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This thesis is devoted to the study of excitations in atomic and polariton Bose-Einstein condensates (BEC). These two specimens are prime examples for equilibrium and non equilibrium BEC. The corresponding condensate wave function of each system satisfies a particular partial differential equation (PDE). These PDEs are discussed in the beginning of this thesis and justified in the context of the quantum many-body problem. For high occupation numbers and when neglecting quantum fluctuations the quantum field operator simplifies to a semiclassical wave. It turns out that the interparticle interactions can be simplified to a single parameter, the scattering length, which gives rise to an effective potential and introduces a nonlinearity to the PDE. In both cases, i.e. equilibrium and non equilibrium, the main model corresponding to the semiclassical wave is the Gross-Pitaevskii equation (GPE), which includes certain mathematical adaptions depending on the physical context of the consideration and the nature of particles/quasiparticles, such as additional complex pumping and growth terms or terms due to motion. In the course of this work I apply a variety of state-of-the-art analytical and numerical tools to gain information about these semiclassical waves. The analytical tools allow e.g. to determine the position of the maximum density of the condensate wave function or to find the critical velocities at which excitations are expected to be generated within the condensate. In addition to analytical considerations I approximate the GPE numerically. This allows to gain the condensate wave function explicitly and is often a convenient tool to study the emergence of excitations in BEC. It is in particular shown that the form of the possible excitations significantly depends on the dimensionality of the considered system. The generated excitations within the BEC include quantum vortices, quantum vortex rings or solitons. In addition multicomponent systems are considered, which enable more complex dynamical scenarios. Under certain conditions imposed on the condensate one obtains dark-bright soliton trains within the condensate wave function. This is shown numerically and analytical expressions are found as well. In the end of this thesis I present results as part of an collaborative effort with a group of experimenters. Here it is shown that the wave function due to a complex GPE fits well with experiments made on polariton condensates, statically and dynamically.

付記する学位プログラム名: 京都大学卓越大学院プログラム「先端光・電子デバイス創成学」 / 京都大学 / 新制・課程博士 / 博士(理学) / 甲第24402号 / 理博第4901号 / 新制||理||1700(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 川上 則雄, 教授 佐々 真一, 教授 高橋 義朗 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

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.

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.

In this thesis, we study quantum phase transitions and topological phases in low dimensional fermionic systems. In the first part, we study quantum phase transitions and the nature of currents in one-dimensional systems, using eld theoretic techniques like bosonization and renormalization group. This involves the study of currents in Luttinger liquids, and the fate of a persistent current in a 1D system. In the second part of the thesis, we study the different types of Majorana edge modes in a 1D p-wave topological superconductor. Further we extend our analysis to the e ect of an additional s-wave pairing and a Zeeman field on the topological properties, and present a detailed phase diagram and symmetry classification for each of the cases. In the third part, we concentrate on the topological phases in two-dimensional systems. More specifically, we study the experimental realization of SU(3) topological phases in optical lattice experiments, which is characterized by the presence of gapless edge modes at the boundaries of the system. We discuss the specific characteristics required by a such a three component Hamiltonian to have a non-zero Chern number, and discuss a schematic lattice model for a possible experimental realization. The thesis is divided into three chapters, as discussed below: In the first chapter, we study the effect of a boost (Fermi sea displaced by a finite momentum) on one dimensional systems of lattice fermions with short-ranged interactions. In the absence of a boost such systems with attractive interactions possess algebraic superconducting order. Motivated by physics in higher dimensions, one might naively expect a boost to weaken and ultimately destroy superconductivity. However, we show that for one dimensional systems the e ect of the boost can be to strengthen the algebraic superconducting order by making correlation functions fall o more slowly with distance. This phenomenon can manifest in interesting ways, for example, a boost can produce a Luther-Emery phase in a system with both charge and spin gaps by engendering the destruction of the former. In the second chapter, we study the type of Majorana modes and the topological phases that can appear in a one-dimensional spinless p-wave superconductor. We have considered two types of p-wave pairing, 4"" = 4## and 4"" = 4##., and show that in both cases two types of Majorana bound states (MBS) with different spatial dependence emerge at the edges: one purely decaying and one damped oscillatory. Even in the presence of a Zeeman term B, this nature of the MBS persists in each case, where the value of chemical potential and magnetic field B decides which type will appear. We present a corresponding phase diagram, indicating the number and type of MBS in the -B space. Further, we identify the possible symmetry classes for the two cases (based on the ten-fold classification), and also in the presence of perturbations like a s-wave pairing and various terms involving magnetic field. It is seen that in the presence of a s-wave perturbation, the MBS will now have only one particular nature, the damped oscillating behaviour, unlike that for the unperturbed p-wave case. In the third chapter, we study SU(3) topological phases in two dimension. It is shown by Barnett et.al that N copies of the Hofstadter model with 2N Abelian ux per plaquette is equivalent to an N-component atom coupled to a homogeneous non-Abelian SU(N) gauge field in a square lattice. Such models have non-zero Chern number and for N = 3, can be written in terms of the SU(3) generators. In our work, we uncover two salient ingredients required to express a general three-component lattice Hamiltonian in a SU(3) format with non-trivial topological invariant. We nd that all three components must be coupled via a gauge eld, with opposite Bloch phase (in momentum space, if the NN hopping between two components is teik, then for the other two components, this should be te ik) between any two components, and there must be band inversion between all three components in a given eigenstate. For spinless particles, we show that such states can be obtained in a tripartite lattice with three inequivalent lattice sites, in which the Bloch phase associated with the nearest neighbor hopping acts as k-space gauge eld. The second criterion is the hopping amplitude t should have an opposite sign in the diagonal element for one of the two components, which can be introduced via a constant phase ei along the direction of hopping. The third and a more crucial criterion is that there must also be an odd-parity Zeeman-like term (as k ! k, the term changes sign), i.e. sin(k) z term, where z is the third Pauli matrix defined with any two components of the three component basis. In the presence of a constant vector potential, the kinetic energy of the electron gets modified when the vector potential causes a flux to be enclosed. This can generate the desired odd parity Zeeman term, via a site-selective polarization of the vector potential. This can be achieved in principle by suitable modifications of techniques used in Sisyphus cooling, and with a suitable arrangement of polarizer plates, etc. The topological phase is a firmed by edge state calculation, obeying the bulk-boundary correspondence.

Ultracold Fermi gas is an exciting field benefiting from atomic physics, optical physics and condensed matter physics. It covers many aspects of quantum mechanics. Here I introduce some of my work during my graduate study. We proposed an optical spectroscopic method based on electromagnetically-induced transparency (EIT) as a generic probing tool that provides valuable insights into the nature of Fermi paring in ultracold Fermi gases of two hyperfine states. This technique has the capability of allowing spectroscopic response to be determined in a nearly non-destructive manner and the whole spectrum may be obtained by scanning the probe laser frequency faster than the lifetime of the sample without re-preparing the atomic sample repeatedly. Both quasiparticle picture and pseudogap picture are constructed to facilitate the physical explanation of the pairing signature in the EIT spectra. Motivated by the prospect of realizing a Fermi gas of 40 K atoms with a synthetic non-Abelian gauge field, we investigated theoretically BEC-HCS crossover physics in the presence of a Rashba spin-orbit coupling in a system of two-component Fermi gas with and without a Zeeman field that breaks the population balance. A new bound state (Rashba pair) emerges because of the spin-orbit interaction. We studied the properties of Rashba pairs using a standard pair fluctuation theory. As the two-fold spin degeneracy is lifted by spin-orbit interaction, bound pairs with mixed singlet and triplet pairings (referred to as rashbons) emerge, leading to an anisotropic superfluid. We discussed in detail the experimental signatures for observing the condensation of Rashba pairs by calculating various physical observables which characterize the properties of the system and can be measured in experiment. The role of impurities as experimental probes in the detection of quantum material properties is well appreciated. Here we studied the effect of a single classical impurity in trapped ultracold Fermi superfluids. Although a non-magnetic impurity does not change macroscopic properties of s-wave Fermi superfluids, depending on its shape and strength, a magnetic impurity can induce single or multiple mid-gap bound states. The multiple mid-gap states could coincide with the development of a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase within the superfluid. As an analog of the Scanning Tunneling Microscope, we proposed a modified radio frequency spectroscopic method to measure the focal density of states which can be employed to detect these states and other quantum phases of cold atoms. A key result of our self consistent Bogoliubov-de Gennes calculations is that a magnetic impurity can controllably induce an FFLO state at currently accessible experimental parameters.

Submitted by Emanuel Varela Cardoso (emanuel.varela@ufcg.edu.br) on 2018-10-09T17:21:02Z No. of bitstreams: 1 Dissertação-Aline Nascimento Lins (ok).pdf: 569104 bytes, checksum: 74a65b4ab8272bfcc42b2378df62b4ec (MD5) / Made available in DSpace on 2018-10-09T17:21:02Z (GMT). No. of bitstreams: 1 Dissertação-Aline Nascimento Lins (ok).pdf: 569104 bytes, checksum: 74a65b4ab8272bfcc42b2378df62b4ec (MD5) Previous issue date: 2017-02 / O conceito de Matéria Escura surgiu para explicar uma anomalia na curva de rotação de galáxias espirais, desde então, pesquisadores de todo o mundo vêm tentando detectar a partícula que compõe tal matéria. Sem sucesso nas detecções, surgiram algumas teorias para explicar a existência tanto da Matéria Escura quanto da Energia Escura, responsável pela expansão acelerada do Universo. Neste trabalho apresentamos uma dessas teorias. Aqui sugerimos que o Universo se comporta de uma maneira diferente com a qual estamos acostumados, para isso tzemos uso da relatividade geral e da equação não-linear de Klein-Gordon, e então admitimos que o Universo está imerso em um superfluido relativístico para assim podermos explicar teoricamente Matéria Escura e Energia Escura. / The concept of Dark Matter arises to explain an anomaly in the rotation curve of spiral galaxies, since then, researchers around the world have been trying to detect the particle that composes this matter. Without success in the detections, some theories appeared to explain the existence of both Dark Matter and Dark Energy, the last one is responsible for the accelerated expansion of the Universe. In this dissertation we present one of these theories. Here we suggest that the Universe behaves in a di erent way with which we are accustomed, for this we have made use of the general relativity and the Klein-Gordon's nonlinear equation, and then we admit that the Universe is immersed in a relativistic superfluid so we can theoretically explain Dark Matter and Dark Energy.

Les atomes froids sont des outils uniques pour sonder la physique de la matière quantique. Hautement contrôlables, les gaz de Bose et de Fermi ultrafroids sont des systèmes idéaux pour la simulation quantique et pour explorer des manifestations spectaculaires des effets quantiques, comme la superfluidité. Avec des gaz froids de 6Li et de 7Li, nous avons produit le premier mélange de superfluides bosonique-fermionique, et étudié ses propriétés en initiant un contre-flot entre les nuages de Bose et de Fermi (mode dipolaire). La vitesse critique de superfluidité a été mesurée dans le crossover BEC-BCS et elle est trouvée proche de la vitesse du son dans le gaz de Fermi. Nous comparons nos mesures avec des prédictions théoriques récentes. En élevant la température du mélange, nous avons aussi observé une synchronisation inattendue entre les mouvements des deux nuages, interprétée comme un effet Zénon induit par la dissipation. Finalement, ce mélange de bosons et de fermions offre la possibilité unique de créer un piège homogène pour le gaz de Fermi. En ajustant finement les interactions, nous proposons d’utiliser la répulsion entre les bosons et les fermions pour compenser la courbure du piège harmonique pour les fermions. Pour des fermions présentant une polarisation de spin, nous prédisons théoriquement l’existence d’un superfluide avec une structure en “coquille” et fournissons les premières indications expérimentales de l’observation de ce superfluide topologiquement original. / Ultracold atoms are unique tools to probe the physics of quantum matter. Indeed, the high degree of tunability of ultracold Bose and Fermi gases makes them ideal systems for quantum simulation and for exploring macroscopic manifestations of quantum effects, such as superfluidity. In this work, we have realized the first Bose-Fermi superfluid mixture, with ultracold gases of 6Li and 7Li. The properties of the mixture are investigated by initiating a Bose-Fermi counterflow through their dipole modes. The superfluid critical velocity is measured in the BEC-BCS crossover, and is found close to the sound velocity of the Fermi gas near unitarity. We compare our findings to recent theoretical predictions. Raising the temperature of the mixture, we observe an unexpected synchronization of the motion of the two clouds, interpreted with a Zeno-like model induced by dissipation. Finally, this Bose-Fermi mixture offers the unique possibility to create a homogeneoustrap for the Fermi gas. By a fine tuning of the interactions, we propose to use the Bose-Fermi repulsion to compensate the curvature of the harmonic trap for fermions. For a spin-polarized Fermi gas in such a trap, we theoretically predict the existence of a superfluid with a shell structure and we provide first experimental evidence for this topologically new superfluid.