Dissertations / Theses on the topic 'Reconnection of quantum vortices'

2010/2011
La presente tesi concerne la modellizzazione e simulazione numerica, attraverso l'equazione di Gross-Pitaevskii (chiamata anche equazione di Schroedinger non lineare), della dinamica dei vortici quantistici nell'elio superfluido e in particolare del fenomeno della riconnessione. La riconnessione si verifica qualora due vortici approssimativamente antiparalleli, si intersecano e si scambiano le estremità. Questo fenomeno è stato osservato sperimentalmente e risulta essere una caratteristica essenziale della turbolenza quantistica.
XXIII Ciclo
1983

Tableau d’honneur de la Faculté des études supérieures et postdoctorales, 2015-2016
Ce mémoire investigue le phénomène de reconnexion visqueuse de deux tourbillons initialement placés de façon orthogonale ou antiparallèle. Les équations de Navier-Stokes pour un fluide incompressible sont résolues directement (DNS) à l’aide d’un algorithme pseudospectral utilisant des expansions périodiques dans les trois directions cartésiennes. La condition de circulation nulle inhérente à cette méthode numérique est contournée en résolvant les équations dans un repère tournant approprié. Une méthode simple utilisant des lignes de vorticité est proposée afin de calculer le pourcentage de reconnexion instantané η de deux tourbillons. Cette méthode est également employée pour séparer le champ de vorticité en ses composantes reconnectées et non-reconnectées, ce qui facilite l’identification visuelle des différentes étapes du processus de reconnexion. Finalement, l’échelle de temps de la reconnexion tourbillonnaire Trec est calculée pour différents nombres de Reynolds (500 ≤ Re ≤ 10000). Il est trouvé que l’ordre de grandeur de Trec varie de façon continue de [symbol] à mesure que Re augmente.
This work focuses on the viscous reconnection phenomenon of two vortex tubes that are initially antiparallel or orthogonal to each other. The incompressible Navier-stokes equations are solved directly (DNS) using a Fourier pseudospectral algorithm with triply periodic boundary conditions. The associated zero-circulation constraint is circumvented by solving the governing equations in a proper rotating frame of reference. A simple method using vortex lines is proposed to compute the instantaneous reconnection level η of two vortices. The proposed method is also used to split the vorticity field into its reconnected and non-reconnected parts, which allows for a clear and intuitive visual identification of the different reconnection phases. Finally, the Reynolds number dependence of the reconnection timescale Trec is investigated for 500 ≤ Re ≤ 10000. The scaling is found to vary continuously as Re is increased from [symbol].

In this thesis, we numerically investigate quantum turbulence in trapped atomic Bose-Einstein condensates (BECs). We first discuss the appropriate qualitative characterization of turbulence in these systems, showing the limitation of analogies with classical hydrodynamics and turbulence in large superfluid Helium experiments. Due to their lack of available length scales, our investigated systems can only fit the ultraquantum (or Vinen) type of quantum turbulence. Secondly, we propose experimentally feasible schemes for more controlled investigations of turbulence making use of dynamical instability of multicharged vortices as an onset for complex vortex dynamics. In two dimensions, our suggested scheme allows control over vortex polarization in the harmonically trapped system. This setup is then used to study how turbulence decays in such a scenario, through the phenomenological modeling of a vortex-number rate equation. As a consequence, we were able to identify that vortex annihilation in these trapped systems happens through a four-vortex process. For three dimensions, we have first provided a study on the decay of a quadruply-charged vortex, also in a harmonically trapped BEC. Having this setting as a comparison point, we propose a quasi-isotropic turbulent system, starting from a phase-imprinted initial state of two doubly-charged, anti-parallel vortices. The vortex turbulence arisen from such configuration was shown to agree with the Vinen turbulent regime, after we characterized specific features of its decay, such as the energy spectrum [E(k) ∼ k1] and the time evolution of the vortex-line density [L(t) ∼ t1]. Although these features have been frequently verified in the context of superfluid Helium turbulence, here this identification was for the first time done for realistic, trapped atomic BECs.
Nesta tese, investigamos numericamente a turbulência quântica em condensados de Bose- Einstein (BECs) aprisionados. Discutimos, inicialmente, a caracterização qualitativa apropriada para estes sistemas, mostrando a limitação de analogias tipicamente feitas com hidrodinâmica clássica e turbulência em grandes sistemas com Hélio superfluido. Devido às suas limitadas escalas espaciais, os sistemas investigados somente podem exibir o tipo de turbulência conhecida como ultra-quântica (ou de Vinen). Em seguida, propomos sistemas experimentalmente factíveis que permitem investigações mais controladas da turbulência, fazendo uso da instabilidade dinâmica de vórtices multi-carregados como ponto de partida para geração de dinâmicas complexas. Em duas dimensões, nossa proposta permite controle sobre a polarização de vórtices em sistemas aprisionados em potencial harmônico. Este arranjo é então utilizado no estudo do decaimento da turbulência nesse contexto, através de um modelo fenomenológico para equação que descreve a taxa de variação do número de vórtices. Como consequência, pudemos verificar que a aniquilação de vórtices dá-se através de um processo que envolve quatro vórtices. Em três dimensões, apresentamos um estudo do decaimento de um vórtice de carga topológica quatro, também em potencial harmônico. Mantendo em mente esse sistema a título de comparação, propomos um cenário turbulento, quase-isotrópico, partindo de um estado inicial formado por dois vórtices duplamente carregados, mas orientados anti-paralelamente. Verificamos que a turbulência decorrente desse arranjo coincide com a regime de Vinen analisando características do seu decaimento, especificamente obtendo o espectro de energia [E(k) ∼ k1] e evolução temporal da densidade de linhas de vórtices [L(t) ∼ t1]. Apesar de que essas características são comumente encontradas no contexto de Hélio superfluido, apresentamos pela primeira vez essa identificação no cenário realístico de BEC aprisionados.

Quantum uids possess amazing properties of which two are particularly striking. Firstly they exhibit super uid ow, with the total absence of viscosity. Secondly, there are no excitations when the uid velocity (relative to some obstacle or surface) is slower than a critical value; above this velocity the ow becomes dissipative and macroscopic excitations are created in the form of quantised vortices with xed circulation proportional to Planck’s constant. In this thesis we numerically study the dynamics of quantum uids in the vicinity of obstacles and surfaces, from the production of a single vortex pair to the complex and chaotic motion of turbulent vortex tangles. This approach provides quantitative predictions for atomic Bose-Einstein condensates (BEC) and qualitative insight for super uid helium. We give detailed descriptions of the numerical schemes and present extensive numerical simulation of the Gross-Pitaevskii equation (GPE) and its variants at zero temperature and beyond, in both two and three dimensions. We study the wake that forms behind obstacles in the presence of a super uid ow, modelling atomic BEC experiments with moving laser-induced potentials, and explore the dependence on obstacle shape and size. We nd that suitable obstacles produce classicallike wakes consisting of clusters of vortices of the same polarity. Remarkably, symmetric wakes resemble those observed in classical viscous ow at low Reynolds number, despite the constrained vorticity. The structures are unstable, forming time-dependent asymmetric wakes similar to a classical Bénard–von Kármán vortex street. Motivated by the recentwork of Kwon et al. (Phys. Rev. A 90, 063627 (2014)), we model an atomic BEC experiment in which a trapped, oblate condensate is translated past a stationary, laser-induced obstacle. The critical velocity is exceeded and so vortices nucleate, forming a state of two-dimensional quantum turbulence. We explore the system at both zero-temperature and with thermal dissipation, modelled through a phenomenological term in the GPE. Our simulations provide insight into early stage evolution, not accessible experimentally, and into the decay of vortices by annihilation or passage out of the condensate. We use classical eld methods to simulate homogeneous Bose gases at nite temperature, from strongly non-equilibrium initial distributions to thermalised equilibrium states. We introduce a moving cylindrical potential and study how the thermal component of the gas a ects vortex nucleation. We have found that the critical velocity decreases with increasing temperature and scales with the speed of sound of the condensate. Above the critical velocity, vortices are nucleated as irregular vortex lines, rings, or vortex tangles. Finally we model the surfaces of walls and moving objects (such as wires, grids, propellers or spheres) in the presence of super uid ow, using a real rough boundary obtained via atomic force microscopy. We nd evidence pointing to the formation of a thin ‘super- uid boundary layer’ consisting of vortex loops and rings. As boundary layers usually arise from viscous forces, this is a surprising and intriguing result.

In this paper we discuss some of our most important results in quantum nanomagnets in the last twenty years. We start with the tunnelling of the magnetic moment in single domain particles, then we will move to molecular magnets to explain both resonant spin tunnelling and quantum magnetic deflagration and we will finish discussing the quantum phenomena recently observed in vortices of two dimensional disks and in type I superconductors. Probably the most important question to answer in the cases presented in this paper refers to the possibility to detect both coherent phonons and photons from the demagnetization process of molecular magnets as well as the fact to go deeper in the quantum phenomena observed in vortices of two dimensional disks and in type I superconductors. When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/35358

This thesis concerns the mathematical analysis of some hydrodynamic models describing quantum fluids, namely fluids whose macroscopic behavior still exhibits quantum effects. The prototype model for such fluids is the quantum hydrodynamic (QHD) system arising as model in the description of phenomena like superfluidity, Bose-Einstein condensation (BEC), superconductivity, quantum plasmas and semi conductor devices. From a mathematical point of view, the QHD system is given by a compressible Euler system augmented by a stress tensor accounting for the quantum features in the fluid and which depends on the density and its derivatives. Stress tensors of this kind also appear in the theory of capillarity developed by Korteweg, one refers to these systems as Euler–Korteweg and Navier–Stokes-Korteweg systems when inviscid or viscous respectively. Motivated by the analysis of some physically relevant solutions like quantized vortices, we consider the system on the whole space complemented with non-zero boundary conditions at infinity. The Cauchy problem for finite and infinite energy weak solutions (including vortex solutions) is investigated. Our method relies on the equivalence between QHD systems and NLS type equations through the Madelung transforms and the polar factorization method that renders the equivalence rigorous for rough solutions. We are thus led to study the well-posedness theory in the energy space for a class of nonlinear Schrödinger equations with non-zero boundary conditions at infinity that we develop ad-hoc. Moreover, we consider the asymptotic behavior of weak solutions to the QHD system in a suitable scaling regime that is linked to the study of quantized vortices and can be interpreted as quantum counterpart of the low Mach number limit of classical fluid dynamics. The dispersion relation of acoustic waves turns out to be characterized by the Bogoliubov dispersion relation. In the scaling regime, the dynamics of vortex solutions can be asymptotically described by the Kirchhoff-Onsager ODE system. Secondly, we study the quantum Navier-Stokes (QNS) equations that can be understood as a viscous regularization of the QHD system with density dependent viscosity tensor. Physically, it is motivated by applications in the modeling of dissipative quantum fluids and as a showcase model for capillary fluids. We introduce global existence of finite energy weak solutions of the quantum Navier-Stokes system with non-trivial far-field behavior. In contrast to the results for the QHD system, the analysis of the QNS system is entirely based on techniques from fluid dynamics. Finally, we investigate the low Mach number limit and prove strong convergence to weak solutions of the incompressible Navier-Stokes equations for general ill-prepared data.

Optical vortices represent a particular class of wavefront dislocations characterized by a topological charge l. The surface of constant phase of an electromagnetic wave carrying an optical vortex has a helical structure and presents a singularity along the axis of the helicoid, where the phase is undefined. As a consequence, the intensity distribution of a vortex light beam contains a central dark region, where the intensity is zero due to destructive interference. Optical vortices can be produced by using phase modifying devices, i.e. particular optical elements possessing an optical singularity generally located at their center. The most efficient of them are fork holograms and spiral phase plates. In the last decade, the properties of optical vortices have found interesting applications in optical physics. Among these, the most promising are those in optical communications, nanotechnologies and biology. Optical vortices are attracting increasing attention also in astronomy, where the properties of such features of the electromagnetic radiation could provide a new approach to the study of astrophysical phenomena. The purpose of this Thesis is to present some possible applications of optical vortices to instrumental astronomy. In particular, this work is focused on the development and testing of new techniques to improve the performances of optical systems. Firstly, a method is proposed to improve the resolving power of a diffraction-limited telescope by means of an l = 1 fork hologram. Both the experiments and numerical simulations reveal that the superposition of the optical vortices produced by two light beams characterized by equal Airy intensity distributions present a detectable asymmetry even for separations that are one order of magnitude below the limit of the Rayleigh criterion. It is shown that this result can be achieved both with monochromatic and white light beams. We then present the first astronomical experiment in which we produced optical vortices in starlight beams with an l = 1 fork hologram placed at the focal plane of the Galileo 122 cm telescope in Asiago. By using the Lucky Imaging approach to reduce the effects of mediocre seeing conditions, we were able to observe the images of the optical vortices produced by the two main components of the multiple system alpha Her, in non-monochromatic light, and by the single star alpha Boo, using a narrow bandpass. In both cases, the intensity profiles of the observed optical vortices are in agreement with numerical simulations. Detailed analytical models and numerical simulations confirm that the spatial structure of an optical vortex produced by a phase modifying device is extremely sensitive to off-axis displacements of the input beam, especially when high values of the topological charge l are used. This property could be used to perform ground-based astrometric measurements with a precision competitive to standard PSF-fitting astrometry. The sensitivity to small off-axis displacements might also help to improve the tip/tilt correction of the wavefront for a small field of view. We discuss also the possible application of the nulling property of even-charged optical vortices to perform high-contrast coronagraphy. In this case, the nulling of the light of an on-axis star is achieved by using a spiral phase plate and a circular diaphragm as Lyot stop. In principle, this coronagraphic design is one of the very few that might allow direct imaging of extrasolar terrestrial planets. However, such remarkable performance is still strongly limited by the current techniques used to manufacture spiral phase plates. In the framework of projecting an l = 2 optical vortex coronagraph for visible wavelengths, we present the results of numerical simulations obtained considering a spiral phase plate with a surface subdivided in N discrete levels. A description of the experimental procedures used to test spiral phase plates manufactured with PMMA (polymethyl methacrylate) material is also given.
I vortici ottici rappresentano una particolare classe di dislocazioni dei fronti d'onda caratterizzate da una carica topologica l. La superficie di fase costante di un'onda elettromagnetica che trasporta un vortice ottico ha una struttura elicoidale. Lungo l'asse di questa elica è presente una singolarità in cui la fase non può essere definita. Di conseguenza, la distribuzione d'intensità di un fascio di luce contenente un vortice ottico presenta una zona centrale dove l'intensità è nulla per effetto dell’interferenza distruttiva. I vortici ottici possono essere prodotti utilizzando particolari elementi ottici detti phase modifying devices che modificano la fase di un'onda incidente. I più efficienti tra questi sono i fork holograms (ologrammi) e le spiral phase plates (maschere di fase). Negli ultimi anni, le proprietà dei vortici ottici hanno trovato interessanti applicazioni nei campi della fisica e dell'ottica. Tra queste, le più promettenti sono quelle in comunicazioni ottiche, nelle nanotecnologie ed in biologia. Recentemente, i vortici ottici stanno suscitando un crescente interesse anche nella comunità astronomica. Infatti, queste particolari proprietà della radiazione elettromagnetica potrebbero permettere di studiare diversi fenomeni astrofisici da un punto di vista completamente nuovo. In questa Tesi vengono presentate alcune possibili applicazioni dei vortici ottici in strumentazione astronomica. In particolare, lo scopo principale di questo lavoro è lo sviluppo di nuove tecniche che permetteranno di migliorare le prestazioni di sistemi ottici. In primo luogo, viene proposto un metodo per aumentare il potere risolutivo di un telescopio limitato dalla diffrazione che prevede l'utilizzo di un fork hologram con una singola dislocazione. I risultati di esperimenti e simulazioni numeriche rivelano che la sovrapposizione dei vortici ottici prodotti da due fasci di luce con una distribuzione d'intensità di Airy mostra già un'evidente asimmetria quando la separazione è di un ordine di grandezza inferiore rispetto al limite posto dal criterio di Rayleigh. Questo risultato è stato ottenuto sia in luce monocromatica, sia in luce bianca. Viene poi presentato il primo esperimento astronomico in cui sono stati prodotti vortici ottici con un fork hologram avente una singola dislocazione posto al piano focale del telescopio Galileo da 122 cm di Asiago. Utilizzando i principi del Lucky Imaging per ridurre gli effetti provocati da condizioni di seeing mediocre, sono state osservate le immagini dei vortici ottici prodotti dalle due componenti principali del sistema multiplo alfa Her, in luce non monocromatica, e dalla stella singola alfa Boo, utilizzando un filtro spaziale a banda stretta. In entrambi i casi, i profili d’intensità dei vortici ottici osservati sono riproducibili con simulazioni numeriche. La sensibilità dell'immagine di un vortice ottico prodotto con un phase modifying device rispetto a spostamenti fuori asse del fascio entrante è confermata da modelli analitici dettagliati e anche da simulazioni numeriche, specialmente nel caso in cui vengano utilizzati elevati valori della cariche topologica l. Questa proprietà potrebbe essere utilizzata per fare misure astrometriche da terra con una precisione che potrebbe competere con quella fornita dalle tecniche standard di astrometria di PSF. La sensibilità rispetto a piccoli spostamenti fuori asse potrebbe anche essere sfruttata per migliorare la correzione dell’aberrazione di tip/tilt del fronte d'onda in un piccolo campo di vista. Viene poi discussa la possibile applicazione di vortici ottici con carica topologica pari nella coronografia ad alto contrasto. In questo caso, l'azione combinata di una spiral phase plate e di un diaframma circolare utilizzato come stop di Lyot permette di annullare totalmente la luce di una stella in asse. Studi teorici indicano che il coronografo a vortici ottici è uno dei pochi che potrebbe realmente permettere l'osservazione diretta di pianeti extrasolari di tipo terrestre. Purtroppo, questa notevole proprietà è fortemente limitata dalle attuali tecniche usate per produrre le spiral phase plate. Nell'ambito di un progetto di costruzione di un coronografo a vortici ottici con l = 2 ottimizzato per lunghezze d'onda visibili, vengono presentati i risultati di simulazioni numeriche ottenuti considerando una spiral phase plate la cui superficie è suddivisa in N livelli discreti. Infine, vengono discusse le procedure sperimentali utilizzate per testare spiral phase plates in PMMA (polimetil-metacrilato).

This dissertation presents several experimental methods that were devised to generate or manipulate quantized vortices in highly oblate dilute-gas Bose-Einstein condensates (BECs). Studies that involve single vortex dynamics, vortex-vortex interactions, and vortex-impurity interactions are essential in developing a deeper understanding of the nature of superfluidity and in particular, superfluid turbulence. In highly oblate systems, vortex dynamics have a two-dimensional (2D) nature and the resulting superfluid characteristics may be substantially different from those in three-dimensional (3D) superfluids. However, there have been remarkably few experimental studies of 2D vortex dynamics in superfluids. Therefore, to study 2D vortex dynamics and interactions, it is necessary to first develop experimental methods that can generate vortices and vortex distributions in nominally 2D systems, such as highly oblate BECs. Four main experiments are discussed in this dissertation. Two of these experiments generate multiple singly quantized vortices in a relatively stochastic manner leading to disordered vortex distributions. From these two vortex methods, the physics of high vorticity and highly disordered systems may be observed and studied in a highly oblate system. These methods may prove useful in studies of 2D quantum turbulence. The other two experiments involve newly developed techniques for controlled generation and manipulation of vortices. One of these methods creates multiply quantized pinned vortices with a control in the generated vorticity. The other method reliably creates a pair of singly quantized vortices of opposite circulation, whose positions can be easily manipulated after creation, such that they can be placed in any location within the BEC. The two techniques may be scalable to higher number of vortices and may prove useful in superfluid dynamics and vortex interactions that require repeatable vortex distributions. Taken together, these tools and methods may be applicable to many further studies of vortex physics in highly oblate BECs.

The aim of this thesis was to study quantum phenomena and excitation modes in type-I superconductors and magnetic vortices. The intermediate state in type-I superconductors is characterized by the gradual penetration of magnetic flux and the coexistence of normal and superconducting domains. This thermodynamic phase shows a magnetic irreversibility of topological origin, even in the case of defect-free samples. This irreversibility has been explored in disk-shaped samples made of lead (the prototype of a type-I superconductor) using a magnetic field applied perpendicularly to the disk plane, by means of the measurement of hysteresis cycles at different temperatures, zero-field-cooled and field-cooled magnetization curves at different magnetic fields and magnetic relaxations along the descending branch of the hysteresis cycles. Non-thermal magnetic relaxations have been observed in these samples at low temperatures, which have been attributed to the tunnel effect of normal-superconductor interfaces through pinning energy barriers. A quantum model based on the Caldeira-Leggett theory for dissipative systems have been developed to explain these experimental observations. The interface is described as a 2D elastic manifold that is pinned by a planar defect. The pinning barrier can be controlled by a supercurrent that exerts a force on the interface. The vortex state turns out to be the ground state of magnetic disks for a wide variety of thicknesses and radii. It is characterized by the curling of the magnetization in the plane of the disk, leaving virtually no magnetic ‘charges’. The very weak uncompensated magnetic moment of the disk sticks out of a small area confined to the vortex core. The low-frequency dynamics of the vortex state is characterized by the spiral-like precessional motion of the vortex core as a whole (gyrotropic mode), which can be induced by the application of an in-plane magnetic field. The presence of structural defects in these magnetic disks affects the dynamics of the vortex state, which is indicative of the elastic nature of the vortex core along the axial direction of the disk. It has been studied whether the gyrotropic mode allows spatial dispersion similar to spin waves of a finite wavelength in ferromagnets. The excitation spectrum splits into two branches, one related to the gyrotropic mode with a gap given by the gyrofrequency of the disk and the other related to the existence of an effective mass associated to the vortex core. The magnetic irreversibility of the vortex state has been also explored by means of an experimental protocol analogous to that used in the case of type-I superconductors. Non-thermal magnetic relaxations have been observed again at low temperatures, which is attributed to the tunnel effect of a segment of the vortex core line through pinning barriers. A quantum model based on the Caldeira-Leggett theory for dissipative systems have been developed to explain these experimental observations. The interface is described as a 1D elastic manifold that is pinned by a linear defect. To conclude, the effect of the vortex state on the supercurrent of a Josephson junction has been studied in the case where the non-superconducting layer consists of a magnetic disk with the vortex as the ground state. It has been concluded that the variation of the Josephson current with tiny displacements of the vortex core can be detected experimentally.
El objetivo de esta tesis ha sido estudiar fenómenos cuánticos y modos de excitación en superconductores tipo-I y vórtices magnéticos. La irreversibilidad magnética en muestras de plomo con forma de disco en el estado intermedio ha sido explorada mediante medidas de ciclos de histéresis a diferentes temperaturas, medidas de las curvas de magnetización zero-field-cooled y field-cooled a diferentes campos y relajaciones magnéticas a lo largo de la rama descendiente de los ciclos de histéresis. Se han observado relajaciones magnéticas independientes de la temperatura en estas muestras, las cuales se atribuyen al efecto túnel de las interficies normal-superconductor a través de barreras de anclaje. Un modelo de efecto túnel basado en la teoría de Caldeira-Leggett para sistemas disipativos se ha construido para explicar estas observaciones experimentales, donde la interfície se trata como una variedad 2D elástica que se ancla a defectos planares. La barrera de anclaje se puede controlar mediante la inyección de supercorriente en el sistema. El núcleo del estado vórtice muestra una naturaleza elástica a lo largo de la dirección axial de los discos magnéticos que lo presentan como estado fundamental. Se ha estudiado bajo qué condiciones el modo girótropo es compatible con una dispersión espacial semejante a las ondas de espín de longitud de onda finita presentes en un ferromagneto. El espectro de excitaciones axiales presenta dos ramas bien definidas, una asociada al modo girótropo y la otra originada por la existencia de una masa efectiva asociada al núcleo. También se ha explorado la irreversibilidad magnética del estado vórtice mediante un protocolo análogo al de los superconductores tipo-I. De nuevo se ha observado un comportamiento no térmico a bajas temperaturas en las relajaciones magnéticas, el cual es atribuido al efecto túnel de un segmento del núcleo vorticial a través de las barreras de anclaje. Un modelo de efecto túnel basado en la teoría de Caldeira-Leggett para sistemas disipativos se ha construido para explicar estas observaciones experimentales, donde el núcleo vorticial se trata como una variedad 1D elástica anclada a un defecto lineal. Por último, se ha estudiado cuál sería el efecto del estado vórtice sobre la supercorriente de una unión Josephson si como capa no superconductora se escogiera un disco magnético con este estado fundamental. Se ha concluido que la variación de la corriente Josephson con desplazamientos pequeños del núcleo vorticial es detectable experimentalmente.

Turbulence in classical fluids has been the subject of scientific study for centuries, yet there is still no complete general theory of classical turbulence connecting microscopic physics to macroscopic fluid flows, and this remains one of the open problems in physics. In contrast, the phenomenon of quantum turbulence in superfluids has well-defined theoretical descriptions, based on first principles and microscopic physics, and represents a realm of physics that can connect the classical and quantum worlds. Studies of quantum turbulence may thus be viewed as a path for progress on the long-standing problem of turbulence.A dilute-gas Bose-Einstein condensate (BEC) is, in most cases, a superfluid that supports quantized vortices, the primary structural elements of quantum turbulence. BECs are particularly convenient systems for the study of vortices, as standard techniques allow the microscopic structure and dynamics of the vortices to be investigated. Vortices in BECs can be created and manipulated using a variety of techniques, hence BECs are potentially powerful systems for the microscopic study of quantum turbulence.This dissertation focuses on quantized vortices in BECs, specifically experimental and numerical studies of their formation, dynamics, and decay, in an effort to understand the microscopic nature of vortices as elements of quantum turbulence. Four main experiments were performed, and are described in the main chapters of this dissertation, after introductions to vortices, experimental methods, and turbulence are presented. These experiments were aimed at understanding various aspects of how vortices are created and behave in a superfluid system. They involved vortex dipole nucleation in the breakdown of superfluidity, persistent current generation from a turbulent state in the presence of energy dissipation, decay of angular momentum of a BEC due to trapping potential impurities, and exploration of the spontaneous formation of vortices during the BEC phase transition. These experiments represent progress towards enhanced understanding of the formation, dynamics, and decay of vortices in BECs and thus may be foundational to more general studies of quantum turbulence in superfluids.

This thesis is mainly concerned with theoretical studies of two types of models:  quantum mechanical Bose-Hubbard models and (semi-)classical discrete nonlinear Schrödinger (DNLS) models. Bose-Hubbard models have in the last few decades been widely used to describe Bose-Einstein condensates placed in periodic optical potentials, a hot research topic with promising future applications within quantum computations and quantum simulations. The Bose-Hubbard model, in its simplest form, describes the competition between tunneling of particles between neighboring potential wells (`sites') and their on-site interactions (can be either repulsive or attractive). We will also consider extensions of the basic models, with additional interactions and tunneling processes. While Bose-Hubbard models describe the behavior of a collection of particles in a lattice, the DNLS description is in terms of a classical field on each site. DNLS models can also be applicable for Bose-Einstein condensates in periodic potentials, but in the limit of many bosons per site, where quantum fluctuations are negligible and a description in terms of average values is valid. The particle interactions of the Bose-Hubbard models become  nonlinearities in the DNLS models, so that the DNLS model, in its simplest form, describes a competition between on-site nonlinearity and tunneling to neighboring sites. DNLS models are however also applicable for several other physical systems, most notably for nonlinear waveguide arrays, another rapidly evolving research field. The research presented in this thesis can be roughly divided into two parts: 1) We have studied certain families of solutions to the DNLS model. First, we have considered charge flipping vortices in DNLS trimers and hexamers. Vortices represent a rotational flow of energy, and a charge flipping vortex is one where the rotational direction (repeatedly) changes. We have found that charge flipping vortices indeed exist in these systems, and that they belong to continuous families of solutions located between two stationary solutions. Second, we have studied discrete breathers, which are spatially localized and time-periodic solutions, in a DNLS models with the geometry of a ring coupled to an additional, central site. We found under which parameter values these solutions exist, and also studied the properties of their continuous solution families. We found that these families undergo different bifurcations, and that, for example, the discrete breathers which have a peak on one and two (neighboring) sites, respectively, belong to the same family below a critical value of the ring-to-central-site coupling, but to separate families for values above it. 2) Since Bose-Hubbard models can be approximated with DNLS models in the limit of a large number of bosons per site, we studied signatures of certain classical solutions and structures of DNLS models in the corresponding Bose-Hubbard models. These studies have partly focused on quantum lattice compactons. The corresponding classical lattice compactons are solutions to an extended DNLS model, and consist of a cluster of excited sites, with the rest of the sites exactly zero (generally localized solutions have nonzero `tails'). We find that only one-site classical lattice compactons remain compact for the Bose-Hubbard model, while for several-site classical compactons there are nonzero probabilities to find particles spread out over more sites in the quantum model. We have furthermore studied the dynamics, with emphasize on mobility, of quantum states that correspond to the classical lattice compactons. The main result is that it indeed is possible to see signatures of the  classical compactons' good mobility, but that it is then necessary to give the quantum state a `hard kick' (corresponding to a large phase gradient). Otherwise, the time scales for quantum fluctuations and for the compacton to travel one site become of the same order. We have also studied the quantum signatures of a certain type of instability (oscillatory) which a specific solution to the DNLS trimer experiences in a parameter regime. We have been able to identify signatures in the quantum energy spectrum, where in the unstable parameter regime the relevant eigenstates undergo many avoided crossings, giving a strong mixing between the eigenstates. We also introduced several measures, which either drop or increase significantly in the regime of instability. Finally, we have studied quantum signatures of the charge flipping vortices mentioned above, and found several such, for example when considering the correlation of currents between different sites.

Thesis (Ph.D.); Submitted to the Department of Physics and Astronomy, Rice University, Houston, TX (US); 1 Sep 2004.
Published through the Information Bridge: DOE Scientific and Technical Information. "LA-14149-T" J. Koller. US DOE (US) 09/01/2004. Report is also available in paper and microfiche from NTIS.

Cette thèse est consacrée à la description de la physique à une particule ainsi qu'à celle de fluides quantiques bosoniques dans des systèmes topologiques. Les deux premiers chapitres sont introductifs. Dans le premier, nous introduisons des éléments de théorie des bandes et les quantités géométriques et topologiques associées : tenseur métrique quantique, courbure de Berry, nombre de Chern. Nous discutons différents modèles et réalisations expérimentales donnant lieu à des effets topologiques. Dans le second chapitre, nous introduisons les condensats de Bose-Einstein ainsi que les excitons-polaritons de cavité.La première partie des résultats originaux discute des phénomènes topologiques à une particule dans des réseaux en nid d'abeilles. Cela permet de comparer deux modèles théoriques qui mènent à l'effet Hall quantique anormal pour les électrons et les photons dû à la présence d'un couplage spin-orbite et d'un champ Zeeman. Nous étudions aussi l'effet Hall quantique de vallée photonique à l'interface entre deux réseaux de cavités avec potentiels alternés opposés.Dans une seconde partie, nous discutons de nouveaux effets qui émergent due à la présence d'un fluide quantique interagissant décrit par l’équation de Gross-Pitaevskii dans ces systèmes. Premièrement, il est montré que les interactions spin anisotropes donnent lieu à des transitions topologiques gouvernées par la densité de particules pour les excitations élémentaires d’un condensat spineur d’exciton-polaritons.Ensuite, nous montrons que les tourbillons quantifiés d'un condensat scalaire dans un système avec effet Hall quantique de vallée, manifestent une propagation chirale le long de l'interface contrairement aux paquets d'ondes linéaires. La direction de propagation de ces derniers est donnée par leur sens de rotation donnant lieu à un transport de pseudospin de vallée protégé topologiquement, analogue à l’effet Hall quantique de spin.Enfin, revenant aux effets géométriques linéaires, nous nous sommes concentrés sur l’effet Hall anormal. Dans ce contexte, nous présentons une correction non-adiabatique aux équations semi-classiques décrivant le mouvement d’un paquet d’ondes qui s’exprime en termes du tenseur géométrique quantique. Nous proposons un protocole expérimental pour mesurer cette quantité dans des systèmes photonique radiatifs
This thesis is dedicated to the description of both single-particle and bosonic quantum fluid Physics in topological systems. After introductory chapters on these subjects, I first discuss single-particle topological phenomena in honeycomb lattices. This allows to compare two theoretical models leading to quantum anomalous Hall effect for electrons and photons and to discuss the photonic quantum valley Hall effect at the interface between opposite staggered cavity lattices.In a second part, I present some phenomena which emerge due to the interplay of the linear topological effects with the presence of interacting bosonic quantum fluid described by mean-field Gross-Pitaevskii equation. First, I show that the spin-anisotropic interactions lead to density-driven topological transitions for elementary excitations of a condensate loaded in the polariton quantum anomalous Hall model (thermal equilibrium and out-of-equilibrium quasi-resonant excitation configurations). Then, I show that the vortex excitations of a scalar condensate in a quantum valley Hall system, contrary to linear wavepackets, can exhibit a robust chiral propagation along the interface, with direction given by their winding in real space, leading to an analog of quantum spin Hall effect for these non-linear excitations. Finally, coming back to linear geometrical effects, I will focus on the anomalous Hall effect exhibited by an accelerated wavepacket in a two-band system. In this context, I present a non-adiabatic correction to the known semiclassical equations of motion which can be expressed in terms of the quantum geometric tensor elements. We also propose a protocol to directly measure the tensor components in radiative photonic systems

L'étude des vortex trouve sa justification dans le rôle que ces derniers jouent dans la turbulence quantique. L'équation de Gross-Pitaevskii ne peut pas nous permettre de modéliser convenablement l'Hélium superfluide, mais on peut l'utiliser pour obtenir le paramètre d'ordre d'un superfluide modèle, ayant le maximum de propriétés en commun avec l'Hélium, notamment une courbe de dispersion identique, par la modification du terme d'interactions.En supposant que le minimum roton influence l'essentiel de la physique, on détermine la forme du paramètre d'ordre loin de la perturbation créée par le vortex rectilinéaire axisymétrique par deux approches différentes - il apparaît alors que seuls deux paramètres sont nécessaires pour caractériser entièrement le profil.Le modèle proposé par Pomeau-Rica, qui offre la possibilité d'étudier le superfluide près de la cristallisation, met en lumière l'impact de la profondeur du minimum roton sur l'amplitude des oscillations. Par comparaison avec les résultats obtenus ab initio par Reatto, les résultats donnés par le modèle de Berloff-Roberts exhibent un déphasage marqué, qui semble être une conséquence non-physique de la forme du spectre d'excitation. Les calculs énergétiques laissent à penser que les oscillations portent une faible fraction de l'énergie du vortex, l'énergie cinétique dominant.Le calcul du paramètre d'ordre est effectué pour un anneau de grande taille par rapport à la distance interatomique, à vitesse nulle et à vitesse non-nulle. La détermination des énergies potentielle et cinétique permet d'accéder à la vitesse maximale atteinte par l'anneau en fonction de son rayon et de la comparer à la vitesse critique de Landau.

A extensão dos fenômenos quânticos em escala macroscópica é responsável por toda uma classe de efeitos como a supercondutividade, superfluidez, e condensação de Bose-Einstein, as quais desempenham um papel central na física ao longo do século passado. A produção dos primeiros condensados de Bose-Einstein tornou possível a realização de experimentos envolvendo fenômenos quânticos macroscópicos com um nível sem precedentes de controle dos parâmetros externos. As correntes persistentes em condensados estão intimamente relacionados com a nucleação de vórtices quantificados, que são defeitos topológicos como resposta à transferência de quanta de momento angular. Um método convencional para geração de tais defeitos consiste em confinar a nuvem atômica condensada em uma armadilha com rotação. Acontece que, para velocidades angulares acima de um valor crítico, estados de vórtice se tornam energeticamente favoráveis, induzindo assim a criação de vórtices quânticos. Realizações experimentais de condensados de átomos de metais alcalinos confinados por potenciais dependentes do tempo permitiram a observação não só de redes de vórtices, mas também de turbulência quântica. Uma vez que a turbulência quântica é caracterizada pela presença de um emaranhado de vórtices quânticos interagindo entre si, uma correta compreensão da dinâmica, formação e estabilidade de vórtices tem se mostrado de grande importância sendo objeto de muitos trabalhos teóricos. Em particular, o papel das excitações acústicas geradas pelo decaimento de vórtices de multipla carga no desenvolvimento de turbulência ainda é uma questão em aberto. Este trabalho tem como objetivo fornecer um conjunto de ferramentas que ajude a identificar a presença, como também a carga de vórtices em nuvens (não turbulentas) observadas utilizando imagens de tempo-de-voo. Temos feito um estudo detalhado de condensados contendo vórtices carga múltipla colocados no seu centro, onde a dinâmica do tempo-de-voo é apenas de nossos pontos de interesse. Devido ao controle que este sistema fornece experimentalmente, os modos coletivos tornam-se uma descrição importante, uma vez que podem ser excitadas usando métodos experimentais bem estabelecido tal como a modulação do comprimento de espalhamento de ondas-s, e que também pode ser responsável pelo decaimento do vórtice. Para tais fins, temos utilizado o método variacional (semi-analítico), e o cálculo totalmente numérico da equação de Gross-Pitaevskii. Assim, descrevemos os modos coletivos que acoplam a dinâmica do vórtice com as oscilações das componentes externas do condensado, bem como os efeitos em tempo-de-voo. O momento angular atua aumentando a energia cinética em torno do núcleo de vórtice, que implica em um aumento mais rápido da direção perpendicular a este. Esta situação desloca as freqüências de oscilações coletivas de um estado livre de vórtice, e gera modos coletivos mais ricos devido ao acoplamento. Agora, existem quatro modos possíveis, sendo dois tipos de modo monopolar e dois tipos de modos de quadrupolo. A diferença dentre tais modos é a fase de oscilação do vórtice. Quando se considera flutuações sem simetria polar, seus modos coletivos resultam no decaimento do vórtice. A fim de controlar e prevenir estes processos propusemos três mecanismos dinâmicos, tais como a modulação de comprimento de espalhamento, a modulação das frequências da armadilha harmônica e modulação da amplitude do potencial de Laguerre-Gauss. O último tem provado ser mais eficaz.
The extension of quantum phenomena into macroscopic scales is responsible for a whole class of effects such as superconductivity, superfluidity, and Bose-Einstein condensation, which played central roles in physics throughout the last century. The production of the first Bose-Einstein condensates made possible the realization of experiments involving macroscopic quantum phenomena with an unprecedented level of control of the external parameters. The persistent currents in condensates are intimately related to the nucleation of quantized vortices, which are topological defects as response to transference of quanta of angular momentum. A conventional method for generation of such defects consists in confining the condensed atomic cloud into a rotating trap. It turns out that, for angular velocities higher than a critical value, vortex states become energetically favorable, thus inducing the creation of quantized vortices. Experimental realizations of condensed alkali-metal atoms confined by more general time-dependent potentials allowed the observation not only of vortex lattices but also of quantum turbulence. Since quantum turbulence is characterized by the presence of a self-interacting tangle of quantized vortices, the correct understanding of dynamics, formation, and stability of vortices has shown to be of paramount importance being the subject of many theoretical works. In particular, the role of acoustic excitations generated by decaying multi-charged vortices in the development of turbulence is still an open question. This work aims to provide a set of tools that helps to identify the presence as well as the charge of vortices in non-turbulent clouds observed using time-of-flight pictures. We have done a detailed study of condensates containing multi-charged vortices placed at its center where time-of-flight dynamics is only one point of our interest. Due to the control that this system provides experimentally, the collective modes become an important description since they can be excited using well stablished experimental methods as such as modulation of the s-wave scattering length, and they can also be responsible to vortex decaying. For such purposes we have used the semi-analytical variational method, and the fully numerical calculation of Gross-Pitaevskii equation. Thus we have describes the collective modes that couples dynamics of vortex with the oscillation of external components of condensed atomic cloud as well as the effects in time-of-flight. The angular momentum acts increasing the kinetic energy around the vortex core, which results in a faster expansion of perpendicular direction to it. This situation shifts the frequencies of collective oscillations of a vortex-free state, and generates richer collective modes due the coupling. Now there are four possible modes, being two types of monopole mode and two types of quadrupole modes. The difference among these types is the phase of vortex oscillation. When one considers fluctuations without polar symmetry, their collective modes result in the vortex decaying. In order to control and prevent these processes we have proposed three dynamical mechanisms such as modulation of s-wave scattering length, modulation of frequencies of harmonic trap, and modulation of the amplitude of Laguerre-Gauss potential. The last one has proven to be more effective.

Neste trabalho descrevemos a produção e estudo de excitações topológicas em um condensado de Bose-Einstein em átomos de Rubídio-87. O condensado é produzido através de resfriamento evaporativo forçado por rádio-freqüência em uma armadilha puramente magnética do tipo QUIC. A armadilha magnética é carregada por um sistema de duplo-MOT. A temperatura de transição é de cerca de 150nK. Condensados puros com 1 - 2 × 10^5 átomos de Rb-87 são observados. Realizamos uma caracterização da amostra em relação às suas características fundamentais. Fração condensada, expansão anisotrópica, distribuição espacial e efeitos de temperatura finita são descritos. Com o objetivo de observar excitações coerentes do condensado entre os estados da armadilha, adicionamos um campo magnético do tipo quadrupolo esférico oscilante no tempo. Observamos, no entanto, a transferência de momento angular para a amostra com a formação de vórtices e arranjos de vórtices. Definimos regiões de amplitude que geram números de vórtices crescentes. Observamos a formação de estruturas de três vórtices não convencionais donde supusemos a possibilidade de excitação conjunta de vórtices e anti-vórtices. Observamos evidência de turbulência quântica, um estado onde os arranjos dos vórtices não são regulares nem as linhas de vórtices têm um eixo de rotação comum.
In this work we describe the production and investigation of topological excitations in a Bose-Einstein condensate in Rubidium-87 atoms. The condensate is produced through forced evaporative cooling by radio-frequency in a QUIC-type purely magnetic trap. The magnetic trap is loaded from a double-MOT system. Transition temperature is about 150nK. Pure condensates containing 1-2×105 87Rb atoms are observed. We performed the characterization of the sample in relation to its fundamental aspects. Condensed fraction, anisotropic expansion, spacial distribution and finite temperature effects are described. Aiming to observe coherent topological excitations of the condensate between two states of the trap, we added a spherical quadrupole magnetic fields oscillating in time. We observe, instead, angular momentum tranference to the sample and the formation of vortices and arrays of vortices. We define amplitude regions where an increasing number of vortices are observed. We observe the formation of non-usual three-vortex structures from which we infer the existence of vortices and anti-vortices together in the sample. We observe evidence of quantum turbulence, a state where non-regular vortex arrays appear as well as vortex lines have no preferred direction to form.

L’objet de cette thèse est l’étude expérimentale des comportements hydrodynamiques d’un faisceau laser se propageant dans un milieu à réponse non-linéaire. Pour un milieu presentant un indice de réfraction qui depend de l’intensité du laser, ainsi que dans le cadre de l’approximation paraxiale, l’intensité du faisceau est assimilée à une densité de fluide. L’axe de propagation représente le temps d’évolution du fluide, le gradient de phase du faisceau définit sa vitesse d’écoulement et la variation de l’indice de réfraction permet de définir une vitesse du son dans le fluide. Dans le cadre de cette analogie, nous appelons le faisceau qui se propage, fluide de lumière. Dans cette thèse, nous étudions la notion de superfluidité de la lumière dans un régime non linéaire auto-défocalisant (self-defocusing). Cette notion est définie par l’absence de diffraction lorsque le fluide de lumière se propage en présence d’un obstacle. Les paramètres permettant de contrôler la transition superfluide sont : la vitesse du fluide de lumière ainsi que la vitesse du son. La première est pilotée par la direction du vecteur d’onde, ainsi que la deuxième est contrôlée par l’intensité du laser. Nous étudions aussi dans le cadre de cette analogie, le régime d’émission de vortex quantifiés suite à l’interaction entre le fluide de lumière et l’obstacle, considéré dans ce cas comme étant fort. Quand deux fois la vitesse d’écoulement aux pôles de l’obstacle dépasse la vitesse du son, des paires de vortex/anti-vortex sont émises, démontrant ainsi un comportement hydrodynamique de la lumière. Dans le but de comprendre l’effet non linéaire mis en jeu, nous présentons également dans cette thèse, une étude de l’effet photoréfractif non-linéaire en exploitant l’effet de l’automodulation de phase (self-phase modulation)
This thesis presents an experimental study of hydrodynamical phenomena of a laser propagating nonlinearly. For a medium presenting an intensity-dependent refractive index, and in the frame of the paraxial approximation, The intensity of the laser beam is equivalent to a density of a fluid, the propagation direction is seen as a time evolution of the fluid as well as the phase gradient of the laser beam defines a flow velocity and the nonlinear refractive index change allows defining a sound velocity of the fluid. Under this analogy, we call the propagating laser beam a fluid of light. In this thesis, we provide a study of the superfluidity concept of a fluid of light in a selfdefocusing regime of the nonlinearity. It is defined as the absence of diffraction when the fluid of light encounters an obstacle. The parameters which control the superfluid transition are: the flow velocity as well as the sound velocity. They are controlled respectively through the wave vector and the intensity of the laser beam. In the frame of this analogy, we also present in this thesis a study of vortex shedding regime as a result of the interaction between the fluid of light and the obstacle. Here, the obstacle is considered to be strong. When twice the flow velocity at the poles of the obstacle is larger than the sound velocity, pairs of vortex/anti-vortex are emitted demonstrating a hydrodynamical behaviour of the fluid of light. In order to underline the nonlinear refractive index change, we also report in this thesis a study of the photorefractive effect using the self-phase modulation effect

No
Vortices circulating in a ring made from a Josephson array in the insulating phase are studied. The ring contains a `dual Josephson junction' through which the vortices tunnel. External non-classical microwaves are coupled to the device. The time evolution of this two-mode fully quantum mechanical system is studied, taking into account the dissipation in the system. The effect of the quantum statistics of the photons on the quantum statistics of the vortices is discussed. Entropic calculations quantify the entanglement between the two systems. Quantum phenomena in the system are also studied through Wigner functions. After a certain time (which depends on the dissipation parameters) these quantum phenomena are destroyed due to dissipation.

Quantum trajectories are investigated within the complex quantum Hamilton-Jacobi formalism. A unified description is presented for complex quantum trajectories for one-dimensional time-dependent and time-independent problems. Complex quantum trajectories are examined for the free Gaussian wave packet, the coherent state in the harmonic potential, and the the barrier scattering problems. We analyze the variations of the complex-valued kinetic energy, the classical potential, and the quantum potential along the complex quantum trajectories. For one-dimensional time-independent scattering problems, we demonstrate general properties and similar structures of the complex quantum trajectories and the quantum potentials. In addition, it is shown that a quantum vortex forms around a node in the wave function in complex space, and the quantized circulation integral originates from the discontinuity in the real part of the complex action. Although the quantum momentum field displays hyperbolic flow around a node, the corresponding Polya vector field displays circular flow. Moreover, local topologies of the quantum momentum function and the Polya vector field are thoroughly analyzed near a stagnation point or a pole (including circular, hyperbolic, and attractive or repulsive structures). The local structure of the quantum momentum function and the Polya vector field around a stagnation point are related to the first derivative of the quantum momentum function. However, the magnitude of the asymptotic structures for these two fields near a pole depends only on the order of the node in the wave function. Finally, quantum interference is investigated and it leads to the formation of the topological structure of quantum caves in space-time Argand plots. These caves consist of the vortical and stagnation tubes originating from the isosurfaces of the amplitude of the wave function and its first derivative. Complex quantum trajectories display helical wrapping around the stagnation tubes and hyperbolic deflection near the vortical tubes. Moreover, the wrapping time for a specific trajectory is determined by the divergence and vorticity of the quantum momentum field. The lifetime for interference features is determined by the rotational dynamics of the nodal line in the complex plane. Therefore, these results demonstrate that the complex quantum trajectory method provides a novel perspective for analysis and interpretation of quantum phenomena.
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