archives@tulane.edu / The fundamental problem of thermalization in quantum systems with long-range interactions is a target of the present study. This problem is relevant for the vast number of phenomena ranging from thermal conductivity of materials to error propagation in quantum computers. Two types of quantum systems are studied analytically in this work with a support from numerical simulations. Spin chains with power-law interactions are chosen as an example system that represents behavior of qubits in a quantum computer while the vibrational problem with non-linear interactions is a toy model of a polymer molecule with anharmonic bonding. The analytical results developed for both models within the framework of resonant counting method allow one to predict the integrability-chaos transitions for the future experimental verification. / 1 / Andrii Makysmov

This thesis presents a series of studies of thermalization and dissipation in a variety of strongly coupled systems. The main tool for these investigations is the Gauge/Gravity duality, which establishes a correspondence between a d+1-dimensional quantum theory of gravity and a d-dimensional quantum field theory. We study the decay rates of fluctuations around the thermal equilibrium in theories in non-commutative geometry. Rapid thermalization of such fluctuations is found and motivates the conjecture that the phenomena at the black hole horizon is described by non-local physics. In the same type of environment, we analyze the Langevin dynamics of a heavy quark, which undergoes Brownian motion. We find that the late-time behavior of the displacement squared is unaffected by the non-commutativity of the geometry. In a different scenario, we study the correlation functions in theories with quantum critical points. We compute the response of these quantum critical points to a disturbance caused by a massive charged particle and analyze its late time behavior. Finally, we analyze systems far-from-equilibrium as they evolve towards a thermal state. We characterize this evolution for systems with chemical potential by focusing on the ``strong subadditivity" property of their entanglement entropy. This is achieved on the gravity side by using time dependent functions for mass and charge in an AdS-Vaydia metric. / text

In a thermally equilibrated system, electron behavior in a metal is described by the Fermi-Dirac equation. With ultrafast lasers, electrons can be excited into temporary distributions which are not described by the Fermi-Dirac equation and are therefore not at a well-defined temperature. These nonthermal distributions quickly equilibrate through two primary processes: electron-electron scattering and electron-phonon scattering. In most situations, these effects are unnoticeable, since they are completed within 5 ps. A probabilistic numerical model for electron-electron scattering is presented. The model is robust, scaleable, and requires only one parameter. The success of the model suggests future work on a similar electron-phonon scattering model, which would provide a complete description of the elctron distribution during thermalization. Once complete, this model can be tested by measuring the amount of second harmonic light generated by an ultrafast laser in a pump-probe experiment.

This work investigates diffusion in nonlinear Hamiltonian systems. The diffusion, more precisely subdiffusion, in such systems is induced by the intrinsic chaotic behavior of trajectories and thus is called chaotic diffusion''. Its properties are studied on the example of one- or two-dimensional lattices of harmonic or nonlinear oscillators with nearest neighbor couplings. The fundamental observation is the spreading of energy for localized initial conditions. Methods of quantifying this spreading behavior are presented, including a new quantity called excitation time. This new quantity allows for a more precise analysis of the spreading than traditional methods. Furthermore, the nonlinear diffusion equation is introduced as a phenomenologic description of the spreading process and a number of predictions on the density dependence of the spreading are drawn from this equation. Two mathematical techniques for analyzing nonlinear Hamiltonian systems are introduced. The first one is based on a scaling analysis of the Hamiltonian equations and the results are related to similar scaling properties of the NDE. From this relation, exact spreading predictions are deduced. Secondly, the microscopic dynamics at the edge of spreading states are thoroughly analyzed, which again suggests a scaling behavior that can be related to the NDE. Such a microscopic treatment of chaotically spreading states in nonlinear Hamiltonian systems has not been done before and the results present a new technique of connecting microscopic dynamics with macroscopic descriptions like the nonlinear diffusion equation. All theoretical results are supported by heavy numerical simulations, partly obtained on one of Europe's fastest supercomputers located in Bologna, Italy. In the end, the highly interesting case of harmonic oscillators with random frequencies and nonlinear coupling is studied, which resembles to some extent the famous Discrete Anderson Nonlinear Schroedinger Equation. For this model, a deviation from the widely believed power-law spreading is observed in numerical experiments. Some ideas on a theoretical explanation for this deviation are presented, but a conclusive theory could not be found due to the complicated phase space structure in this case. Nevertheless, it is hoped that the techniques and results presented in this work will help to eventually understand this controversely discussed case as well. / Diese Arbeit beschäftigt sich mit dem Phänomen der Diffusion in nichtlinearen Systemen. Unter Diffusion versteht man normalerweise die zufallsmäss ige Bewegung von Partikeln durch den stochastischen Einfluss einer thermodynamisch beschreibbaren Umgebung. Dieser Prozess ist mathematisch beschrieben durch die Diffusionsgleichung. In dieser Arbeit werden jedoch abgeschlossene Systeme ohne Einfluss der Umgebung betrachtet. Dennoch wird eine Art von Diffusion, üblicherweise bezeichnet als Subdiffusion, beobachtet. Die Ursache dafür liegt im chaotischen Verhalten des Systems. Vereinfacht gesagt, erzeugt das Chaos eine intrinsische Pseudo-Zufälligkeit, die zu einem gewissen Grad mit dem Einfluss einer thermodynamischen Umgebung vergleichbar ist und somit auch diffusives Verhalten provoziert. Zur quantitativen Beschreibung dieses subdiffusiven Prozesses wird eine Verallgemeinerung der Diffusionsgleichung herangezogen, die Nichtlineare Diffusionsgleichung. Desweiteren wird die mikroskopische Dynamik des Systems mit analytischen Methoden untersucht, und Schlussfolgerungen für den makroskopischen Diffusionsprozess abgeleitet. Die Technik der Verbindung von mikroskopischer Dynamik und makroskopischen Beobachtungen, die in dieser Arbeit entwickelt wird und detailliert beschrieben ist, führt zu einem tieferen Verständnis von hochdimensionalen chaotischen Systemen. Die mit mathematischen Mitteln abgeleiteten Ergebnisse sind darüber hinaus durch ausführliche Simulationen verifiziert, welche teilweise auf einem der leistungsfähigsten Supercomputer Europas durchgeführt wurden, dem sp6 in Bologna, Italien. Desweiteren können die in dieser Arbeit vorgestellten Erkenntnisse und Techniken mit Sicherheit auch in anderen Fällen bei der Untersuchung chaotischer Systeme Anwendung finden.

The project concerns the study of the interplay among quantum mechanics, statistical mechanics and thermodynamics, in isolated quantum systems. The goal of this research is to improve our understanding of the concept of thermal equilibrium in quantum systems. First, I investigated the role played by observables and measurements in the emergence of thermal behaviour. This led to a new notion of thermal equilibrium which is specific for a given observable, rather than for the whole state of the system. The equilibrium picture that emerges is a generalization of statistical mechanics in which we are not interested in the state of the system but only in the outcome of the measurement process. I investigated how this picture relates to one of the most promising approaches for the emergence of thermal behaviour in quantum systems: the Eigenstate Thermalization Hypothesis. Then, I applied the results to study the equilibrium properties of peculiar quantum systems, which are known to escape thermalization: the many-body localised systems. Despite the localization phenomenon, which prevents thermalization of subsystems, I was able to show that we can still use the predictions of statistical mechanics to describe the equilibrium of some observables. Moreover, the intuition developed in the process led me to propose an experimentally accessible way to unravel the interacting nature of many-body localised systems. Then, I exploited the "Concentration of Measure" and the related "Typicality Arguments" to study the macroscopic properties of the basis states in a tentative theory of quantum gravity: Loop Quantum Gravity. These techniques were previously used to explain why the thermal behaviour in quantum systems is such an ubiquitous phenomenon at the macroscopic scale. I focused on the local properties, their thermodynamic behaviour and interplay with the semiclassical limit. The ultimate goal of this line of research is to give a quantum description of a black hole which is consistent with the expected semiclassical behaviour. This was motivated by the necessity to understand, from a quantum gravity perspective, how and why an horizon exhibits thermal properties.

La polarisation dynamique nucléaire (DNP pour son acronyme en anglais) est une des techniques les plus prometteuses d’amélioration de l’IRM. En pratique, on voudrait utiliser la résonance magnétique nucléaire (RMN) sur d’autres noyaux que ceux d’hydrogène, par exemple le carbone. Pour pouvoir détecter le carbone, sa polarisation de spin doit être augmentée. À l’équilibre thermodynamique — à basse température et forts champs magnétiques — les électrons sont bien plus polarisés que tout système de spin nucléaires, ce qui est dû à leur plus petite masse. La technique de DNP consiste à amener le système hors d’équilibre avec une irradiation par des microondes. Cette irradiation va induire le transfer de polarisation des spins électroniques vers les spins nucléaires. Pendant ma thèse, j’ai étudié, par des méthodes analytiques et numériques, la compétition entre les interactions dipolaires présentes entre les spins électroniques (qui peuvent se régler expérimentalement) et le désordre naturellement présent dans l’échantillon. Pour ce faire, j’ai proposé deux modèles : une chaîne de spins d’Heisenberg et un système de fermions libres dans le modèle d’Anderson. J’ai trouvé l’existence de deux régimes : Pour le régime de fortes interactions, l’état stationnaire a des traces d’un comportement thermodynamique, étant caractérisé par une température effective. Dans le régime de faibles interactions, il n’est pas possible de définir une température effective, et l'on peut le relier à une phase de many-body localization (ou localisation d'Anderson). Mes recherches portent sur l’étude des propriétés deux phases en relation avec la performance de la DNP et j’ai trouvé qu’elle est optimale à la transition entre les deux phases. Ce résultat intéressant a récemment été confirmé par des expériences menées à l’École Normale Supérieure de Paris. / Dynamic nuclear polarization (DNP) is one of the most promising techniques towards a new generation of Magnetic Resonance Imaging (MRI). The idea is to use the Nuclear Magnetic Resonance (NMR) in other nuclei rather than the traditional hydrogen, such as carbon. For the carbon signal to be detected, one needs to enhance its spin polarization. In thermal equilibrium — at low temperature and high magnetic field — electron spins are far more polarized than any system of nuclear spins, which is due to their smaller mass. With the DNP technique we bring the system out-of-equilibrium irradiating it with microwaves. This triggers polarization transfer from the electron spins to the nuclear ones. During my Ph.D, I have studied both analytically and numerically the competition between the dipolar interactions among electron spins (which can be tuned experimentally) and the disorder naturally present in the sample. I proposed two models to study DNP: a Heisenberg spin-chain and a system free-fermions in the Anderson model. Two different regimes were found : For strongly interacting electron spins, the out-of-equilibrium steady state displays an effective thermodynamic behavior characterised by a very low spin temperature. In the weakly interacting regime, it is not possible to define a spin temperature, and it is associated to a many-body localized phase (or an Anderson-localized phase). My research was focused on the properties of the two phases with respect to the performance of DNP, and I found it to be optimal at the transition between the two. This is a very important result that has been verified by recent experiments carried in École Normale Supérieure de Paris.

No description available.

Les théories de champ moyen quantique représentent une base robuste pour la description de la dynamique de nombreux systèmes physiques, des noyaux aux systèmes moléculaires et aux agrégats. Cependant, le traitement incomplet des corrélations électroniques au niveau du champ moyen empêche de donner une description propre de la dynamique, en particulier la dynamique dans les régimes dissipatifs. La dissipation est intrinsèquement liée à la thermalisation qui représente le phénomène cible à d'écrire dans ce travail. Nous avons exploré un schéma purement quantique en terme des matrices densités et qui consiste en l'inclusion des corrélations de type collisions, responsables de la thermalisation dans les systèmes quantiques finis. Ceci est fait en traitant les corrélations entre deux particules avec la théorie des perturbations dépendantes du temps tout au long d'un intervalle de temps. Ceci permet de créer un ensemble d'états de type champ moyen pur pour les différentes configurations. Ces états sont traités stochastiquement dans la dynamique et fournissent en moyenne un état corrélé. Nous proposons dans ce travail une reformulation de cette théorie en terme des fonctions d'ondes où les corrélations sont traitées comme des transitions multiples de type particule-trou, limitées aux transitions deux-particules-deux-trous dans notre cas. On applique le schéma obtenu à un modèle unidimensionnel simulant de petites molécules. La capacité de cette théorie à introduire les effets dissipatifs dans le cadre du champ moyen est illustrée à travers plusieurs observables tels que les matrices à un et deux corps, les nombres d'occupation et l'entropie à un corps / Quantum mean field theories represent a robust basis for the description of many dynamical situations from nuclei to molecular systems and clusters. However, the missing of electronic correlations on top of mean field prevents them to give a proper description of the dynamics, in particular dissipative dynamics. Dissipation is intrinsically linked to thermalization which represents the target phenomenon to be described in this thesis. We thus explore a fully quantum mechanical strategy proposed in terms of density matrices in the case of nuclear collisions and which consists in the inclusion of collisional correlations responsible of thermalization in quantum finite systems. This is done by treating two body correlations in time dependent perturbation theory along a certain time span that allows to create an ensemble of pure mean field states for different configurations. These states are used into the dynamics, stochastically, providing in the average one correlated state. We propose in this work a reformulation of this theory in term of wave functions where correlations are translated into multiple particle-hole transitions, restricted to two-particles-two-holes transitions in our case. We apply the obtained scheme to a one dimensional model simulating small molecules. The ability of this theory to include dissipative effects on top of mean field is illustrated through several observables such as the one and two body density matrices, the occupation numbers and the one body entropy.

Le comportement collectif des atomes de Rydberg est au cœur de nombreux protocoles d'information quantique, notamment de répéteurs quantiques. Cette thèse traite de deux sujets distincts: la dynamique collective de nuages d'atomes de Rydberg et l'utilisation de répéteurs quantiques dans des réseaux complexes. Dans la première partie, nous étudions un système simple composé d'une chaîne 1D d'atomes de Rydberg couplée à un laser résonnant sur la transition vers un niveau de Rydberg dans le régime contenant quelques excitations. Les atomes de Rydberg sont soumis à une forte interaction dipolaire qui tend à empêcher l'excitation simultanée de deux atomes proches l'un de l'autre. C'est ce phénomène de blocage de Rydberg qui fait des atomes de Rydberg d'éminents candidats pour des protocoles d'information quantique. Ce blocage induit une distribution spatiale particulière des excitations le long de la chaîne d'atomes. Le calcul exact de cette distribution est souvent impossible en pratique même numériquement, et des approximations sont a priori nécessaires:- l'approximation des sphères de Rydberg dures: l'interaction dipolaire est modélisée par une sphère centrée autour de chaque excitation, à l'intérieur de laquelle toute autre excitation est impossible;- l'hypothèse de thermalisation: le système est supposé thermaliser, c'est-à-dire qu'après suffisamment de temps, même sans effets dissipatifs, le système tendra vers un état quasi-thermique qui peut être décrit par la physique statistique et plus précisément l'ensemble microcanonique. Cette thèse présente une étude de la thermalisation d'un ensemble 1D d'atomes de Rydberg et, plus particulièrement, de l'acuité des prédictions de l'ensemble microcanonique en supposant l'hypothèse des sphères dures. Nous avons simulé numériquement la dynamique d'un tel système composé de 100 atomes, dans le régime contenant au plus deux excitations dans l'ensemble. De plus, un modèle analytique à 6 dimensions est présenté. Comparant les trois approches, nous montrons que le modèle analytique corrobore la simulation numérique, tandis que simulation et modèle mis ensemble contredisent les prédictions microcanoniques. Dans ce régime, l'utilisation de cet ensemble est donc inadaptée. La seconde partie de cette thèse porte sur la distribution d'intrication dans un réseau de répéteurs quantiques. Ces derniers devraient permettre la communication quantique de deux parties distantes. Ces répéteurs quantiques sont presque toujours connectés en un réseau linéaire. Dans cette thèse, nous explorons les possibilités offertes par des réseaux arbitraires constitués de ces répéteurs connectant une multitude de clients. Nous avons représenté ces réseaux à l'aide de graphes non orientés. Nous avons étudié deux scénarios de routage:- le routage classique d'intrication qui corresponds au cas où des clients, très limités par leurs dispositifs quantiques, souhaitent partager des paires intriqués. Sur ces réseaux, les problèmes de communication sont équivalents à des problèmes de chemins disjoints. Lorsque les clients souhaitant communiquer ensemble (les terminaux) sont choisis par un adversaire, nous avons obtenu deux bornes: l'une proportionnelle au genre topologique, et l'autre au degré minimal du graphe. Nous proposons deux architectures de réseau saturant la plus contraignante, celle due au degré minimal. D'autre part, lorsque les clients sont répartis dans un espace à 2-3 dimensions, nous avons montré une limitation géométrique sur la fraction de clients pouvant communiquer simultanément.- le routage quantique utilisant le codage de réseau, qui correspond au cas où le réseau quantique est composé de petits processeurs quantiques capable d'effectuer des opérations locales. Nous avons étudié un problème de communication, le réseau papillon, où le routage classique de l'intrication entre deux paires de clients est impossible. Grâce au codage de réseau, nous avons résolu ce problème de communication. / The collective behavior of Rydberg gases is at the heart of many proposals for quantum information. This thesis treats two distinct topics: the collective dynamic of a Rydberg ensemble and the use of quantum repeaters across quantum networks.In the first part of this thesis, we choose to focus on a simple system involving Rydberg atoms: a 1-dimensional Rydberg gas coupled to a laser resonant with the Rydberg transition. Rydberg atoms interact together through the dipole-dipole interaction. This particular feature is used for quantum information purposes, like applying multi-qubits gates for example. This interaction is strong enough so that the dynamic of such system in the regime of few excitations in the gas ensemble is already intractable without any assumptions. One of them is the hardcore Rydberg sphere assumption: we approximate this interaction by a sphere around each excitation inhibiting any second excitation within it. Another one is to suppose that the system thermalizes in such regime; a statistical treatment could then be applied. We have investigated the thermalization of a 1D-Rydberg gas and evaluated the accuracy of the microcanonical ensemble predictions under the first assumption. To do so, we have numerically simulated the dynamic of such system constituted by 100 atoms, in the regime of at most two excitations in the chain, in the initial excitation-less state. Furthermore, we constructed a 6-dimensional analytical model. Comparing the three approaches together, we have concluded that the numerical simulation and the analytical model both agree together but contradicts the microcanonical treatment. In this regime, the microcanonical ensemble is unadapted.In the second part of this thesis, we have studied the distribution of entanglement across a generic quantum network. We have mapped these quantum networks to undirected graphs and studied two different routing scenarios:- the classical routing of quantum entanglement corresponding to the scenario where clients of the network can perform only a single Bell measurement or keep a single qubit. This is the usual model of quantum repeaters. On these networks, peer-to-peer communication problems are equivalent to the vertex disjoint path problem. When the peers are chosen by an adversary, we have found two limitations due to the topological genus and the minimum degree of the graph. We have found two network architectures (almost) saturating the most constraining one, the minimum degree inequality. For the case where the peers are chosen at random, we have studied a specific graph lying in a 2- or 3-dimensional manifold and investigated the trade-off between the quantum links and the number of peers that can communicate simultaneously through the network.- true quantum routing problem (using network coding) corresponding to the situation where the quantum network is composed by small quantum processors that could apply local gates. We focus on a particular communication problem, namely the butterfly network, where classical routing is impossible. Using network coding, this communication is solved.

Based on the works [R1] - [R10], this thesis tackles various aspects of the dynamics of interacting quantum many-body systems. Particular emphasis is given to the understanding of transport and thermalization phenomena in isolated (quasi) one-dimensional quantum spin models. Employing a variety of methods, these phenomena are studied both, close to equilibrium where linear response theory (LRT) is valid, as well as in far-from-equilibrium situations where LRT is supposed to break down. The main results of this thesis can be summarized as follows. First, it is shown that conventional hydrodynamic transport, i.e., diffusion, occurs in a number of (integrable and nonintegrable) quantum models and can be detected by looking at different signatures in position and momentum space as well as in the time and the frequency domain. Furthermore, the out-of-equilibrium dynamics resulting from a realistic class of initial states is explored. These states are thermal states of the model in the presence of an additional static force, but become nonequilibrium states when this force is eventually removed. Remarkably, it is shown that in some cases, the full time-dependent relaxation process can become independent of whether the initial state is prepared close to or far away from equilibrium. In this context, a new connection between the eigenstate thermalization hypothesis and linear response theory is unveiled. Finally, this thesis also reports progress on the development and improvement of numerical and (semi-)analytical techniques to access the dynamics of quantum many-body systems. Specifically, a novel combination of dynamical quantum typicality and numerical linked cluster expansions is employed to study current-current correlation functions in chain and ladder geometries in the thermodynamic limit.