Rozprawy doktorskie na temat „Closed Interacting Quantum Systems”

Dans cette thèse nous étudions au niveau théorique le comportement des particules quantiques (électrons, atomes, photons, etc.) se mouvant dans un milieu désordonné et sujets à la localisation d’Anderson. Pour des particules non interagissantes, le spectre de l’énergie peut posséder un ou plus points critiques, où les fonctions d’onde étendues deviennent localisées, en donnant lieu à une transition de phase métal-isolant connue comme Transition d’Anderson.Une question fondamentale est si et comment les transitions d’Anderson survivent dans des systèmesquantiques interagissants. Dans cet ouvrage, nous étudions un modèle simple décrivant le cas de deux particules dans un réseau désordonné et sujettes à des interactions mutuelles à courte portée. En combinant des simulations numériques sur une grande échelle avec des techniques à la fonction de Green, nous montrons que les transitions d’Anderson à deux particules se produisent en trois dimensions et explorons le diagramme de phase dans l’espace de l’énergie, du désordre et de l’interaction.Cette dernière présente une structure riche, caractérisée par un double renfoncement de la limite de phase, engendrée par la compétition entre les états de diffusion et les états liés de la paire. Nous prouvons aussi que les annonces précédentes concernant l’apparition de transitions d’Anderson en deux dimensions étaient essentiellement dues à des effets de taille finie.Un deuxième problème que nous abordons dans cette thèse est celui de l’occurrence de transitions métal-isolant en deux dimensions pour une particule en la présence d’un potentiel spatialement corrélé et sujette à des interactions spin-orbite, modélisées par les couplages Rashba-Dresselhaus. On éclaire que, indépendamment des propriétés du désordre, il y a un régime où l’énergie critique dépend linéairement du paramètre de désordre. La pente et l’intercepte sont étudiées en voisinage du point de symétrie spin-hélice persistant, dans lequel la symétrie SU(2) est restaurée et la transition métal-isolant disparaît
In this thesis we theoretically investigate the behaviour of quantum particles (electrons, atoms, photons, etc.) moving in a random medium and undergoing Anderson localization. For noninteractingparticles, the energy spectrum can possess one or more critical points, where the nature of the single-particle wavefunctions changes from extended to localized leading to a undergoes a metal-insulator phase transition, also known as Anderson transition.A fundamental question is whether and how Anderson transitions survive in interacting quantum systems. Here we study a minimal model of two particles moving in a disordered lattice and subject to short-range mutual interactions. By combining large-scale numerics with Green’s functions techniques, we show that two-particle Anderson transitions do occur in three dimensions and explore the phase diagram in the space of energy, disorder and interaction strength. The latter presents a rich structure, characterized by a doubly reentrant behavior, caused by the competition between scattering and bound states of the pair. We also show that previous claims of 2D Anderson transitions of the pair are essentially due to finite-size effects.A second problem that we address in this thesis is the occurrence of 2D metal-insulator transitions for a single particle in the presence of a spatially correlated potential and subject to spin-orbit interactions, described by Rashba-Dresselhaus couplings. We illustrate that, irrespective of the properties of the disorder, there is a regime where the critical energy depends linearly on the disorder strength. The slope and the intercept are studied in the vicinity of the spin-helix point, where the SU(2) symmetry is restored and the 2D metal-insulator transition disappears

Thesis (PhD)--University of Stellenbosch, 2011.
ENGLISH ABSTRACT: The fractional quantum Hall effect represents a true many-body phenomenon in which the collective behaviour of interacting electrons plays a central role. In contrast to its integral counterpart, the appearance of a mobility gap in the fractional quantum Hall regime is due entirely to the Coulomb interaction and is not the result of a perturbed single particle gap. The bulk of our theoretical understanding of the underlying many-body problem is based on Laughlin’s ansatz wave function and the composite fermion picture proposed by Jain. In the latter the fractional quantum Hall effect of interacting electrons is formulated as the integral quantum Hall effect of weakly interacting quasiparticles called composite fermions. The composite fermion picture provides a qualitative description of the interacting system’s low-energy spectrum and leads to a generalisation of Laughlin’s wave functions for the electron ground state. These predictions have been verified through extensive numerical tests. In this work we present an alternative formulation of the composite fermion picture within a more rigorous mathematical framework. Our goal is to establish the relation between the strongly interacting electron problem and its dual description in terms of weakly interacting quasiparticles on the level of the microscopic Hamiltonian itself. This allows us to derive an analytic expression for the interaction induced excitation gap which agrees very well with existing numerical results. We also formulate a mapping between the states of the free particle and interacting descriptions in which the characteristic Jastrow-Slater structure of the composite fermion ansatz appears naturally. Our formalism also serves to clarify several aspects of the standard heuristic construction, particularly with regard to the emergence of the effective magnetic field and the role of higher Landau levels. We also resolve a long standing issue regarding the overlap of unprojected composite fermion trial wave functions with the lowest Landau level of the free particle Hamiltonian.
AFRIKAANSE OPSOMMING: Die fraksionele kwantum Hall-effek is ’n veeldeeltjie verskynsel waarin die kollektiewe gedrag van wisselwerkende elektrone ’n sentrale rol speel. In teenstelling met die heeltallige kwantum Hall-effek is die ontstaan van ’n energie gaping in die fraksionele geval nie ’n enkeldeeltjie effek nie, maar kan uitsluitlik aan die Coulomb wisselwerking toegeskryf word. Die teoretiese raamwerk waarbinne hierdie veeldeeltjie probleem verstaan word is grootliks gebaseer op Laughlin se proefgolffunksie en die komposiete-fermion beeld van Jain. In laasgenoemde word die fraksionele kwantum Hall-effek van wisselwerkende elektrone geformuleer as die heeltallige kwantum Hall-effek van swak-wisselwerkende kwasi-deeljies wat as komposiete-fermione bekend staan. Hierdie beeld lewer ’n kwalitatiewe beskrywing van die wisselwerkende sisteem se lae-energie spektrum en lei tot ’n veralgemening van Laughlin se golffunksies vir die elektron grondtoestand. Hierdie voorspellings is deur verskeie numeriese studies geverifieer. In hierdie tesis ontwikkel ons ’n alternatiewe formulering van die komposiete-fermion beeld binne ’n strenger wiskundige raamwerk. Ons doel is om die verband tussen die sterk-wisselwerkende elektron sisteem en sy duale beskrywing in terme van swak-wisselwerkende kwasi-deeltjies op die vlak van die mikroskopiese Hamilton-operator self te realiseer. Hierdie konstruksie lei tot ’n analitiese uitdrukking vir die opwekkingsenergie wat baie goed met bestaande numeriese resultate ooreenstem. Ons identifiseer ook ’n afbeelding tussen die vrye-deeltjie en wisselwerkende toestande waarbinne die Jastrow-Slater struktuur van die komposiete-fermion proefgolffunksies op ’n natuurlike wyse na vore kom. Verder werp ons formalisme nuwe lig op kwessies binne die standaard heuristiese konstruksie, veral met betrekking tot die oorsprong van die effektiewe magneetveld en die rol van ho¨er effektiewe Landau vlakke. Ons lewer ook uitspraak oor die vraagstuk van die oorvleueling van ongeprojekteerde komposiete-fermion golffunksies met die laagste Landau vlak van die vrye-deeltjie Landau probleem.

Each year, the gap between theoretical proposals and experimental endeavours to create quantum computers gets smaller, driven by the promise of fundamentally faster algorithms and quantum simulations. This occurs by the combination of experimental ingenuity and ever simpler theoretical schemes. This thesis comes from the latter perspective, aiming to find new, simpler ways in which components of a quantum computer could be built. We first search for ways to create quantum gates, the primitive building blocks of a quantum computer. We find a novel, low-control way of performing a two-qubit gate on qubits encoded in a decoherence-free subspace, making use of many-body interactions that may already be present. This includes an analysis of the effect of control errors and magnetic field fluctuations on the gate. We then present novel ways to create three-qubit Toffoli and Fredkin gates in a single step using linear arrays of qubits, including an assessment of how well these gates could perform, for quantum or classical computation, using state-of-the-art ion trap and silicon donor technology. We then focus on a very different model from the normal circuit model, combining ideas from measurement-based quantum computation (MBQC) and holonomic quantum computation. We generalise an earlier model to show that all MBQC patterns with a property called gflow can be converted into a holonomic computation. The manifestation of the properties of MBQC in this adiabatically driven model is then explored. Finally, we investigate ways in which quantum information can be communicated between distant parties, using minimally engineered spin chains. The viability of using 1D Wigner crystals as a quantum channel is analysed, as well as schemes using ideal uniform spin chains with nextneighbour interactions, and edge-locking effects.

This thesis approaches questions concerning the thermalisation of subsystems of closed quantum systems, prepared in pure states of definite energy but far from equilibrium, under exact unitary evolution. Taking motivation from experiments in the field of ultracold atoms, an extensive study of relaxation to a thermal state in the Hubbard model is presented. The study of small local subsystems in Hubbard-model lattice clusters has led to some interesting findings. Explored are the effects of interactions between fermions, the initial-state energy and the energy uncertainty in the initial state and their effects on relaxation dynamics and thermalisation. The most significant finding is that while subsystem thermalisation is seen for a large range of subsystem-bath coupling strengths, the temporal form of the relaxation varies markedly from exponential decay for weak couplings with a crossover to Gaussian behaviour with increased coupling strength. This is found to hold more generally for random couplings between the subsystem and bath and for bosons as well as fermions, thus demonstrating generality. As well as being demonstrated numerically, this behaviour is derived for a generic class of bi-partite quantum systems which may be described with the use of random matrices. A Brownian motion model is employed to show the exponential to Gaussian crossover when the subsystem-bath coupling matrix takes a banded form. This result agrees well with numerical Hubbard-model results, and yields identical results at short times to those from straight-forward perturbative methods. It is demonstrated that the non-Markovian Gaussian behaviour should also be observable in the limit of macroscopic baths.

In this thesis, we discuss many-particle systems on general compact quantum graphs. The results cover systems of distinguishable particles as well as systems of bosons or fermions. The main focus lies on the introduction of many-particle interactions in order to establish a useful model regarding many-particle quantum chaos 811d onc-dimensional Bose-Einstein condensation (BEC). Using suitable quadratic forms, we will characterise self-adjoint realisations of the two- and many-particle Laplacian which incorporate two different types of interactions, i.e. singular interactions localised at the vertices of the graph and contact interactions which are also present along the edges. In that context, we will establish regularity results in order to characteristic the domains of the self-adjoint realisations explicitly. We will also discuss spectral properties of the constructed operators by establishing discreteness of their spectra and Weyl laws for the corresponding eigenvalue counts. Finally, based on the introduced models of interacting particles, we discuss BoseEinstein condensation on general quantum graphs. We will distinguish between systems of bosons for which BEC occurs and such for which no BEC is present at any finite temperature. As a final result, we prove that no Bose-Einstein condensation occurs (in the sense of phase transitions) in a system of bosons interacting via repulsive hard-core interactions.

This thesis contains four studies of the effects of disorder and randomness on strongly correlated quantum phases of matter. Starting with an itinerant ferromagnet, I first use an order-by-disorder approach to show that adding quenched charged disorder to the model generates new quantum fluctuations in the vicinity of the quantum critical point which lead to the formation of a novel magnetic phase known as a helical glass. Switching to bosons, I then employ a momentum-shell renormalisation group analysis of disordered lattice gases of bosons where I show that disorder breaks ergodicity in a non-trivial way, leading to unexpected glassy freezing effects. This work was carried out in the context of ultracold atomic gases, however the same physics can be realised in dimerised quantum antiferromagnets. By mapping the antiferromagnetic model onto a hard-core lattice gas of bosons, I go on to show the importance of the non-ergodic effects to the thermodynamics of the model and find evidence for an unusual glassy phase known as a Mott glass not previously thought to exist in this model. Finally, I use a mean-field numerical approach to simulate current generation quantum gas microscopes and demonstrate the feasibility of a novel measurement scheme designed to measure the Edwards-Anderson order parameter, a quantity which describes the degree of ergodicity breaking and which has never before been experimentally measured in any strongly correlated quantum system. Together, these works show that the addition of disorder into strongly interacting quantum systems can lead to qualitatively new behaviour, triggering the formation of new phases and new physics, rather than simply leading to small quantitative changes to the physics of the clean system. They provide new insights into the underlying physics of the models and make direct connection with experimental systems which can be used to test the results presented here.

La première partie de la thèse est dédiée à la contrôlabilité exacte globale de l'équation de Schrödinger bilinéaire (BSE).Nous montrons comment construire un voisinage de toute fonction propre du Laplacien Dirichlet où la contrôlabilité exacte locale est satisfaite à un temps explicit. Ensuite, pour tout couple de telles fonctions propres, nous étudions comment construire des contrôles et des temps tels que le flot de (BSE) envoie la première sur un voisinage de la seconde arbitrairement petit. Finalement, en regroupant les deux résultats précédents, nous définissons une dynamique entre états propres et nous fournissons un temps explicite requis pour atteindre l'état propre ciblé.Dans la deuxième partie, nous étudions la contrôlabilité exacte globale en projection d'une infinité d'équation de type (BSE) et nous prouvons la contrôlabilité exacte locale en projection à des termes dephases près pour tout temps positif. Dans la démonstration, nous adoptons différentes techniques provenant de la méthode du retour de Coron habituellement utilisée pour ces types de résultats. La principale nouveauté de ce travail est le fait que nous fournissons un ensemble de conditions en le champ de contrôle, impliquant la validité du résultat. Pour un champs de contrôle donné, nous pouvons vérifier si ces hypothèses sont satisfaites.La troisième partie du travail traite de la contrôlabilité de l'équation de Schrödinger bilinéaire (BSE) sur des graphiques compactes. Considérer (BSE) sur un telle structure est utile quand nous devons étudier la dynamique des paquets d'ondes sur un modèle de type graphes. Nous étudions les hypothèses sur le graphe et le champ de contrôle implique que (BSE) soit bien posée dans des espaces appropriés que nous caractérisons en utilisant les méthodes d'interpolation. Ensuite, nous fournissons la contrôlabilité exacte globale dans ces espaces en étudiant comment la structure du graphe et des conditions de bords affectent le résultat. Nous donnons également des exemples de graphes et de champ de contrôle, tels que les hypothèses spectrales de la contrôlabilité exacte globale soient vérifiées, par exemple les graphes en étoile, graphe dit « têtard » et graphe à double anneau. Enfin, quand nos hypothèses de la contrôlabilité exacte globale ne sont pas vérifiées, nous définissons une notion plus faible de contrôlabilité appelée « contrôlabilité énergétique » qui assure l'existence d'un ensemble d'états liés pour lesquels la contrôlabilité exacte est vérifiée. En d'autres termes, nous prouvons l'existence de niveaux d'énergie pour lesquelles il est possible de changer l'état du système. Cette technique permet de traiter un grand nombre de problèmes intéressants. En effet, pour des graphes complexes, il n'est pas possible de vérifier les hypothèses spectrales donnant la contrôlabilité exacte globale. Cependant, la contrôlabilité énergétique permet d'obtenir des résultats intéressants en regardant seulement des sous-graphes particuliers
The first part of the research is dedicated to the global exact controllability of the bilinear Schrödinger equation (BSE).We show how to construct a neighborhood of some eigenfunctions of the Dirichlet Laplacian where the local exact controllability is satisfied in a specific time. Then, for any couple of those eigenfunctions, we study how to construct controls and times such that the relative dynamics of (BSE) drives the first close to the second as much desired. Third, by gathering the two previous results, we define a dynamics steering eigenstates in eigenstates and we provide an explicit time required to reach the target.In the second part, we study the simultaneous global exact controllability in projection of infinitely many (BSE) and we prove the simultaneous local exact controllability in projection up to phases for any positive time. In the proof, we use different techniques from the Coron's return method usually adopted for those types of results. The main novelty of the work is the fact that it provides a set of conditions implying the validity of the result. Given any control field, one can verify if those assumptions are satisfied.The third part of the work treats the controllability of the bilinear Schrödinger equation (BSE) on compact graph. Considering (BSE) on such a complex structure is useful when one has to study the dynamics of wave packets on graph type model. We investigate assumptions on the graph and on the control field implying the well-posedness of (BSE) in suitable spaces that we characterize by providing peculiar interpolation features.Then, we provide the global exact controllability in those spaces by studying how the structure of the graph and the boundary conditions affect the result. We also provide examples of graphs and control fields so that the spectral assumptions of the global exact controllability are satisfied, e.g. star graphs, tadpole graphs and double-ring graphs.Afterwards, when the hypothesis for the global exact controllability fail, we define a weaker notion of controllability, the so-called “energetic controllability" which ensures the existence of a set of bounded states for which the exact controllability is verified. In other words, we prove the existence of energy levels in which it is possible to change the energy of the system.This technique allows to treat a large number of interesting problems. Indeed, for complex graphs, it is not possible to verify the spectral hypothesis of the global exact controllability. However, the energetic controllability allows to obtain interesting results only by looking for particular substructure contained in the graph

The goal of research in quantum information is to investigate how quantum systems can be used to store, transmit and elaborate information and how the non-classical nature of their correlations allows defining protocols that outperform their classical counterparts. Despite of the many progresses, both theoretical and experimental, made in this field in the latest decades, many challenges lie ahead for practical implementations of quantum technologies. One of the most important ones is caused by the unavoidable interaction of quantum systems with their surroundings: The coupling to the environment is generally detrimental to the quantum information contained in the system as the system undergoes decoherence. In the quest for quantum technologies, it is fundamental to overcome the problem of decoherence and loss of information. Different physical implementations of qubits, such as superconducting and solid-state devices, are affected by the interaction with the environment in a way that can be described in terms of classical stochastic noise. The classical noise model can also be used to give an approximate, sometimes equivalent, description of full quantum models of system-environment interaction. This thesis contains my personal contribution to the study of the dynamics of discrete-variable quantum systems affected by classical noise. It covers in particular single- and two-qubit systems affected by Gaussian and non-Gaussian noise. It also discusses the dynamics of a quantum walk affected by spatially correlated classical noise. Analytical solutions for particular forms of noise and interactions, and a general numerical method for simulation of the dynamics are presented. Moreover, the thesis presents the experimental implementation of a quantum optical simulator of noisy dynamics of single-qubit systems. Finally, the use of quantum systems as probes of the spectral properties of large classical environments is discussed, showing that entanglement is a resource for improvements in the precision of the estimation.

In this thesis, we study the physical properties of several ultracold few-boson systems depending on the interactions between their constituents. Nowadays, experimentally, it is possible to have great control with high precision over the geometry and the interactions between the particles, making them an excellent setup to test directly the principles of quantum mechanics. A very interesting point is to study the evolution of their properties with the number of particles. The theoretical study of these systems pretends to microscopically understand the current experimental results and give support to new experimental developments. The method that will be used is the exact diagonalization of the Hamiltonian of the system. As we will see, in spite of the attempts to improve it, the method is limited by the fact that, in practice, it is only useful to study few-particle systems. The method has several advantages. First of all, one has access to both the ground and the excited states. In second place, the method is variational and converges to the exact solution as long as the Hilbert space in which we diagonalize is enlarged. Moreover, since we have access to the states of the system, it is possible to calculate any observable quantity of interest. First, we will study a system of spinless bosons trapped in a two-dimensional harmonic potential. The effect of the trap is to keep the system bound. It will be seen how the presence of a repulsive interaction changes the energy spectrum and other properties of the system. For instance, the density profile, which is usually measurable, and also the two-body distribution function, which is intimately related to the existence of correlations. Afterwards, the focus will be on the particular case of having only two bosons in the system interacting through a strong repulsive force. Inspired by the one-dimensional case where the fermionization phenomenon takes place in the strongly-interacting limit, we will study whether in two dimensions there is a resembling reminiscent effect. In other words, we will analyze if there are properties of the two strongly-interacting bosons in two dimensions that are like the ones of two noninteracting fermions. After that, we will tackle the localization phenomenon in a one-dimensional system that is caused by an external speckle potential that introduces disorder in the system. We will show that the localization is a robust phenomenon against repulsive contact interactions. Finally, we will study the influence of the spin-orbit coupling in a system of bosons with two possible pseudospin components, associated, for instance, to two hyperfine levels, confined in a two-dimensional harmonic trap. We will present an exhaustive analysis of the combined effects of the interaction and the spin-orbit coupling in the spectrum and the properties of the system. In particular we show the existence of a crossover in the ground state of the system susceptible to be experimentally identified.
En aquesta tesi, estudiarem les propietats físiques de diversos sistemes de pocs bosons ultrafreds depenent de les interaccions entre els seus constituents. Avui dia, a nivell experimental, es té un gran control amb una gran precisió de la geometria i les interaccions entre les partícules, fet que fa aquest sistemes excel·lents per comprovar de forma directa els principis de la mecànica quàntica. Un punt d'interès és comprovar l'evolució de les seves propietats amb el nombre de partícules. L'estudi teòric d'aquests sistemes pretén entendre a nivell microscòpic els resultats experimentals actuals i donar suport pels nous avenços experimentals. El mètode que farem servir serà la diagonalització exacta del hamiltonià del sistema. Com veurem, malgrat les millores que es poden implementar, ens trobarem amb la limitació de no poder estudiar sistemes de més d'unes quantes partícules. Els avantatges d'aquest mètode són diversos. En primer lloc, podrem obtenir no només l'estat fonamental del sistema sinó que també els primers estats excitats. En segon lloc, el mètode és variacional i sabem que convergeix cap a la solució exacta a mesura que ampliem l'espai de Hilbert en que diagonalitzem. A més a més, en tenir accés als estats del sistema, podem calcular qualsevol quantitat observable que sigui d'interès. Primerament, estudiarem un sistema de bosons sense espín atrapats en un potencial harmònic bidimensional. L'efecte de la trampa és de mantenir el sistema lligat. En haver-hi una interacció repulsiva, veurem com canvia l'espectre d'energia del sistema i també altres propietats. Per exemple, la seva densitat, que habitualment es pot mesurar, i també la funció de distribució de dos cossos, que va íntimament lligada a l'existència de correlacions. Tot seguit, ens centrarem en el cas particular de tenir només dos bosons en el sistema interaccionant a través d'una gran força repulsiva. Inspirats pel cas unidimensional en que té lloc el fenomen de la fermionització en el limit d'interacció molt forta, estudiarem si en el cas bidimensional hi queda cap reminiscència d'aquest efecte. En altres paraules, analitzarem si hi ha propietats dels dos bosons fortament interactuants en dues dimensions que siguin com les de fermions no interactuants en el mateix sistema. A continuació, tractarem el fenomen de la localització en un sistema unidimensional en el qual hi ha un potencial extern de tipus speckle que introdueix desordre en el sistema. Veurem que la localització és un fenomen robust en front de les interaccions repulsives. Per últim, estudiarem la influència de l'espín-òrbita en un sistema de bosons amb dues components de pseudoespín, associades, per exemple, a dos nivells hiperfins, atrapats en un potencial harmònic bidimensional. Presentarem un anàlisi exhaustiu dels efectes conjunts de la interacció i l'espín- òrbita en l'espectre i en les propietats del sistema. En particular, mostrarem l'existència d'un encreuament en l'estat fonamental del sistema susceptible de ser identificat experimentalment.

This thesis is a study of three categories of problems in fermionic systems for which topology plays an important role: (i) The properties of zero modes arising in systems of fermions interacting with a bosonic background, with a special focus on Majorana modes arising in the superconductor state. We propose a method for counting Majorana modes and we study a mechanism for controlling their number parity in lattice systems, two questions that are of relevance to the protection of quantum bits. (ii) The study of dispersionless bands in two dimensions as a platform for correlated physics, where it is shown the possibility of stabilizing the fractional quantum Hall effect in a flat band with Chern number. (iii) The extension of the hierarchy of quantum Hall fluids to the case of time-reversal symmetric incompressible ground states describing a phase of strongly interacting topological insulators in two dimensions.
Physics

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2010.
Cataloged from PDF version of thesis.
Includes bibliographical references (p. 131-136).
Strongly correlated systems present interesting challenges in condensed matter physics. On the one hand, the theoretical work in the last two decades suggests that strong interactions may lead to new phases and phase transitions of matter that don't fit paradigms such as Fermi liquid theory or Landau's theory of phase transitions. On the other hand, there are actual materials which are undoubtedly governed by strong interactions and indeed do not fit the conventional paradigms but whose behavior often doesn't quite match our theoretical expectations. This gap between theory and experiments is slowly narrowing owing to the discovery of new materials and recent advances in numerical simulations. As an example, the material K - (ET)2Cu 2(CN) 3 exhibits metallic specific heat in its insulating phase. This is indicative of the theoretically proposed phenomena of 'fractionalization' where elementary excitations in a phase carry quantum numbers that are fractions of that corresponding to an electron. Similarly, there is growing numerical evidence of the theoretical phenomena of 'deconfined quantum criticality', where quantum Berry phases lead to emergence of fractionalized particles right at the phase transition. In this thesis we study phenomena where the concept of fractionalization is a useful tool to explore new phases and phase transitions. Most of our examples are in the context of frustrated quantum magnets. Along the way, we also explore topics such as quantum numbers of topological defects and non-abelian phases of matter. Whenever possible, we compare theoretical predictions with experimental and numerical data. We also discuss deconfined quantum criticality in the context of metallic systems where it opens the route to phase transitions very different from the conventional spin-density wave instability of Fermi surface.
by Tarun Grover.
Ph.D.

In this thesis we study collective, emergent and optical properties of interacting quantum systems both in equilibrium and nonequilibrium situations from a microscopic modelling. This orientation steams from both the fact there is a profound need to design, characterise and set up control strategies for realistic systems in which quantum technologies could be conceived and the interest to grasp and identify fundamental principles for the emergence of macroscopic behaviour. The thesis is divided into three parts: I Optical and Collective Phenomena; II Equilibrium many-body systems and III Nonequilibrium many-body systems. Part I includes four complementary contributions to the optics emerging from the collective behaviour of microscopic quantum systems. In part II (Equilibrium many-body systems) of the thesis I have addressed the physics of quantum phase transitions from the perspective of nonequilibrium thermodynamics. We have shown that such an approach captures the essential features of finite order transitions that have a strong connection to thermodynamical and energetic figures of merit, but does not capture infinite order transitions that are of a much more subtle nature. Motivated by these exotic infinite order transitions we have looked at quantum phases and phase transitions through an informational and operational perspective based on pure state conversions restricted by local operations. In the third and last part (Nonequilibrium many-body systems) of the thesis I have laid out a project on the closed evolution of quantum spin chains focussing on the emergent nonequilibrium laws that depart from equilibrium physics.

In the first part of the Thesis we mostly concentrate on spectral properties of strongly correlated systems and on their equilibrium properties. This is accomplished by the general concept of imaginary-time dynamics which we apply to a number of different problems in which different strengths of this approach emerge. In Chapter 1 we introduce the formalism that allows for a connection between the quantum and the classical worlds. The connection is established by means of the imaginary-time quantum evolution which, under certain circumstances, is shown to be equivalent to a classical stochastic process. It is further shown that exact static and spectral properties of correlated systems can be obtained when this mapping is feasible. The relationship between the imaginary-time dynamics in different frameworks such as the path-integral and the perturbative one is also underlined. In Chapter 2 we present a specific implementation of the general ideas previously presented. In particular we introduced an extension to lattice systems of the Reptation Monte Carlo algorithm [30] which benefits of a sampling scheme based on directed updates. Specific improvements over the existing methodologies consist in the unbiased evaluation of the imaginary-time path integrals for bosons and a systematic scheme to improve over the Fixed-node approximation for fermions. Applications to the Hubbard and the Heisenberg models are presented. In Chapter 3 we demonstrate the application of the imaginary-time dynamics to the exact study of spectral properties. Subject of our attention is a highly anharmonic and correlated quantum crystal such as Helium 4 at zero temperature.[33] Concerning this system, we have obtained the first ab-initio complete phonon dispersion in good agreement with neutron spectroscopy experiments. Moreover, we have also studied the density excitations of solid helium in a region of wave-vectors in between the collective (phonon) and the single-particle regimes, where the presence of residual coherence in the dynamics shows analogies between the highly anharmonic crystal and the superfluid phase. In Chapter 4 we introduce a novel method, based on the imaginary-time dynamics, to obtain unbiased estimates of fermionic properties.[34] By means of this method and of a very accurate variational state, we provide strong evidence for the stability of a saturated ferromagnetic phase in the high-density regime of the two-dimensional infinite-U Hubbard model. By decreasing the electron density, we observe a discontinuous transition to a paramagnetic phase, accompanied by a divergence of the susceptibility on the paramagnetic side. This behavior, resulting from a high degeneracy among different spin sectors, is consistent with an infinite-order phase transition scenario. In Chapter 5 the use of imaginary-time dynamics in the context of finite-temperature response functions is highlighted. As an application, we study an intriguing quantum phase featuring both glassy order and Bose-Einstein condensation. [35] We introduce and validate a model for the role of geometrical frustration in the coexistence of off-diagonal long range order with an amorphous density profile. The exact characterization of the response of the system to an external density perturbation is what allows here to establish the existence of a spin-glass phase. The differences between such a phase and the otherwise insulating "Bose glasses" are further elucidated in the Chapter. In the second part of the Thesis we focus our attention on the dynamics of closed systems out of equilibrium. This is accomplished by both non-stochastic exact methods for the dynamics and the introduction of a novel time-dependent Variational Monte Carlo scheme. In Chapter 6 exact diagonalization schemes and renormalization-based methods for one-dimensional systems are introduced. We identify key phenomenological traits resulting from the many-body correlation in closed systems driven sufficiently away from equilibrium.[31] We provide evidences that the dynamics of interacting lattice bosons away from equilibrium can be trapped into extremely long-lived inhomogeneous metastable states. The slowing down of incoherent density excitations above a threshold energy, much reminiscent of a dynamical arrest on the verge of a glass transition, is identified as the key feature of this phenomenon. In Chapter 7 we present an extension to dynamical properties of the Variational Quantum Monte Carlo method.[32] This is accomplished by introducing a general class of time-dependent variational states which is based on the mapping of the many-body dynamics onto an instantaneous ground-state problem. The application of the method to the experimentally relevant quantum quenches of interacting bosons reveals the accuracy and the reliability of the introduced numerical scheme. We indeed obtain for the first time a consistent variational description of the approach to the equilibrium of local observables and underline the origin of the metastability and glassy behavior previously identified. In the very last part we draw our conclusions and show some possible paths for stimulating future research.

In this thesis we study three examples of interacting many-body systems undergoing a non equilibrium time evolution. Firstly we consider the time evolution in an integrable system: the sine-Gordon field theory in the repulsive regime. We will focus on the one point function of the semi-local vertex operator e^iβφ(x)/2 on a specific class of initial states. By analytical means we show that the expectation value considered decays exponentially to zero at late times and we determine the decay time. The method employed is based on a form-factor expansion and uses the "Representative Eigenstate Approach" of Ref. [73] (a.k.a. "Quench Action"). In a second example we study the time evolution in models close to "special" integrable points characterised by hidden symmetries generating infinitely many local conservation laws that do not commute with one another, in addition to the infinite commuting family implied by integrability. We observe that both in the case where the perturbation breaks the integrability and when it breaks only the additional symmetries maintaining integrability, the local observables show a crossover behaviour from an initial to a final quasi stationary plateau. We investigate a weak coupling limit, identify a time window in which the effects of the perturbations become significant and solve the time evolution through a mean-field mapping. As an explicit example we study the XYZ spin-1/2 chain with additional perturbations that break integrability. Finally, we study the effects of integrability breaking perturbations on the non-equilibrium evolution of more general many-particle quantum systems, where the unperturbed integrable model is generic. We focus on a class of spinless fermion models with weak interactions. We employ equation of motion techniques that can be viewed as generalisations of quantum Boltzmann equations. We benchmark our method against time dependent density matrix renormalisation group computations and find it to be very accurate as long as interactions are weak. For small integrability breaking, we observe robust prethermalisation plateaux for local observables on all accessible time scales. Increasing the strength of the integrability breaking term induces a "drift" away from the prethermalisation plateaux towards thermal behaviour. We identify a time scale characterising this crossover.

On etudie la contrôlabilité des systèmes quantiques dans deux contextes différents: le cadre standard fermé, dans lequel un système quantique est considéré comme isolé et le problème de contrôle est formulé sur l'équation de Schrödinger; le cadre ouvert qui décrit un système quantique en interaction avec un plus grand, dont seuls les paramètres qualitatifs sont connus, au moyen de l'équation de Lindblad sur les états.Dans le contexte des systèmes fermés on se focalise sur la classe intéressante des systèmes spin-boson, qui décrivent l'interaction entre un système quantique à deux niveaux et un nombre fini de modes distingués d'un champ bosonique. On considère deux exemples prototypiques, le modèle de Rabi et le modèle de Jaynes-Cummings qui sont encore très populaires dans plusieurs domaines de la physique quantique. Notamment, dans le contexte de la Cavity Quantum Electro Dynamics (C-QED), ils fournissent une description précise de la dynamique d'un atome à deux niveaux dans une cavité micro-onde en résonance, comme dans les expériences récentes de S. Haroche. Nous étudions les propriétés de contrôlabilité de ces modèles avec deux types différents d'opérateurs de contrôle agissant sur la partie bosonique, correspondant respectivement – dans l'application à la C-QED – à un champ électrique et magnétique externe. On passe en revue quelques résultats récents et prouvons la contrôlabilité approximative du modèle de Jaynes-Cummings avec ces contrôles. Ce résultat est basé sur une analyse spectrale exploitant les non-résonances du spectre. En ce qui concerne la relation entre l'Hamiltonien de Rabi et Jaynes-Cummings nous traitons dans un cadre rigoureux l'approximation appelée d'onde tournante. On formule le problème comme une limite adiabatique dans lequel la fréquence de detuning et le paramètre de force d'interaction tombent à zero, ce cas est connu sous le nom de régime de weak-coupling. On prouve que, sous certaines hypothèses sur le rapport entre le detuning et le couplage, la dynamique de Jaynes-Cumming et Rabi montrent le même comportement, plus précisément les opérateurs d'évolution qu'ils génèrent sont proches à la norme.Dans le cadre des systèmes quantiques ouverts nous étudions la contrôlabilité de l'équation de Lindblad. Nous considérons un contrôle agissant adiabatiquement sur la partie interne du système, que nous voyons comme un degré de liberté qui peut être utilisé pour contraster l'action de l'environnement. L'action adiabatique du contrôle est choisie pour produire une transition robuste. On prouve, dans le cas prototype d'un système à deux niveaux, que le système approche un ensemble de points d'équilibre déterminés par l'environnement, plus précisément les paramètres qui spécifient l'opérateur de Lindblad. Sur cet ensemble, le système peut être piloté adiabatiquement en choisissant un contrôle approprié. L'analyse est fondée sur l'application de méthodes de perturbation géométrique singulière
We investigate the controllability of quantum systems in two differentsettings: the standard 'closed' setting, in which a quantum system is seen as isolated, the control problem is formulated on the Schroedinger equation; the open setting that describes a quantum system in interaction with a larger one, of which just qualitative parameters are known, by means of the Lindblad equation on states.In the context of closed systems we focus our attention to an interesting class ofmodels, namely the spin-boson models. The latter describe the interaction between a 2-level quantum system and finitely many distinguished modes of a bosonic field. We discuss two prototypical examples, the Rabi model and the Jaynes-Cummings model, which despite their age are still very popular in several fields of quantum physics. Notably, in the context of cavity Quantum Electro Dynamics (C-QED) they provide an approximate yet accurate description of the dynamics of a 2-level atom in a resonant microwave cavity, as in recent experiments of S. Haroche. We investigate the controllability properties of these models, analyzing two different types of control operators acting on the bosonic part, corresponding -in the application to cavity QED- to an external electric and magnetic field, respectively. We review some recent results and prove the approximate controllability of the Jaynes-Cummings model with these controls. This result is based on a spectral analysis exploiting the non-resonances of the spectrum. As far as the relation between the Rabi andthe Jaynes-Cummings Hamiltonians concerns, we treat the so called rotating waveapproximation in a rigorous framework. We formulate the problem as an adiabaticlimit in which the detuning frequency and the interaction strength parameter goes to zero, known as the weak-coupling regime. We prove that, under certain hypothesis on the ratio between the detuning and the coupling, the Jaynes-Cumming and the Rabi dynamics exhibit the same behaviour, more precisely the evolution operators they generate are close in norm.In the framework of open quantum systems we investigate the controllability ofthe Lindblad equation. We consider a control acting adiabatically on the internal part of the system, which we see as a degree of freedom that can be used to contrast the action of the environment. The adiabatic action of the control is chosen to produce a robust transition. We prove, in the prototype case of a two-level system, that the system approach a set of equilibrium points determined by the environment, i.e. the parameters that specify the Lindblad operator. On that set the system can be adiabatically steered choosing a suitable control. The analysis is based on the application of geometrical singular perturbation methods

Ultracold atoms in optical lattices offer an important tool for studying dynamics in many-body interacting systems in a pristine environment. This thesis focuses on three theoretical works motivated by recent optical lattice experiments. In the first, we theoretically study the center of mass dynamics of states derived from the disordered Bose-Hubbard model in a trapping potential. We find that the edge states in the trap allow center of mass motion even with insulating states in the center. We identify short and long-time mechanisms for edge state transport in insulating phases. We also argue that the center of mass velocity can aid in identifying a Bose-glass phase. Our zero temperature results offer important insights into mechanisms of transport of atoms in trapped optical lattices while putting bounds on center of mass dynamics expected at non-zero temperature. In the second work, we study the domain wall expansion dynamics of strongly interacting bosons in 2D optical lattices with disorder in a recent experiment {[}J.-y. Choi et al., Science 352, 1547 (2016)]. We show that Gutzwiller mean-field theory (GMFT) captures the main experimental observations, which are a result of the competition between disorder and interactions. Our findings highlight the difficulty in distinguishing glassy dynamics, which can be captured by GMFT, and many-body localization, which cannot be captured by GMFT, and indicate the need for further experimental studies of this system. The last work features our study of phase diagrams of the 2D Bose-Hubbard model in an optical lattice with synthetic spin-orbit coupling. We investigate the transitions between superfluids with different phase patterns, which may be detected by measuring the spin-dependent momentum distribution.
Ph. D.
Ultracold atoms in optical lattices, a periodic potential generated by laser beams, offer an important tool for quantum simulations in a pristine environment. Motivated by recent optical lattice experiments with the implementation of disorder and synthetic spin-orbit coupling, we utilize Gutzwiller mean-field theory (GMFT) to study the dynamics of disordered state in an optical lattice under the sudden shift of the harmonic trap, the domain wall expansion of strongly interacting bosons in 2D lattices with disorder, and spin-orbit-driven transitions in the Bose-Hubbard model. We argue that the center of mass velocity can aid in identifying a Bose-glass phase. Our findings show that evidence for many-body localization claimed in experiments [J.-y. Choi et al., Science 352, 1547 (2016)] must lie in the differences between GMFT and experiments. We also find that strong spin-orbit coupling alone can generate superfluids with finite momentum and staggered phase patterns.

Thesis (Ph. D.)--Physics, Georgia Institute of Technology, 2007.
Landman, Uzi, Committee Member ; Yannouleas, Constantine, Committee Member ; Bunimovich, Leonid, Committee Member ; Chou, Mei-Yin, Committee Member ; Pustilnik, Michael, Committee Member.

Thermalization, the evolution of an interacting many-body system towards a thermal Gibbs ensemble after initialization in an arbitrary non-equilibrium state, is currently a phenomenon of great interest, both in theory and experiment. As the time evolution of a quantum system is unitary, the proposed mechanism of thermalization in quantum many-body systems corresponds to the so-called eigenstate thermalization hypothesis (ETH) and the typicality of eigenstates. Although this formally solves the contradiction of thermalizing but unitary dynamics in a closed quantum many-body system, it does neither make any statement on the dynamical process of thermalization itself nor in which way the coupling of the system to an environment can hinder or modify the relaxation dynamics. In this thesis, we address both the question whether or not a quantum system driven away from equilibrium is able to relax to a thermal state, which fulfills detailed balance, and if one can identify universal behavior in the non-equilibrium relaxation dynamics. As a first realization of driven quantum systems out of equilibrium, we investigate a system of Ising spins, interacting with the quantized radiation field in an optical cavity. For multiple cavity modes, this system forms a highly entangled and frustrated state with infinite correlation times, known as a quantum spin glass. In the presence of drive and dissipation, introduced by coupling the intra-cavity radiation field to the photon vacuum outside the cavity via lossy mirrors, the quantum glass state is modified in a universal manner. For frequencies below the photon loss rate, the dissipation takes over and the system shows the universal behavior of a dissipative spin glass, with a characteristic spectral density $\\mathcal{A}(\\omega)\\sim\\sqrt{\\omega}$. On the other hand, for frequencies above the loss rate, the system retains the universal behavior of a zero temperature, quantum spin glass. Remarkably, at the glass transition, the two subsystems of spins and photons thermalize to a joint effective temperature, even in the presence of photon loss. This thermalization is a consequence of the strong spin-photon interactions, which favor detailed balance in the system and detain photons from escaping the cavity. In the thermalized system, the features of the spin glass are mirrored onto the photon degrees of freedom, leading to an emergent photon glass phase. Exploiting the inherent photon loss of the cavity, we make predictions of possible measurements on the escaping photons, which contain detailed information of the state inside the cavity and allow for a precise, non-destructive measurement of the glass state. As a further set of non-equilibrium systems, we consider one-dimensional quantum fluids driven out of equilibrium, whose universal low energy theory is formed by the so-called Luttinger Liquid description, which, due to its large degree of universality, is of intense theoretical and experimental interest. A set of recent experiments in research groups in Vienna, Innsbruck and Munich have probed the non-equilibrium time-evolution of one-dimensional quantum fluids for different experimental realizations and are pushing into a time regime, where thermalization is expected. From a theoretical point of view, one-dimensional quantum fluids are particular interesting, as Luttinger Liquids are integrable and therefore, due to an infinite number of constants of motion, do not thermalize. The leading order correction to the quadratic theory is irrelevant in the sense of the renormalization group and does therefore not modify static correlation functions, however, it breaks integrability and will therefore, even if irrelevant, induce a completely different non-equilibrium dynamics as the quadratic Luttinger theory alone. In this thesis, we derive for the first time a kinetic equation for interacting Luttinger Liquids, which describes the time evolution of the excitation densities for arbitrary initial states. The resonant character of the interaction makes a straightforward derivation of the kinetic equation, using Fermi\'s golden rule, impossible and we have to develop non-perturbative techniques in the Keldysh framework. We derive a closed expression for the time evolution of the excitation densities in terms of self-energies and vertex corrections. Close to equilibrium, the kinetic equation describes the exponential decay of excitations, with a decay rate $\\sigma^R=\\mbox\\Sigma^R$, determined by the self-energy at equilibrium. However, for long times $\\tau$, it also reveals the presence of dynamical slow modes, which are the consequence of exactly energy conserving dynamics and lead to an algebraic decay $\\sim\\tau^$ with $\\eta_D=0.58$. The presence of these dynamical slow modes is not contained in the equilibrium Matsubara formalism, while they emerge naturally in the non-equilibrium formalism developed in this thesis. In order to initialize a one-dimensional quantum fluid out of equilibrium, we consider an interaction quench in a model of interacting, dispersive fermions in Chap.~\\ref. In this scenario, the fermionic interaction is suddenly changed at time $t=0$, such that for $t>0$ the system is not in an eigenstate and therefore undergoes a non-trivial time evolution. For the quadratic theory, the stationary state in the limit $t\\rightarrow\\infty$ is a non-thermal, or prethermal, state, described by a generalized Gibbs ensemble (GGE). The GGE takes into account for the conservation of all integrals of motion, formed by the eigenmodes of the Hamiltonian. On the other hand, in the presence of non-linearities, the final state for $t\\rightarrow\\infty$ is a thermal state with a finite temperature $T>0$. . The spatio-temporal, dynamical thermalization process can be decomposed into three regimes: A prequench regime on the largest distances, which is determined by the initial state, a prethermal plateau for intermediate distances, which is determined by the metastable fixed point of the quadratic theory and a thermal region on the shortest distances. The latter spreads sub-ballistically $\\sim t^$ in space with $0<\\alpha<1$ depending on the quench. Until complete thermalization (i.e. for times $t<\\infty$), the thermal region contains more energy than the prethermal and prequench region, which is expressed in a larger temperature $T_{t}>T_$, decreasing towards its final value $T_$. As the system has achieved local detailed balance in the thermalized region, energy transport to the non-thermal region can only be performed by the macroscopic dynamical slow modes and the decay of the temperature $T_{t}-T_\\sim t^$ again witnesses the presence of these slow modes. The very slow spreading of thermalization is consistent with recent experiments performed in Vienna, which observe a metastable, prethermal state after a quench and only observe the onset of thermalization on much larger time scales. As an immediate indication of thermalization, we determine the time evolution of the fermionic momentum distribution after a quench from non-interacting to interacting fermions. For this quench scenario, the step in the Fermi distribution at the Fermi momentum $k\\sub$ decays to zero algebraically in the absence of a non-linearity but as a stretched exponential (the exponent being proportional to the non-linearity) in the presence of a finite non-linearity. This can serve as a proof for the presence or absence of the non-linearity even on time-scales for which thermalization can not yet be observed. Finally, we consider a bosonic quantum fluid, which is driven away from equilibrium by permanent heating. The origin of the heating is atomic spontaneous emission of laser photons, which are used to create a coherent lattice potential in optical lattice experiments. This process preserves the system\'s $U(1)$-invariance, i.e. conserves the global particle number, and the corresponding long-wavelength description is a heated, interacting Luttinger Liquid, for which phonon modes are continuously populated with a momentum dependent rate $\\partial_tn_q\\sim\\gamma |q|$. In the dynamics, we identify a quasi-thermal regime for large momenta, featuring an increasing time-dependent effective temperature. In this regime, due to fast phonon-phonon scattering, detailed balance has been achieved and is expressed by a time-local, increasing temperature. The thermal region emerges locally and spreads in space sub-ballistically according to $x_t\\sim t^{4/5}$. For larger distances, the system is described by an non-equilibrium phonon distribution $n_q\\sim |q|$, which leads to a new, non-equilibrium behavior of large distance observables. For instance, the phonon decay rate scales universally as $\\gamma_q\\sim |q|^{5/3}$, with a new non-equilibrium exponent $\\eta=5/3$, which differs from equilibrium. This new, universal behavior is guaranteed by the $U(1)$ invariant dynamics of the system and is insensitive to further subleading perturbations. The non-equilibrium long-distance behavior can be determined experimentally by measuring the static and dynamic structure factor, both of which clearly indicate the exponents for phonon decay, $\\eta=5/3$ and for the spreading of thermalization $\\eta_T=4/5$. Remarkably, even in the presence of this strong external drive, the interactions and their aim to achieve detailed balance are strong enough to establish a locally emerging and spatially spreading thermal region. The physical setups in this thesis do not only reveal interesting and new dynamical features in the out-of-equilibrium time evolution of interacting systems, but they also strongly underline the high degree of universality of thermalization for the classes of models studied here. May it be a system of coupled spins and photons, where the photons are pulled away from a thermal state by Markovian photon decay caused by a leaky cavity, a one-dimensional fermionic quantum fluid, which has been initialized in an out-of-equilibrium state by a quantum quench or a one-dimensional bosonic quantum fluid, which is driven away from equilibrium by continuous, external heating, all of these systems at the end establish a local thermal equilibrium, which spreads in space and leads to global thermalization for $t\\rightarrow\\infty$. This underpins the importance of thermalizing collisions and endorses the standard approach of equilibrium statistical mechanics, describing a physical system in its steady state by a thermal Gibbs ensemble.

Intrinsic quantum Thermodynamics (IQT) is a theory that unifies thermodynamics and quantum mechanics into a single theory. Its mathematical framework, steepest-entropy-ascent quantum thermodynamics (SEAQT), can be used to model and describe the non-equilibrium phenomenon of diffusion based on the principle of steepest-entropy ascent. The research presented in this dissertation demonstrates the capability of this framework to model and describe diffusion at atomistic levels and is used here to develop a non-equilibrium-based model for an isolated system in which He3 diffuses in He4. The model developed is able to predict the non-equilibrium and equilibrium characteristics of diffusion as well as capture the differences in behavior of fermions (He3) and bosons (He4). The SEAQT framework is also used to develop the transient and steady-state model for an open system in which oxygen diffuses through a tin anode. The two forms of the SEAQT equation of motion are used. The first, which only involves a dissipation term, is applied to the state evolution of the isolated system as its state relaxes from some initial non-equilibrium state to stable equilibrium. The second form, the so-called extended SEAQT equation of motion, is applied to the transient state evolution of an open system undergoing a dissipative process as well mass-interactions with two mass reservoirs. In this case, the state of the system relaxes from some initial transient state to steady state. Model predictions show that the non-equilibrium thermodynamic path that the isolated system takes significantly alters the diffusion data from that of the equilibrium-based models for isolated atomistic-level systems found in literature. Nonetheless, the SEAQT equilibrium predications for He3 and He4 capture the same trends as those found in the literature providing a point of validation for the SEAQT framework. As to the SEAQT results for the open system, there is no data in the literature with which to compare since the results presented here are completely original to this work.
Ph. D.

This thesis is concerned with static properties of large bosonic systems in two dimensions. These systems at very low-temperatures are expected to exhibit emph{Bose-Einstein condensation}. From a mathematical and physical, point of view it is interesting to provide conditions for the occurrence of Bose-Einstein condensation. Obviously, studying a system of N particles, where N is large, is very challenging. However, to overcome this problem we can rely on effective theories, which describe the collective behaviour of the particles. The aim of the manuscript is to present new results regarding the occurrence of Bose-Einstein condensation in two-dimensional bosonic systems in suitable scaling limits. Our first result consists of the rigorous derivation of complete Bose-Einstein condensation of low-energy states in a regime where the interaction potential scales as N^2bV(N^b ), for b >0 such that log (N^b) <

La simulation quantique consiste à réaliser expérimentalement des systèmes artificiels équivalent à des modèles proposés par les théoriciens. Pour réaliser ces systèmes, il est possible d'utiliser des atomes dont les états individuels et les interactions sont contrôlés par la lumière. En particulier, une fois excités dans un état de haute énergie (appelé état de Rydberg), les atomes peuvent être contrôlés individuellement et leurs interactions façonnées arbitrairement par des faisceaux laser. Cette thèse s'intéresse à deux types de simulateurs quantiques à base d'atomes de Rydberg, et en particulier à leurs potentielles limitations.Dans l'expérience du Joint Quantum Institute (USA), nous observons la décohérence dans une structure cubique contenant jusqu'à 40000 atomes. A partir d'atomes préparés dans un état de Rydberg bien défini, nous constatons l'apparition spontanée d'états de Rydberg voisins et le déclenchement d'un phénomène d'avalanche. Nous montrons que ce mécanisme émane de l'émission stimulée produite par le rayonnement du corps noir. Ce phénomène s'accompagne d'une diffusion induite par des interactions de type dipole-dipole résonant. Nous complétons ces observations avec un modèle de champ moyen en état stationnaire. Dans un second temps, l'étude de la dynamique du problème nous permet de mesurer les échelles de temps caractéristiques. La décohérence étant globalement néfaste pour la simulation quantique, nous proposons plusieurs solutions pour en atténuer les effets. Nous évaluons notamment la possibilité de travailler dans un environnement cryogénique, lequel permettrait de réduire le rayonnement du corps noir.Dans l'expérience du Laboratoire Charles Fabry à l'Institut d'Optique (France), nous analysons les limites d'un simulateur quantique générant des structures bi- et tridimensionnelles allant jusqu'à 70 atomes de Rydberg piégés individuellement dans des pinces optiques. Le système actuel étant limité par le temps de vie des structures, nous montrons que l'utilisation d'un cryostat permettrait d'atteindre des tailles de structures jusqu'à 300 atomes. Nous présentons les premiers pas d'une nouvelle expérience utilisant un cryostat à 4K, et en particulier les études amont pour le développement de composants optomécaniques placés sous vide et à froid
Quantum simulation consists in engineering well-controlled artificial systems that are ruled by the idealized models proposed by the theorists. Such toy models can be produced with individual atoms, where laser beams control individual atomic states and interatomic interactions. In particular, exciting atoms into a highly excited state (called a Rydberg state) allows to control individual atoms and taylor interatomic interactions with light. In this thesis, we investigate experimentally two different types of Rydberg-based quantum simulators and identify some possible limitations.At the Joint Quantum Institute, we observe the decoherence of an ensemble of up to 40000 Rydberg atoms arranged in a cubic geometry. Starting from the atoms prepared in a well-defined Rydberg state, we show that the spontaneous apparition of population in nearby Rydberg states leads to an avalanche process. We identify the origin of the mechanism as stimulated emission induced by black-body radiation followed by a diffusion induced by the resonant dipole-dipole interaction. We describe our observations with a steady-state mean-field analysis. We then study the dynamics of the phenomenon and measure its typical timescales. Since decoherence is overall negative for quantum simulation, we propose several solutions to mitigate the effect. Among them, we discuss the possibility to work at cryogenic temperatures, thus suppressing the black-body induced avalanche.In the experiment at Laboratoire Charles Fabry (Institut d'Optique), we analyze the limitation of a quantum simulator based on 2 and 3 dimensional arrays of up to 70 atoms trapped in optical tweezers and excited to Rydberg states. The current system is limited by the lifetime of the atomic structure. We show that working at cryogenic temperatures could allow to increase the size of the system up to N=300 atoms. In this context, we start a new experiment based on a 4K cryostat. We present the early stage of the new apparatus and some study concerning the optomechanical components to be placed inside the cryostat

On étudie dans cette thèse le comportement en temps long de solutions d’équations aux dérivées partielles. Celles-ci modélisent des systèmes à grand nombre de particules dont la dynamique est due à des forces externes, internes et à l’interaction entre ces particules. Cependant, on considère différentes échelles. On voyage ainsi du niveau quantique des atomes au niveau macroscopique des étoiles, et l’on voit que des différences apparaissent bien que certaines propriétés soient conservées. Dans ce voyage, on croise le chemin de diverses applications telles que l’astrophysique, les plasmas,les semi-conducteurs, la biologie et l’économie. Ce travail est divisé en trois parties.Dans la première, on étudie le comportement semi-classique de l’équation de Hartree en mécanique quantique et sa limite vers l’équation de Vlasov. On quantifie uniformément en la constante de Planck des propriétés telles que la propagation des moments et de normes de Lebesgue à poids et la dispersion. On les utilise ensuite pour établir des estimées de stabilité entre les deux équations au moyen d’un analogue semi-classique des distances de Wasserstein. Dans la deuxième partie, on regarde le comportement en temps long d’équations cinétiques dont l’opérateur de collision est linéaire et a un équilibre local avec peu de moments, tel que l’opérateur de Fokker-Planck, sa version fractionnaire et un opérateur de Boltzmann linéaire. Deux principales techniques sont utilisées, l’une consistant à construire des entropies et la seconde à utiliser la positivité.Enfin, la dernière partie s’intéresse à des modèles macroscopiques inspirés de l’équation de Keller-Segel et l’on regarde les paramètres sous lesquels ce type de système s’effondre sur lui-même, se disperse ou se stabilise. Le premier effet se voit en introduisant des poids appropriés, le deuxième avec des distances de Wasserstein et le troisième au moyen des normes de Lebesgue
In this thesis, we study the behavior of solutions of partial differential equations that arise from the modeling of systems with a large number of particles. The dynamic of all these systems is driven by interaction between the particles and external and internal forces. However, we will consider different scales and travel from the quantum level of atoms to the macroscopic level of stars. We will see that differences emerge from the associated dynamics even though the main properties are conserved. In this journey, we will cross the path of various applications of these equations such as astrophysics, plasma, semi-conductors, biology, economy. This work is divided in three parts.In the first one, we study the semi classical behavior of the quantum Hartree equation and its limit to the kinetic Vlasov equation. Properties such as the propagation of moments and weighted Lebesgue norms and dispersive estimates are quantified uniformly in the Planck constant and used to establish stability estimates in a semiclassical analogue of the Wasserstein distance between the solutions of these two equations.In the second part, we investigate the long time behavior of macroscopic and kinetic models where the collision operatoris linear and has a heavy-tailed local equilibrium, such as the Fokker-Planck operator, the fractional Laplacian with a driftor a Linear Boltzmann operator. This let appear two main techniques, the entropy method and the positivity method.In the third part, we are interested in macroscopic models inspired from the Keller-Segel equation, and we study therange of parameters under which the system collapses, disperses or stabilizes. The first effect is studied using appropriate weights, the second using Wasserstein distances and the third using Lebesgue norms

Ce travail présente quelques résultats sur les systèmes de particules en interaction pour l'interprétation probabiliste des équations aux dérivées partielles, avec des applications à des questions de dynamique moléculaire et de chimie quantique. On présente notamment une méthode particulaire permettant d'analyser le processus de la force biaisante adaptative, utilisé en dynamique moléculaire pour le calcul de différences d'énergies libres. On étudie également la sensibilité de dynamiques stochastiques par rapport à un paramètre, en vue du calcul des forces dans l'approximation de Born-Oppenheimer pour rechercher l'état quantique fondamental de molécules. Enfin, on présente un schéma numérique basé sur un système de particules pour résoudre des lois de conservation scalaires, avec un terme de diffusion anormale se traduisant par une dynamique de sauts sur les particules

Periodically-driven (Floquet) quantum systems are ubiquitous in science and technology. For example, when a laser illuminates a material or an AC voltage is applied to a device, the system is well-described by a time-periodic Hamiltonian. In recent years, periodic driving has been proposed, not just as a tool to excite and probe devices, but actually as a mechanism of engineering new phases of matter, some of which have no equilibrium analog. However, with this promise comes a serious problem. Intuitively, if energy is injected into and distributed throughout a system, it is no surprise that it tends to heat up indefinitely to infinite temperature.

In this thesis, we study the mechanisms of heating, i.e. the process of thermalization, in Floquet systems and propose methods to control them. Specifically, for non-interacting Floquet systems that are coupled to external bosonic and fermionic baths (e.g. laser-driven electrons in a semiconductor that interact with phonons and an external lead), we classify the relevant scattering processes that contribute to cooling/heating in the Floquet bands and suggest methods to suppress heating via bandwidth-restrictions on the baths. We find that is possible, with appropriate dissipative engineering, to stabilize a controlled incompressible nonequilibrium steady-state resembling a ground state - a state we term the "Floquet insulator." We extend this analysis to include short-range interactions that contribute additional heating processes and show, under the same framework, that heating can be controlled with dissipation. In the process, we develop a simple effective model for the Floquet band densities that captures the essence of all the Floquet scattering processes and that is useful for ballparking experimentally-relevant estimates of heating. Next, we turn our attention to strongly-interacting closed Floquet systems and study how heating emerges through a proliferation of resonances. We find a novel integrable point governing the strong-interaction limit of the Floquet system and examine the breakdown of integrability via the proliferation of resonances. We observe two distinct scaling regimes, attributed to non-thermal and thermal behavior, and discover a power-law scaling of the crossover between them as a function of system size. The lingering ergodicity-breaking effects of the conserved quantities in the vicinity (in parameter space) of the integrable point at finite size is a phenomena we term "near-integrability." These results suggest that small quantum systems, which are accessible currently in many platforms (e.g. trapped ions, cold atoms, superconducting devices), intrinsically host non-thermal states that one may be able to utilize to avoid heating. Furthermore, our results suggest a "dual" interpretation, in the thermodynamic limit, that a periodically-driven system exhibits prethermalization as a power-law in interaction strength.

Recently there has been a lot of interest in the potential applications of performing computations on systems whose governing physical laws are quantum, rather than classical in nature. These quantum computers would have the ability to perform some calculations which would not be feasible for their classical counterparts. To date, however, a quantum computer large enough to perform useful calculations has yet to be built. Before this can be accomplished, a method must be developed to control the underlying quantum systems. This is a problem which can naturally be formulated in the language of control theory. This report outlines the basic control-theoretic approach to time-optimally controlling quantum systems evolving under the dynamics of the Schr\"{o}dinger operator equation. It is found that under the assumption of non-singularity, the controls which produce time-optimal trajectories are bang-bang. With this in mind, a switching time algorithm is implemented to find optimal bang-bang controls.

Impurities have a wide range of effects in interacting quantum many-body systems. They can interplay with interactions and lead to new electronic states of matter. They can also serve as a probe of an "intrinsic" many-body system. In this thesis, we consider the effects of impurities in three quantum many-body systems. First, we study the transport properties of a two-dimensional interacting electronic system with dilute quenched disorder. We find that the ground state is in fact a metallic state and in-plane magnetic-field can drive it to an insulating one. Second, we address the orthogonality catastrophe in Bose-Einstein condensate with a local impurity at its center. It is shown that the orthogonality effect in a Bose system has a stretched-exponential form, stronger than the algebraic orthogonality of a Fermi counterpart. The corresponding absorption spectrum is also determined. Finally, we analyze the effects of a spin resonance mode on the scattering tunneling microscopy(STM) spectra of a d-wave superconductor near a potential scattering center. We identify a counterintuitive two-unit-cell spatial modulation, at o ≃ +/-(Delta0 + O0)/h, where Delta 0 is the energy gap and O0 is the resonance mode energy. This effect can be tested by the Fourier-transformed STM technique.

Quantum chaos entails an entropic and computational obstruction to describing a system and thus is intrinsically difficult to characterize. An understanding of quantum chaos is fundamentally related to the mechanism of thermalization in many-body systems and the quantum nature of black holes. In this thesis we adopt the view that quantum information theory provides a powerful framework in which to elucidate chaos in strongly-interacting quantum systems.

We first push towards a more precise understanding of chaotic dynamics by relating different diagnostics of chaos, studying the time-evolution of random matrix Hamiltonians, and quantifying random matrix behavior in physical systems. We derive relations between out-of-time ordered correlation functions, spectral quantities, and frame potentials to relate the scrambling of quantum information, decay of correlators, and Haar-randomness. We give analytic expressions for these quantities in random matrix theory to explore universal aspects of late-time dynamics. Motivated by our random matrix results, we define k-invariance in order to capture the onset of random matrix behavior in physical systems.

We then refine our diagnostics in order to study chaotic systems with symmetry by considering Haar-randomness with respect to quotients of the unitary group, and in doing so we generalize our quantum information machinery. We further consider extended random matrix ensembles in the context of strongly-interacting quantum systems dual to black holes. Lastly, we study operator growth in classes of random quantum circuits.

The question whether and how closed quantum systems equilibrate is still debated today. In this thesis a generic spin system is analysed and criteria to classify unique equilibration dynamics are developed. Furthermore, the eigenstate thermalization hypothesis is investigated as a possible cause for the unique equilibrium. For both problems novel numerical methods for solving the time-dependent Schroedinger equation based on series expansions and typicality are developed. Furthermore, the problem of markovian dynamics on the level of single measurements is discussed.

abstract: In this dissertation two kinds of strongly interacting fermionic systems were studied: cold atomic gases and nucleon systems. In the first part I report T=0 diffusion Monte Carlo results for the ground-state and vortex excitation of unpolarized spin-1/2 fermions in a two-dimensional disk. I investigate how vortex core structure properties behave over the BEC-BCS crossover. The vortex excitation energy, density profiles, and vortex core properties related to the current are calculated. A density suppression at the vortex core on the BCS side of the crossover and a depleted core on the BEC limit is found. Size-effect dependencies in the disk geometry were carefully studied. In the second part of this dissertation I turn my attention to a very interesting problem in nuclear physics. In most simulations of nonrelativistic nuclear systems, the wave functions are found by solving the many-body Schrödinger equations, and they describe the quantum-mechanical amplitudes of the nucleonic degrees of freedom. In those simulations the pionic contributions are encoded in nuclear potentials and electroweak currents, and they determine the low-momentum behavior. By contrast, in this work I present a novel quantum Monte Carlo formalism in which both relativistic pions and nonrelativistic nucleons are explicitly included in the quantum-mechanical states of the system. I report the renormalization of the nucleon mass as a function of the momentum cutoff, an Euclidean time density correlation function that deals with the short-time nucleon diffusion, and the pion cloud density and momentum distributions. In the two nucleon sector the interaction of two static nucleons at large distances reduces to the one-pion exchange potential, and I fit the low-energy constants of the contact interactions to reproduce the binding energy of the deuteron and two neutrons in finite volumes. I conclude by showing that the method can be readily applied to light-nuclei.
Dissertation/Thesis
Doctoral Dissertation Physics 2018

The first part of the present Phd thesis is devoted to transport investigations in disordered quantum systems. We aim at quantitatively determining transport parameters like conductivity, mean free path, etc., for simple models of spatially disordered and/or percolated quantum systems in the limit of high temperatures and low fillings using linear response theory. We find the transport behavior for some models to be in accord with a Boltzmann equation, i.e., long mean free paths, exponentially decaying currents although there are no band-structures to start from, while this does not apply to other models even though they are also almost completely delocalized. The second part of the present PhD thesis addresses the issue of initial state independence (ISI) in closed quantum system. The relevance of the eigenstate thermalization hypothesis (ETH) for the emergence of ISI equilibration is to some extent addressed. To this end, we investigate the Heisenberg spin-ladder and check the validity of the ETH for the energy difference operator by examining the scaling behavior of the corresponding ETH-fluctuations, which we compute using an innovative numerical method based on typicality related arguments. While, the ETH turns out to hold for the generic non-integrable models and may therefore serve as the key mechanism for ISI for this cases, it does not hold for the integrable Heisenberg-chain. However, close analysis on the dynamic of substantially out-of-equilibrium initial states indicates the occurrence of ISI equillibration in the thermodynamic limit regardless of whether the ETH is violated. Thus, we introduce a new parameter $v$, which we propose as an alternative of the ETH to indicate ISI equillibration in cases, in which the ETH does not strictly apply.

Au-Yeung, Kin Chung = 以光學模擬量子自旋和機械鐘擺的相互作用 / 歐陽健聰.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2013.
Includes bibliographical references (leaves 80-81).
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Abstracts also in Chinese.
Au-Yeung, Kin Chung = Yi guang xue mo ni liang zi zi xuan he ji xie zhong bai de xiang hu zuo yong / Ouyang Jiancong.