Many Body Quantum Chaos

This editorial remembers Shmuel Fishman, one of the founding fathers of the research field "quantum chaos", and puts into context his contributions to the scientific community with respect to the twelve papers that form the special issue

Stability of Topological States and Crystalline Solids

From the alignment of magnets to the melting of ice, the transition between different phases of matter underpins our exploitation of materials. Both a quantum and a classical phase can undergo an instability into another state. In this thesis, we study the stability of matter in both contexts: topological states and crystalline solids. We start with the stability of fractional quantum Hall states on a lattice, known as fractional Chern insulators. We investigate, using exact diagonalization, fractional Chern insulators in higher Chern bands of the Harper-Hofstadter model, and examine the robustness of their many-body energy gap in the effective continuum limit. We report evidence of stable states in this regime; comment on two cases associated with a bosonic integer quantum Hall effect; and find a modulation of the correlation function in higher Chern bands. We next examine the stability of molecules using variational and diffusion Monte Carlo. By incorporating the matrix of force constants directly into the algorithms, we find that we are able to improve the efficiency and accuracy of atomic relaxation and eigenfrequency calculation. We test the performance on a diverse selection of case studies, with varying symmetries and mass distributions, and show that the proposed formalism outperforms existing restricted Hartree-Fock and density functional theory methods. Finally, we analyze the stability of three-dimensional crystals. We note that for repulsive Coulomb crystals of point nuclei, cubic systems have a zero matrix of force constants at second order. We investigate this by constructing an analytical model in the tight-binding approximation, and present a phase diagram of the most stable crystal structures, as we tune core and valence orbital radii. We reconcile our results with calculations in the nearly free electron regime, as well as current research in condensed matter and plasma physics.Funded by the Engineering and Physical Sciences Research Council under grant no. EP/M506485/1

Fermionic Quantum Information in Surface Acoustic Waves

Quantum computers are on the verge of revolutionising modern technology by providing scientists with unparalleled computational resources. With quantum-mechanical phenomena such as the superposition principle and entanglement, these computers could solve certain computational problems that are otherwise impossible for even the most powerful classical supercomputers. One of the major challenges standing in the way of this computing revolution is the accurate control of quantum bits. Quantum systems are extremely fragile and, by their nature, cannot be measured without destroying their quantum state. I wrote a numerical program to solve the time-dependent Schrödinger equation, the differential equation that describes the evolution of wave functions. The advantage of my code over other solvers is its speed. I used graphics processing units (GPUs), a technology that has only recently matured, to accelerate high-performance computing. Hardware- acceleration allows me to solve complex time-evolution problems within days rather than years. Such an exceptional speedup has enabled me to calculate the behaviour of single electrons in semiconductor devices. Electrons are particularly interesting because they are ubiquitous in modern technology, as well as being fundamental quantum particles. Using the simulations produced by my code, I track the time evolution of an electron wave function as it propagates along quantum circuits. By animating the evolution of the wave function, I am able to visualise the wave function of electrons propagating in space and time. This is a remarkable tool for studying the behaviour of quantum particles in nanodevices. I focused my thesis on the realistic modelling of devices that are readily available in a laboratory or on designs that could be fabricated in the near future. I began by modelling single electrons as quantum bits. I provide a definition for an optimal qubit and lay out the set of operations required to manipulate the quantum information carried by the electron. In all my simulations, I aim to model experimentally realistic devices. I calculated the electrostatic potential of a real nanodevice and simulated the time-evolution of a single electron. I show that it is possible to create a single-electron beam splitter by tuning the voltages applied to various parts of the device and I calculate the range of voltages in which quantum information is preserved and manipulated accurately. These results were verified experimentally by collaborators at the Institut Néel and were published in Nature Communications 10, 4557 (2019). Using my code, I developed a framework for general measurements of electron qubits and provided a design for a semiconductor device capable of performing positive-operator valued measures (POVMs). A POVM is a powerful measurement technique in quantum mechanics that allows quantum information to be manipulated in interesting ways. The proposed setup is suggested as an implementation of entanglement distillation, which is a useful error correction tool that transforms an arbitrary entangled state into a pure Bell pair. Entanglement is one of the most fascinating aspects of quantum mechanics and it remains a challenge to generate perfectly entangled particle pairs. An experimentally viable method for distilling – or perfecting – entanglement is crucial for the design of quantum computers or quantum communication systems. Using this design, I introduced a protocol to use electrons, rather than photons, in quantum-optics-like systems. These results were published in Phys. Rev. A 96, 052305 (2017). Going beyond single-particle behaviour, I compare different methods for generating entanglement between electron-spin qubits using the power-of-SWAP operation. By using realistic experimental parameters in my simulations, I demonstrate that generating entan- glement via electron-electron collisions in a harmonic channel cannot be implemented for multidimensional systems. These findings go against what researchers thought was possible and put forward the need for new solutions to particle entanglement. I provide an alternative by demonstrating that a method based on the exchange energy is more viable than previously thought. I present a semiconductor device structure and an electrostatic potential that experi- mental groups can use in order to obtain the most efficient entangling quantum logic gates. These findings were published in Phys. Rev. A 101, 022329 (2020). The results presented in this thesis provide a comprehensive description of the control of single electrons in a surface-acoustic-wave-based quantum circuit. However, work in this field is far from over. I present various research paths for future projects. These include going beyond the time-dependent Schrödinger equation to capture more complicated dynamics, using different hardware solutions to further accelerate numerical problem solving, and studying new systems of interest to extend this project beyond semiconductor physics.In all my simulations, I aim to model experimentally realistic devices. I calculated the electrostatic potential of a real nanodevice and simulated the time-evolution of a single electron. I show that it is possible to create a single-electron beam splitter by tuning the voltages applied to various parts of the device and I calculate the range of voltages in which quantum information is preserved and manipulated accurately. These results were verified experimentally by collaborators at the Institut Néel and were published in Nature Communications 10, 4557 (2019). Using my code, I developed a framework for general measurements of electron qubits and provided a design for a semiconductor device capable of performing positive-operator valued measures (POVMs). A POVM is a powerful measurement technique in quantum mechanics that allows quantum information to be manipulated in interesting ways. The proposed setup is suggested as an implementation of entanglement distillation, which is a useful error correction tool that transforms an arbitrary entangled state into a pure Bell pair. Entanglement is one of the most fascinating aspects of quantum mechanics and it remains a challenge to generate perfectly entangled particle pairs. An experimentally viable method for distilling – or perfecting – entanglement is crucial for the design of quantum computers or quantum communication systems. Using this design, I introduced a protocol to use electrons, rather than photons, in quantum-optics-like systems. These results were published in Phys. Rev. A 96, 052305 (2017). Going beyond single-particle behaviour, I compare different methods for generating entanglement between electron-spin qubits using the power-of-SWAP operation. By using realistic experimental parameters in my simulations, I demonstrate that generating entan- glement via electron-electron collisions in a harmonic channel cannot be implemented for multidimensional systems. These findings go against what researchers thought was possible and put forward the need for new solutions to particle entanglement. I provide an alternative by demonstrating that a method based on the exchange energy is more viable than previously thought. I present a semiconductor device structure and an electrostatic potential that experi- mental groups can use in order to obtain the most efficient entangling quantum logic gates. These findings were published in Phys. Rev. A 101, 022329 (2020). The results presented in this thesis provide a comprehensive description of the control of single electrons in a surface-acoustic-wave-based quantum circuit. However, work in this field is far from over. I present various research paths for future projects. These include going beyond the time-dependent Schrödinger equation to capture more complicated dynamics, using different hardware solutions to further accelerate numerical problem solving, and studying new systems of interest to extend this project beyond semiconductor physics.The Institute of Physics Horizon 2020 Marie Skłodowska Curie Actions Fonds de Recherche du Québec – Nature et technologies St Edmund’s College, Cambridge Canadian Imperial Bank of Commerce Canadian Centennial Scholarship Fund Institute of Engineering and Technolog

Quantum Magnetism, Spin Waves, and Light

Both magnetic materials and light have always played a predominant role in information technologies, and continue to do so as we move into the realm of quantum technologies. In this course we review the basics of magnetism and quantum mechanics, before going into more advanced subjects. Magnetism is intrinsically quantum mechanical in nature, and magnetic ordering can only be explained by use of quantum theory. We will go over the interactions and the resulting Hamiltonian that governs magnetic phenomena, and discuss its elementary excitations, denominated magnons. After that we will study magneto-optical effects and derive the classical Faraday effect. We will then move on to the quantization of the electric field and the basics of optical cavities. This will allow us to understand a topic of current research denominated Cavity Optomagnonics. These notes were written as the accompanying material to the course I taught in the Summer Semester 2018 at the Friedrich-Alexander University in Erlangen. The course is intended for Master or advanced Bachelor students. Basic knowledge of quantum mechanics, electromagnetism, and solid state at the Bachelor level is assumed. Each section is followed by a couple of simple exercises which should serve as to "fill in the blanks" of what has been derived, plus specific references to bibliography, and a couple of check-points for the main concepts developed. The figures are pictures of the blackboard taken during the lecture.Comment: Class notes, revised version, typos corrected, figures adde

Investigations on superradiant phases in Landau-quantized graphene

This thesis considers the effect of collective light-matter interaction of Landau-quantized charge carriers in graphene embedded in an optical cavity. Thereby, the focus is on the controversially discussed possible existence of an equilibrium superradiant quantum phase transition in this system. This quantum effect was initially investigated within the framework of the Dicke model but it has never been observed experimentally in equilibrium since then. This is due to the so called no-go theorem which prohibits the emergence of an equilibrium superradiant phase in systems with parabolic dispersion. However, there are no restrictions from similar arguments for systems with linear dispersion. Thus according to the remarkable properties of the band structure, graphene serves as an ideal candidate for theoretical and also experimental investigations on the equilibrium superradiant quantum phase transition. The quantum critical behavior of Landau-quantized graphene interacting with a single cavity mode is considered by means of two different analytical approaches within the framework of this thesis. The analytical results are partially underpinned by independent numerical tight-binding simulations of the system. For the analysis of the critical behavior a selection of characteristic observables is considered. Thereby, distinct signatures of a superradiant quantum phase transition are found. The analytic prediction of the critical coupling strength is in convincing agreement with the tight-binding simulation and tunable by means of the Fermi level and the magnetic field. The resulting phase diagram defines the relevant parameter range for which an equilibrium superradiant quantum phase is predicted

Etude mathématique de modèles quantiques et classiques pour les matériaux aléatoires à l'échelle atomique

Les contributions de cette thèse portent sur deux sujets.La première partie est dédiée à l'étude de modèles de champ moyen pour la structure électronique de matériaux avec des défauts.Dans le chapitre~ref{chap:ergodic_crystals}, nous introduisons et étudions le modèle de Hartree-Fock réduit (rHF) pour des cristaux désordonnés. Nous prouvons l'existence d'un état fondamental et établissons, pour les interactions de Yukawa (à courte portée), certaines propriétés de cet état. Dans le chapitre~ref{chap:défauts_étendus}, nous considérons des matériaux avec des défauts étendus. Dans le cas des interactions de Yukawa, nous prouvons l'existence d'un état fondamental, solution de l'équation auto-cohérente. Nous étudions également le cas de cristaux avec une faible concentration de défauts aléatoires. Dans le chapitre~ref{chap:numerical_simuation}, nous présentons des résultats de simulations numériques de systèmes aléatoires en dimension un.Dans la deuxième partie, nous étudions des modèles Monte-Carlo cinétique multi-échelles en temps. Nous prouvons, pour les trois modèles présentés au chapitre~ref{chap:kMC}, que les variables lentes convergent, dans la limite de la grande séparation des échelles de temps, vers une dynamique effective. Nos résultats sont illustrés par des simulations numériques.The contributions of this thesis concern two topics.The first part is dedicated to the study of mean-field models for the electronic structure of materials with defects. In Chapter~ref{chap:ergodic_crystals}, we introduce and study the reduced Hartree-Fock (rHF) model for disordered crystals. We prove the existence of a ground state and establish, for (short-range)Yukawa interactions, some properties of this ground state. In Chapter~ref{chap:défauts_étendus}, we consider crystals with extended defects. Assuming Yukawa interactions, we prove the existence of an electronic ground state, solution of the self-consistent field equation. We also investigate the case of crystals with low concentration of random defects. In Chapter~ref{chap:numerical_simuation}, we present some numerical results obtained from the simulation of one-dimensional random systems.In the second part, we consider multiscale-in-time kinetic Monte Carlo models. We prove, for the three models presented in Chapter~ref{chap:kMC}, that in the limit of large time-scale separation, the slow variables converge to an effective dynamics. Our results are illustrated by numerical simulations.CERGY PONTOISE-Bib. electronique (951279901) / SudocSudocFranceF

Octopus, a computational framework for exploring light-driven phenomena and quantum dynamics in extended and finite systems

Over the last few years, extraordinary advances in experimental and theoretical tools have allowed us to monitor and control matter at short time and atomic scales with a high degree of precision. An appealing and challenging route toward engineering materials with tailored properties is to find ways to design or selectively manipulate materials, especially at the quantum level. To this end, having a state-of-the-art ab initio computer simulation tool that enables a reliable and accurate simulation of light-induced changes in the physical and chemical properties of complex systems is of utmost importance. The first principles real-space-based Octopus project was born with that idea in mind, i.e., to provide a unique framework that allows us to describe non-equilibrium phenomena in molecular complexes, low dimensional materials, and extended systems by accounting for electronic, ionic, and photon quantum mechanical effects within a generalized time-dependent density functional theory. This article aims to present the new features that have been implemented over the last few years, including technical developments related to performance and massive parallelism. We also describe the major theoretical developments to address ultrafast light-driven processes, such as the new theoretical framework of quantum electrodynamics density-functional formalism for the description of novel light-matter hybrid states. Those advances, and others being released soon as part of the Octopus package, will allow the scientific community to simulate and characterize spatial and time-resolved spectroscopies, ultrafast phenomena in molecules and materials, and new emergent states of matter (quantum electrodynamical-materials)