeeman effect while positive magnetoresistance is caused by the localization length reduction. Therefore, positive and negative magnetoresistance values have different origins depending on the conduction regime (metallic or insulating). For this reason, our experimental investigation, presented in this work, was separated into two parts: the first one treats the metallic samples (weak-disorder regime) and the other treats the insulating samples (strong-disorder regime). From this division, and using available theoretical models, it was possible to analyze the magnetotransport experimental measurements. As a result, we identify the main interaction mechanisms (inelastic scattering, Zeeman effect, spin-orbit coupling, etc.) that interfere on the transport and localization and antilocalization effects.

In this thesis, we investigate the coherent scattering of light propagating in a random medium. We are interested in phenomena like the super- and subradiance and Anderson localization that are related to waves interferences and spatial disorder. However, the fundamental difference between subradiance and Anderson localization still needs to be clarified. This thesis gives new elements for the understanding of these phenomena and we present a new method to observe Anderson localization. A mean-field model that does not contain disorder is developed, and we show that super- and subradiance do not require disorder whereas Anderson localization does. In this theoretical work, the coupling between the light and many atoms is reduced to a coupling matrix between the atoms by tracing over the degrees of freedom of the light, which results in a linear problem for the atomic dipoles. The study of the eigenvalues and eigenmodes of this matrix then allows to determine the super- and subradiant modes, and to probe the Anderson localization phase transition with a scaling analysis. Furthermore, the link to the experiment is realized by showing that the intensity fluctuations present an increase at the localization transition. The system is studied in the steady-state regime when the medium is continuously charged by a laser until reaches a stationary regime, and the decay dynamics, when the laser is switched off, so the cloud releases the energy stored. Finally, we present a preliminary work that shows that the diagonal disorder might be a good strategy to reach Anderson localization. / Nesta tese, investigamos o espalhamento coerente de luz propagando em um meio aleatório. Estamos interessados em fenômenos como superradiância, subradiância e localização de Anderson, os quais estão relacionados com interferências de ondas e desordem espacial. No entanto, as diferenças fundamentais entre subradiância e localização de Anderson ainda precisam ser esclarecidas. Esta tese traz novos elementos na compreensão destes fenômenos e apresentamos um novo método para observar a localização de Anderson. Neste trabalho teórico, estudamos os autovalores e os automodos de uma matriz de acoplamento que permite extrair modos super- e subradiantes, e exibem uma transição de fase de localização de Anderson através de uma análise de escalamento. Além disso, a conexão com o experimento é feita através da intensidade irradiada pela nuvem em todas as direções. Distinguimos dois casos: o regime de estado estacionário, quando o meio é continuamente excitado por um laser e alcança um regime estacionário; e o caso dinâmico, onde o laser é desligado e a nuvem libera a energia armazenada. Desenvolvemos um modelo de campo médio que não inclui desordem, e mostramos que super- e subradiância não precisam da desordem para existir, ao contrário da localização de Anderson. Mostramos também que podemos observar uma transição de fase de localização de Anderson na estatística da intensidade. Finalmente, apresentamos um trabalho preliminar que mostra que a desordem diagonal pode ser uma boa estratégia para alcançar a localização de Anderson.

Orientador: Antonio Vidiella Barranco / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-09-02T14:05:56Z (GMT). No. of bitstreams: 1 Roque_ThalesFigueiredo_D.pdf: 3246905 bytes, checksum: 13e2147eea29b73345cec6df9138b5a1 (MD5) Previous issue date: 2017 / Resumo: Os sistemas optomecânicos são sistemas físicos onde a radiação interage com graus de liberdade mecânicos por meio da pressão de radiação. Estes sistemas destacam-se pela flexibilidade e pelo alto grau de controle sobre a interação optomecânica. Nesta tese, nós apresentamos um estudo teórico acerca dos sistemas optomecânicos, o qual pode ser dividido em duas partes: a primeira parte se dedica ao uso de sistemas optomecânicos na geração de estados não clássicos, e a segunda parte se dedica ao estudo da localização de Anderson em redes optomecânicas. Na primeira parte, nós estudamos, primeiramente, a geração de estados da radiação com estatística sub-Poissoniana no regime quântico não linear. Este tópico já havia sido estudado anteriormente em um sistema convencional composto por uma cavidade ótica acoplada a um oscilador mecânico. Neste trabalho, consideramos um sistema onde duas cavidades óticas interagem com um único oscilador mecânico. Nós mostramos que o sistema estudado permite a geração de estados com estatística sub-Poissoniana significativamente mais acentuada do que o sistema convencional. Além disso, os estados gerados são mais robustos com respeito ao ruído térmico do ambiente. Em seguida, estudamos a geração de estados estacionários comprimidos do oscilador mecânico em sistemas optomecânicos quadráticos operando no regime quântico linear. Nós mostramos que, bombeando a cavidade ótica com \textit{lasers} com frequências e amplitudes específicas, é possível gerar estados estacionários comprimidos do oscilador mecânico. Na segunda parte, nós estudamos a localização de Anderson em redes optomecânicas desordenadas. Redes optomecânicas são arranjos espacialmente ordenados de inúmeros modos óticos e mecânicos que interagem entre si por meio do acoplamento optomecânico. Em um cenário realista, devido a imprecisões na fabricação deste sistema, os parâmetros seriam desordenados. Nossos resultados mostram que os autoestados deste sistema são exponencialmente localizados. Além disso, observamos a existência de dois regimes de operação em redes optomecânicas desordenadas: o regime de acoplamento fraco e o regime de acoplamento forte. A transição entre estes dois regimes apresenta características não triviais que poderiam ser utilizadas para detectar experimentalmente a localização nestas estruturas. Nós estudamos também a dinâmica clássica de redes optomecânicas desordenadas no regime instável. Este é um tópico bastante desafiador, uma vez que a dinâmica é essencialmente não linear neste regime. Nós mostramos que, em um regime de parâmetros específico, é possível utilizar a aproximação linear para tempos curtos, e os resultados desta aproximação fornecem importantes informações sobre a dinâmica não linear. Nós ainda analisamos brevemente a emergência de comportamento caótico neste regime / Abstract: Optomechanical systems are systems in which radiation interacts with mechanical degrees of freedom via radiation pressure. These systems are well known for allowing great flexibility and high control over the optomechanical interaction. In this thesis, we present a theoretical investigation about optomechanical systems. This investigation can be divided in two parts: the first part is devoted to the generation of nonclassical states in optomechanical systems, and the second part is devoted to the study of Anderson localization in optomechanical arrays. In the first part, we study, firstly, the generation of optical sub-Poissonian states in the quantum nonlinear regime. This topic has been previously investigated in a conventional optomechanical system with one optical cavity coupled to one mechanical oscillator. Here, we investigate a system with two optical cavities coupled to one mechanical oscillator. We show that our system allows the generation of stronger sub-Poissonian states in comparison with the conventional system. In addition, the states generated in our system are more robust against thermal noise. Next, we investigate the generation of squeezed steady states of the mechanical oscillator in a quadratic optomechanical system operating in the quantum linear regime. We show that, if the optical cavity is pumped by lasers with specific frequencies and amplitudes, it is possible to generate such states. In the second part, we investigate Anderson localization in disordered optomechanical arrays. Optomechanical arrays are periodic arrays of optical and mechanical modes, which interact with each other via optomechanical coupling. In a realistic scenario, due to imprecisions in the fabrication of such a structure, the parameters of the system will be disordered. We show that the eigenstates in this system are exponentially localized. Furthermore, we show the existence of two regimes in disordered optomechanical arrays: the weak coupling regime and the strong coupling regime. The transition between these regimes displays nontrivial features that could be used to detect localization experimentally. We study also the classical dynamics of disordered optomechanical arrays in the unstable regime. This is a very challenging topic, since the unstable regime is essentially nonlinear. We show that, for a specific regime of parameters, it is possible to use the linear approximation for small times, and the linear results give us important informations about the nonlinear dynamics. We analyze briefly the emergence of chaotic behavior in the regime / Doutorado / Física / Doutor em Ciências / 2012/10476-0 / FAPESP

Neste trabalho, realizamos o estudo da transição metal-isolante e da localização de Anderson na liga de Pb1-xEuxTe do tipo p para x variando de 0 até 0.1. As propriedades de transporte nessa liga (mobilidade, concentração de portadores e resistividade elétrica) foram obtidas utilizando o método de caracterização elétrica por efeito Hall entre as temperaturas de 300 K e 10 K. Nessa região de temperatura, foi possível observar uma transição metal-isolante para x > 0.05. Verificamos que a transição é do tipo Anderson e ocorre devido à desordem presente na liga. Para baixas temperaturas (T < 10 K) e em amostras com x > 0.01, verificamos a presença de magnetorresistência positiva e negativa aplicando campos magnéticos de até 11T. Nas amostras metálicas, a presença de magnetorresistência negativa é causada pelo efeito conhecido como localização de Anderson (efeito de interferência quântica construtiva entre as funções de onda) e a presença de magnetorresistência positiva é causada, principalmente, pelo acoplamento spin-órbita, e é chamada de antilocalização. Nas amostras isolantes, a magnetorresistência negativa é originada pelo efeito Zeeman enquanto que a magnetoresistência positiva é causada pela redução do comprimento de localização. Assim, os valores positivos e negativos da magnetoresistência têm origens diferentes dependendo do regime de condução (metálicoou isolante). Por esse motivo, o estudo dos resultados experimentais apresentados nesse trabalho foi dividido em duas partes: uma parte que trata as amostras metálicas (região de desordem fraca) e outra parte para as amostras isolantes (região de desordem forte). A partir dessa divisão, e utilizando os modelos teóricos disponíveis na literatura, foi possível fazer uma análise das medidas experimentais de magnetotransporte. Como resultado, identificamos os principais mecanismos de interação (espalhamento inelástico, efeito Zeeman, acoplamento spin-órbita, etc.) que interferem no transporte e nos efeitos de localização e antilocalização. / In this work, we investigated Anderson localization and the metal-insulator transition in p-type films of Pb1-xEuxTe for x varying from 0 up to 0.1. The transport properties of this alloy (mobility, carrier concentration and electrical resistivity) were obtained using the Hall method of electrical caracterization for temperatures ranging from 300 K down to 10 K. In this temperature range, it was possible to observe a metal-insulator transition for x > 0:05. The transition is of the Anderson type and is due to the disorder present in the alloy. For low temperatures (T < 10 K) and for samples with x > 0.01, we observed positive and negative magnetoresistance for magnetic fields up to 11 T. For metallic samples, the negative magnetoresistance originates from Anderson’s localization (constructive quantum interference effect between the wave functions) while positive magnetoresistence is caused, mainly, by the spin-orbit scattering, and it is called antilocalization. For insulating samples, negative magnetoresistance is originated from the Zeeman effect while positive magnetoresistance is caused by the localization length reduction. Therefore, positive and negative magnetoresistance values have different origins depending on the conduction regime (metallic or insulating). For this reason, our experimental investigation, presented in this work, was separated into two parts: the first one treats the metallic samples (weak-disorder regime) and the other treats the insulating samples (strong-disorder regime). From this division, and using available theoretical models, it was possible to analyze the magnetotransport experimental measurements. As a result, we identify the main interaction mechanisms (inelastic scattering, Zeeman effect, spin-orbit coupling, etc.) that interfere on the transport and localization and antilocalization effects.

The propagation of ultrasound through two- and three-dimensional strongly scattering media, with either random or ordered internal structures, has been investigated through experiments and finite element simulations. All media investigated have strong scattering resonances, leading to novel transport behaviour. The two-dimensional samples consist of nylon rods immersed in water. When the nylon rods are arranged in a triangular lattice to form two-dimensional phononic crystals, very unusual dispersion properties are observed when the lattice constant is adjusted so that Bragg and hybridization gaps overlap in frequency. This behaviour is attributed to the competition between two co-existing propagating modes, leading to a new method for tuning bandgap properties and adjusting the transmission by orders of magnitude. The scattering resonance of the nylon rods also leads to unusual Dirac cone properties at the K point of the triangular lattice. The three-dimensional media were fabricated by brazing aluminum beads together to form a disordered porous solid network, with either vacuum or air in the pores, depending on the experiment. This system is of particular interest because it has been shown to exhibit Anderson localization of ultrasound. Two experimental approaches were developed to investigate previously unstudied properties of this system. By directly counting the modes in the frequency domain, the density of states was measured. At intermediate frequencies, the density of states was found to be approximately independent of frequency, while at higher frequencies, the frequency dependence was consistent with traditional density-of-states models. The level statistics of the modes was also investigated to determine the conditions under which level repulsion occurs. By using a laser interferometer to measure the ultrasonic displacements on the surface of a large slab-shaped sample, sub-diffusive behaviour was observed, demonstrating the feasibility of using such measurements to investigate the transition to Anderson localization in these samples.

The work in this thesis is motivated by the problem of localization of interacting particles. The qualitative investigation of Thouless-type arguments in Chap.~\ref{ch:interacting-particles} lead us to consider the question of competition between alternative propagation channels, a question which we studied in great detail in the form of a single particle problem with two parallel, coupled channels. The theory also naturally applies for the Anderson localization of hybrid particles such as polaritons. These systems have a common feature: Two or more propagating channels with parametrically different transport properties are coupled and compete with each other. The principal question is: What happens to the localization properties when a less localized lattice is coupled to a more localized one? Will the less localized lattice dominate the localization of the system or the more localized? The qualitative answer to this question depends on the dimensionality of the system. Correspondingly, we exactly solved the Anderson models on a two-leg ladder ($D=1$) and on a two-layer Bethe lattice (formally $D=\infty$). In one dimension, weak disorder has a strong localization effect. In the \emph{weak disorder limit} we have found that under \emph{resonance} conditions the localization lengths of two coupled chains are of the order of the localization length of the more localized, uncoupled leg. We may interpret this phenomenon as a manifestation of the fact that in one dimension the mean free path is the relevant length scale that sets the localization length. It is not surprising that the backscattering rate, and thus the ''worst'' leg of the chains determines the localization properties of a coupled system. If away from resonance the two legs are hardly affected by each other. However, the close relation (proportionality) between mean free path and localization length is special for one-dimensional systems. On coupled Bethe lattices, weak disorder is irrelevant to localization. The localization effect is significant only if the disorder is intermediate or strong. Therefore, resonance conditions, which require weak disorder as compared to the hopping, can not be achieved. In general, we found that the less disordered lattice is not affected much by the more disordered lattice in the presence of coupling, except in the case where the less disordered (delocalized) lattice is very close to the transition and the more disordered lattice is strongly localized, in which case the more disordered lattice can push the less disordered lattice to a localized phase. We believe that these trends persist in high dimensions ($D>2$) where the metal-insulator transition takes places at strong disorder. In two dimensions, the localization length becomes parametrically larger than the mean free path at weak disorder. However, since the proliferation of weak-localization and backscattering leads to complete localization (in the absence of special symmetries), we expect that a well propagating channel becomes more strongly localized upon resonant coupling to a more disordered channel, similarly as in one dimension. It might be interesting to investigate this numerically. Investigating the localization properties of few- or many-particle systems is more complicated. First, we should map an interacting Hamiltonian to the Anderson model in the few- many-particle Fock space [cf. Eq.~\ref{ipfham}]. Thereby, the interaction provides effective hopping among the Fock states. This hopping in Fock space can be organized into channels with rather different propagation characteristics [e.g. Fig.~\ref{four-par} for four particles], namely, faster channels and slower channels. According to our analysis, the slow channel dominates only if it is \emph{resonantly} coupled to the fast channel. If the two channels are away from resonance, the fast channel essentially dominates the localization properties. For the few-particle problems discussed in Chap.~\ref{ch:interacting-particles} we expect that the fast channel, that is, the hierarchical structure we predicted, dominates the delocalization of the interacting particles, since the resonance between the fast and slow channels should be an exception rather than a rule. At this stage, this remains a conjecture which needs to be tested further.