Methods to generate, manipulate, and measure optical and atomic fields with global or local angular momentum have a wide range of applications in both fundamental physics research and technology development. In optics, the engineering of angular momentum states of light can aid studies of orbital angular momentum (OAM) exchange between light and matter. The engineering of optical angular momentum states can also be used to increase the bandwidth of optical communications or serve as a means to distribute quantum keys, for example. Similar capabilities in Bose-Einstein condensates are being investigated to improve our understanding of superfluid dynamics, superconductivity, and turbulence, the last of which is widely considered to be one of most ubiquitous yet poorly understood subjects in physics. The first part of this two-part dissertation presents an analysis of techniques for measuring and manipulating quantized vortices in BECs. The second part of this dissertation presents theoretical and numerical analyses of new methods to engineer the OAM spectra of optical beams. The superfluid dynamics of a BEC are often well described by a nonlinear Schrodinger equation. The nonlinearity arises from interatomic scattering and enables BECs to support quantized vortices, which have quantized circulation and are fundamental structural elements of quantum turbulence. With the experimental tools to dynamically manipulate and measure quantized vortices, BECs are proving to be a useful medium for testing the theoretical predictions of quantum turbulence. In this dissertation we analyze a method for making minimally destructive in situ observations of quantized vortices in a BEC. Secondly, we numerically study a mechanism to imprint vortex dipoles in a BEC. With these advancements, more robust experiments of vortex dynamics and quantum turbulence will be within reach. A more complete understanding of quantum turbulence will enable principles of microscopic fluid flow to be related to the statistical properties of turbulence in a superfluid. In the second part of this dissertation we explore frequency mixing, a subset of nonlinear optical processes in which one or more input optical beam(s) are converted into one or more output beams with different optical frequencies. The ability of parametric nonlinear processes such as second harmonic generation or parametric amplification to manipulate the OAM spectra of optical beams is an active area of research. In a theoretical and numerical investigation, two complimentary methods for sculpting the OAM spectra are developed. The first method employs second harmonic generation with two non-collinear input beams to develop a broad spectrum of OAM states in an optical field. The second method utilizes parametric amplification with collinear input beams to develop an OAM-dependent gain or attenuation, termed dichroism for OAM, to effectively narrow the OAM spectrum of an optical beam. The theoretical principles developed in this dissertation enhance our understanding of how nonlinear processes can be used to engineer the OAM spectra of optical beams and could serve as methods to increase the bandwidth of an optical signal by multiplexing over a range of OAM states.

Three problems are investigated in the context of quantum collective dynamics. First, we examine the optomechanics of a Bose-Einstein condensate trapped in an optical ring cavity and coupled to counter-propagating light fields. Virtual dipole transitions cause the light to recoil elastically from the condensate and to excite its atoms into momentum side modes. These momentum side modes produce collective density oscillations. We contrast the situation to a condensate trapped in a Fabry-Perot cavity, where only symmetric ("cosine") side modes are excited. In the ring cavity case, antisymmetric ("sine") modes can be excited also. We explore the mean field limit and find that even when the counter-propagating light fields are symmetrically pumped, there are parameter regions where spontaneous symmetry breaking occurs and the sine mode becomes occupied. In addition, quantum fluctuations are taken into account and shown to be particularly significant for parameter values near bifurcations of the mean field dynamics. The next system studied is a hybrid composed of a high quality micromechanical membrane coupled magnetically to a spinor condensate. This coupling entangles the membrane and the condensate and can produce position superposition states of the membrane. Successive spin measurements of the condensate can put the membrane into an increasingly complicated state. It is possible in principle to produce nonclassical states of the membrane. We also examine a model of weaker, nonprojective measurements of the condensate's spin using phase contrast imaging. We find an upper limit on how quickly such measurements can be made without severely disrupting the unitary dynamics. The third situation analyzed is the string breaking mechanism in ultrahigh energy collisions. When quark-antiquark pairs are produced in a collision, they are believed to be linked by a tube of chromoelectric field flux, the color string. The energy of the string grows linearly with quark separation. This energy is converted into real particles by the Schwinger mechanism. Screening of the color fields by new particles breaks the string. By quantizing excitations of the string using the conjugate coordinates of field strength and string cross-section, we recover the observed exponential spectrum of outgoing particles.

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Nous présentons l'étude par Résonance Magnétique Nucléaire (RMN) du composé NiCl2-4SC(NH2)2, dit DTN, constitué de chaînes de spins 1 faiblement couplées suivant les directions transverses aux chaînes. A basse température et dans un champ magnétique compris entre les deux valeurs critiques Hc1 et Hc2, ce système s'ordonne dans un état de type Condensat de Bose-Einstein (CBE). Dans cette phase, nous décrivons d'une part la détermination expérimentale du paramètre d'ordre (aimantation transverse), dont l'amplitude est bien décrite par la théorie mais dont la phase (orientation) semble fixée par un terme d'anisotropie. D'autre part nous avons étudié les fluctuations des spins électroniques à basse énergie, par la mesure du taux de relaxation RMN 1/T1, et montré que celui-ci obéit à la loi de puissance 1/T1 propto T^5. Ce comportement peut être associé au processus de 2ème ordre lié à des excitations ayant une dispersion linéaire, tels que les quasiparticules de Bogoliubov, mais sa nature n'est pas encore bien comprise. En dehors de la phase CBE, nous décrivons l'étude des fluctuations de spin dans le régime critique quantique (H ~ Hc2), dans lequel nous établissons une loi d'échelle sur 1/T1, identique à celle que l'on a observé dans un autre composé de description équivalente (échelle de spins BPCB), prouvant ainsi l'universalité de ce régime [S. Mukhopadhyay et al., Phys. Rev. Lett. 109, 177206 (2012)]. Nous avons aussi étudié les effets du désordre induit par la substitution Br-Cl dans le composé Ni(Cl1−xBrx)2-4SC(NH2)2, pour lequel des mesures par des techniques macroscopiques ont suggéré l'existence d'une phase "verre de Bose" [R. Yu et al., Nature 489, 379 (2012)]. Cette phase est caractérisée, pour différentes concentrations du dopage x = 4%, 9%, 13%, par un pic de relaxation RMN 1/T1 au champ Hp = 13.5 T, marquant un regain des fluctuations longitudinales et présentant une forte distribution des valeurs de 1/T1 - probablement due à l'aspect vitreux du système. L'indépendance du Hp en fonction de x démontre que la physique y est dominée par les effets locaux liés aux dopants. / We present a Nuclear Magnetic Resonance (NMR) study of the NiCl2-4SC(NH2)2 compound, called DTN, consisting of spin-1 chains that are weakly coupled along the transverse directions. At low temperatures and for magnetic field values between the two critical fields Hc1 and Hc2, this system enters an ordered phase of the Bose-Einstein Condensate (BEC) type. Within this phase, we first describe the experimental determination of the order parameter (transverse magnetization), the amplitude of which is found to be well described by theory while its phase (orientation) seems to be fixed by an anisotropy term. Second, by NMR relaxation rate 1/T1 we have studied the low-energy fluctuations of the electronic spins and found that they obey the power law 1/T1 ~ T 5. Such a behaviour points to a 2nd order process involving linearly dispersing excitations, such as Bogoliubov quasiparticles, but its nature is not yet well understood. Outside the BEC phase, we report a study of the spin fluctuations in the quantum critical regime (H ~ Hc2), demonstrating a scaling law on 1/T1 similar to the one that has already been observed in another equivalent compound, BPCB spin-ladder, thus proving the universality of this regime [S. Mukhopadhyay et al., Phys. Rev. Lett. 109, 177206 (2012)]. We have also studied the effect of disorder induced by the Br-Cl substitution in the compound Ni(Cl1-xBrx)2-4SC(NH2)2 (doped DTN), for which measurements using macroscopic techniques have suggested the existence of a "Bose glass" phase [R. Yu et al., Nature 489, 379 (2012)]. This phase is characterized, for all studied doping concentrations x = 4%, 9%, 13%, by a peak in the NMR relaxation rate 1/T1 at the field value Hp ~ 13.5 T, evidencing an upsurge of the longitudinal <SzSz> spin fluctuations, and presenting strong inhomogeneity of the 1/T1 values – probably reflecting the glassy character of the system. The observed doping-independence of Hp demonstrates that the corresponding physics is dominated by local effects due to the dopants.

In this thesis we have basically studied two aspects of BoseEinstein condensation (BEC) in trapped dilute gases: (i) superfluidity with the possible onset of quantum turbulence (QT), and (ii) nonuniformity, which suggests the definition of new variables in order to build a global thermodynamic description. Both analyses were performed in a 87Rb BEC magnetically trapped in a QuadrupoleIoffe configuration (QUIC) trap. Concerning the first item, vortices and QT were generated by applying an oscillatory excitation formed by a quadrupole magnetic field superimposed onto the QUIC trapping potential. Scanning both the excitation amplitude and its duration allowed us to observe different regimes, particularly one with regular, welldefined vortices and another, where the onset of QT is believed to have occurred. The transition between this two regimes were explained by considering the finitesize characteristic of trapped gases. Additionally, data analyses on three vortex configurations suggested the presence of both vortices and antivortices (opposite circulation sign), and the vortex nucleation mechanism was proposed to be related to a relative motion between the condensate and thermal components, namely a counterflow. As for the second item, the BEC transition in our experiment was characterized in terms of new global thermodynamic variables. A phase diagram was constructed and compared to the superfluid helium phase transition. Finally, we provide preliminary results on the calculation of a global heat capacity, and briefly discuss the advantages of this new approach over the local density approximation alternative, particularly on BEC clouds in the presence of vortices and QT. / Nesta tese, nós estudamos dois aspectos da condensação de BoseEinstein (CBE) em gases diluídos aprisionados: (i) superfluidez e a possível ocorrência de turbulência quântica (TQ); e (ii) nãouniformidade, o que sugere um tratamento termodinâmico diferente pela definição de novas variáveis globais. Ambos os estudos foram realizados em amostras condensadas de átomos de 87Rb aprisionados magneticamente numa armadilha do tipo QUIC. Em relação ao primeiro item, a geração de vórtices e TQ ocorreu pela aplicação de uma excitação oscilatória gerada pela adição de um campo quadrupolar ao potencial confinante do QUIC. Como dependência da amplitude e duração da excitação, diferentes regimes foram observados. Particularmente, num dos regimes, apenas vórtices bem definidos foram observados e em outro, imagens consistentes com a ocorrência de TQ foram obtidas. A transição entre estes dois regimes foi explicada em termos do tamanho finito característico de gases aprisionados. Além disto, através da análise de dados mostrando configurações com três vórtices, pudemos inferir a presença de vórtices e antivórtices (circulação oposta). Para explicar o mecanismo de nucleação de vórtices, analisamos, como possível causa, um movimento relativo entre as componentes térmicas e condensadas das amostras, conhecido como contrafluxo. Já em relação ao segundo item, a transição de fase da CBE foi descrita em termos de novas variáveis termodinâmicas globais. Um diagrama de fase foi construído ressaltando as semelhanças com a transição observada no hélio superfluido. Por fim, apresentamos resultados preliminares sobre o cálculo de uma capacidade térmica global e discutimos as vantagens desta nova abordagem em relação à alternativa usual baseada na aproximação de densidade local. Estas vantagens são particularmente relevantes no caso de nuvens condensadas que apresentam vórtices e TQ.

Les gaz d'atomes ultrafroids offrent des possibilités sans précédent pour la réalisation et la manipulation des systèmes quantiques. Le contrôle exercé sur les interactions entre particules permet d'atteindre le régime de fortes interactions, pour des espèces d'atomes à la fois fermioniques et bosoniques. Dans la limite unitaire, où la force d'interaction est à son maximum, des propriétés universelles émergent. Pour les atomes bosoniques, celles-ci comprennent l'effet Efimov, l'existance surprenante d'une séquence infinie d'états liés à trois corps. Dans cette thèse, nous avons étudiés un système de bosons unitaires. Partant des cas à deux et à trois corps, nous avons montrés que le modèle choisi capturait correctement les caractéristiques universelles de l'effet Efimov. Pour le modèle à N-corps, nous avons développé un algorithme de Monte Carlo quantique capable de réaliser les différentes phases thermodynamiques du système : gaz normal à haute-température, condensat de Bose-Einstein, et liquide d'Efimov. Un unique composant de notre modèle resterait pertinent à la limite de température infinie, à savoir la répulsion corps dur à trois corps, qui constitue une généralisation du potentiel classique entre sphères dures. Pour ce modèle, nous avons proposé une solution au problème d'empilement compact en deux et trois dimensions, fondée sur une Ansatz analytique et sur la technique de recuit simulé. En étendant ces résultats à une situation de pression finie, nous avons montré que le système présente une transition de fusion discontinue, que nous avons identifié à travers la méthode de Monte Carlo. / Ultracold atomic gases offer unprecedented possibilities to realize and manipulate quantum systems. The control on interparticle interactions allows to reach the strongly-interacting regime, with both fermionic and bosonic atomic species. In the unitary limit, where the interaction strength is at its maximum, universal properties emerge. For bosonic atoms, these include the Efimov effect, the surprising existence of an infinite sequence of three-body bound states. In this thesis, we have studied a system of unitary bosons. Starting from the two- and three-body cases, we have shown that the chosen model correctly captures the universal features of the Efimov effect. For the corresponding many-body problem, we have developed a quantum Monte Carlo algorithm capable of realizing the different thermodynamic phases in which the system may exist: The high-temperature normal gas, Bose-Einstein condensate, and Efimov liquid. A single ingredient of our model would remain relevant in the infinite-temperature limit, namely the three-body hard-core repulsion, which constitutes a generalization of the classical hard-sphere potential. For this model, we have proposed a solution to the two- and three-dimensional packing problem, based on an analytical ansatz and on the simulated-annealing technique. Extending these results to finite pressure showed that the system has a discontinuous melting transition, which we identified through the Monte Carlo method.

The Schrödinger equations have had a profound impact on a wide range of fields of modern science, including quantum mechanics, superfluidity, geometrical optics, Bose-Einstein condensates, and the analysis of dispersive phenomena in the theory of PDE. The main purpose of this thesis is to explore two Schrödinger-type equations appearing in the so-called Bohmian formulation of quantum mechanics and in the study of exciton-polariton condensates. For the first topic, the linear Schrödinger equation is the starting point in the formulation of a phase-space model proposed in [1] for the Bohmian interpretation of quantum mechanics. We analyze this model, a nonlinear Vlasov-type equation, as a Hamiltonian system defined on an appropriate Poisson manifold built on Wasserstein spaces, the aim being to establish its existence theory. For this purpose, we employ results from the theory of PDE, optimal transportation, differential geometry and algebraic topology. The second topic of the thesis is the study of a nonlinear Schrödinger equation, called the complex Gross-Pitaevskii equation, appearing in the context of Bose-Einstein condensation of exciton-polaritons. This model can be roughly described as a driven-damped Gross-Pitaevskii equation which shares some similarities with the complex Ginzburg-Landau equation. The difficulties in the analysis of this equation stem from the fact that, unlike the complex Ginzburg-Landau equation, the complex Gross-Pitaevskii equation does not include a viscous dissipation term. Our approach to this equation will be in the framework of numerical computations, using two main tools: collocation methods and numerical continuation for the stationary solutions and a time-splitting spectral method for the dynamics. After performing a linear stability analysis on the computed stationary solutions, we are led to postulate the existence of radially symmetric stationary ground state solutions only for certain values of the parameters in the equation; these parameters represent the “strength” of the driving and damping terms. Moreover, numerical continuation allows us to show, for fixed parameters, the ground and some of the excited state solutions of this equation. Finally, for the values of the parameters that do not produce a stable radially symmetric solution, our dynamical computations show the emergence of rotating vortex lattices.

This dissertation discusses a number of theoretical models of coupled bosonic modes, all closely related to the Bose-Hubbard dimer. In studying these models, we will repeatedly return to two unifying themes: the classical structure underlying quantum dynamics and the impact of weakly coupling a system to an environment. Or, more succinctly, semiclassical methods and open quantum systems. Our primary motivation for studying models such as the Bose-Hubbard is their relevance to ongoing ultracold atom experiments. We review these experiments, derive the Bose-Hubbard model in their context and briefly discuss its limitations in the first half of Chapter 1. In its second half, we review the theory of open quantum systems and the master equation description of the dissipative Bose-Hubbard model. This opening chapter constitutes a survey of existing results, rather than original work. In Chapter 2, we turn to the mean-field limit of the Bose-Hubbard model. After reviewing the striking localization phenomena predicted by the mean-field (and confirmed by experiment), we identify the first corrections to this picture for the dimer. The most interesting of these is the dynamical tunneling between the self-trapping points of the mean-field. We derive an accurate analytical expression for the tunneling rate using semiclassical techniques. We continue studying the dynamics near the self-trapping fixed points in Chapter 3, focusing on corrections to the mean-field that arise at larger nonlinearities and on shorter time scales than dynamical tunneling. We study the impact of dissipation on coherence and entanglement near the fixed points, and explain it in terms of the structure of the classical phase space. The last chapter of the dissertation is also devoted to a dissipative bosonic dimer model, but one arising in a very different physical context. Abandoning optical lattices, we consider the problem of formulating a quantum model of operation of the cylindrical anode magnetron, a vacuum tube crossed-field microwave amplifier. We derive an effective dissipative dimer model and study its relationship to the classical description. Our dimer model is a first step towards the analysis of solid-state analogs of such devices.