Following the successful localization of dipolar ultracold atoms in harmonic traps, theoretical investigation of their density profiles have attracted very large interest in the literature. In this thesis, density functional theory will be used to investigate the density profiles of ultracold dipolar atoms (e.g. Cr) in harmonic potential traps. Parameters of the problem such as harmonic well depth, magnetic moments of the atoms will be explored in order to understand their effects on the density profile.
Following the successful localization of dipolar ultracold atoms in harmonic traps, theoretical investigation of their density profiles have attracted very large interest in the literature. In this thesis, density functional theory will be used to investigate the density profiles of ultracold dipolar atoms (e.g. Cr) in harmonic potential traps. Parameters of the problem such as harmonic well depth, magnetic moments of the atoms will be explored in order to understand their effects on the density profile.
Recently a new outlook on dealing with dipolar ultracold fermions based on density functional methods has received attention. A Thomas-Fermi treatment coupled with a variational approach has been developed for a collection of fermions trapped in a harmonic potential interacting via dipole-dipole forces. In this thesis, firstly our alternative formalism for Thomas-Fermi method by performing some calculations based on the Kohn-Sham formalism which is one of the main idea of density functional theory is investigated. Furthermore, density distributions are obtained dependent to the parameters; rescaled interaction strength, dipole-dipole energy and the trap parameter which determine the trap geometry based on this theory. The thesis starts with a brief outline of the density functional theory and theory of our system, continues with calculations based on this theory, which are free of any variational assumptions for the density profile. Moreover, results of density graphics for harmonic trap will be followed by discussion of comparison and contrast with Thomas-Fermi method based on the paper of Goral et al.. These discussions are mainly about the shape of the density distribution, variation of the cloud parameters and energy behaviours according to the rescaled interaction strength. The thesis concludes with an analysis of contribution of density functional theory to this fermionic system.
We investigate the behavior of polarized dipolar fermions in a two-dimensional harmonic trap in the framework of the density functional theory (DFT) formalism using the local density approximation. We treat only a few particles interacting moderately. Important results were deduced concerning key characteristics of the system such as total energy and particle density. Our results indicate that, at variance with Coulombic systems, the exchange- correlation component was found to provide a large contribution to the total energy for a large range of interaction strengths and particle numbers. In addition, the density profiles of the dipoles are shown to display important features around the origin that is not possible to capture by earlier, simpler treatments of such systems.
In this thesis we study the effects of the dipole-dipole interaction on the physics of ultracold quantum gases, both bosonic and fermionic, within the theoretical framework provided by the mean-field regime. This kind of interaction takes place in ultracold atomic gases (for instance 52Cr or 164Dy) due to their atomic magnetic dipole moment, and in ultracold molecular gases due to the magnetic or electric dipole moment. \n \nIn the case of quantum gases of bosonic atoms, or Bose-Einstein condensates, the dipole-dipole interaction can be studied within mean-field approximation using the Gross-Pitaevskii equation, which now contains a new non-linear term due to the dipole-dipole interaction. We investigate, on the one hand dipolar condensates confined in harmonic traps, and on the other dipolar condensates confined in toroidal traps. In the harmonic geometry, our focus is on the study of the ground state and the quantized vortex state, where the density profile is characterized as well as some properties leading to the process of vortex formation, such as the critical frequency and the energy barrier that has to be overcome to bring the vortex from the surface to the centre of the gas. We finish the study of dipolar condensates in harmonic traps by dynamically simulating the precession frequency of an off-center vortex in a non-rotating condensate. \n \nIn the toroidal geometry the dipolar effects are strongly magnified when the polarization axis of the dipoles is perpendicular to the trap symmetry axis. In this case, the anisotropic structure of the density can be understood as the response of the system to the double-well effective potential along the ring. We have studied the dynamics of this system when the initial number of atoms in the left and right wells is imbalanced, predicting Josephson and self-trapping oscillations depending on the initial condition. This has led us to name this new system as Self-induced Josephson Junction. We have studied in detail the self-trapping regime and we have seen that the particle flux inversion is closely related to the crossing of vortices across the Josephson junctions. This result opens the door to establishing a more direct connection between the phase-slip regime, widely addressed in superfluid helium, and the self-trapping regime of condensates. \n \nIn the case of quantum gases of fermionic dipolar particles, we have studied how the radial quadrupole mode allows one to distinguish between hydrodynamic and collisionless regimes. We have analytically calculated the frequency of this mode in the mean-field approximation, generalizing the results from the Thomas-Fermi approximation for trapped ideal Fermi gases. On the one hand, we observe that the frequency in the hydrodynamic regime is smaller than in non-dipolar Fermi gases, while in the collisionless regime the frequency is larger or smaller than that corresponding to the non-interacting system depending on the geometry of the harmonic trap. On the other hand, we predict that reducing the trap deformation (aspect ratio) an observable jump in the frequency of the radial quadrupole mode would take place, which would correspond to the transition between the collisionless and hydrodynamic regimes, for instance when the gas undergoes the transition to the superfluid state.
We study a system of ultra-cold dipolar Bose gas atoms confined in a two-dimensional (2D) harmonic trap with a dipolar impurity implanted at the center of the trap. Due to recent experimental progress in dipolar condensates, we focused on calculating properties of dipolar impurity systems that might guide experimentalists if they choose to study impurities in dipolar gases. We used the Gross–Pitaevskii formalism solved numerically via the split-step Crank–Nicolson method. We chose parameters of the background gas to be consistent with dysprosium (Dy), one of the strongest magnetic dipoles and of current experimental interest, and used chromium (Cr), erbium (Er), terbium (Tb), and Dy for the impurity. The dipole moments were aligned by an external field along what was chosen to be the z-axis, and we studied 2D confinements that were perpendicular or parallel to the external field. We show density contour plots for the two confinements, 1D cross-sections of the densities, calculated self-energies of the impurities while varying both number of atoms in the condensate and the symmetry of the trap. We also calculated the time evolution of the density of an initially pure system where an impurity is introduced. Our results show that while the self-energy increases in magnitude with increasing number of particles, it is reduced when the trap anisotropy follows the natural anisotropy of the gas, i.e., elongated along the z-axis in the case of parallel confinement. This work builds upon work conducted in Bose gases with zero-range interactions and demonstrates some of the features that could be found when exploring dipolar impurities in 2D Bose gases.
In this Thesis we will theoretically address some issues concerning the Physics of ultracold Fermi gases, all of them with experimental relevance. The field of ultracold gases, and more recently of ultracold Fermi gases, is gathering a lot of experimental and theoretical interest for two main reasons: the great control on the relevant parameters of the problem and the relative simplicity of the minimal theory which is able to correctly describe the system. Building upon this solid background, ultracold gases provided an ideal laboratory for testing more refined theories and also for addressing fundamental issues of quantum mechanics as well as for simulating more complex physical systems such as those encountered in condensed matter. Even if the effective hamiltonian of ultracold gases can be simple, due to the diluteness of the system and the low temperature, which imply low energy physics, the solution of the quantum mechanic equations governing the state of these systems is not always simple. For the static properties usually mean-field solutions exist or perturbative expansions can be produced in some regimes. However Quantum Monte Carlo (QMC) techniques provide more accurate results especially in the strongly interacting regimes. For confined systems it is possible to use QMC only for a few particles, so that, for large number of particles, a fruitful combined use of Density Functional Theory (DFT) and QMC is necessary. The study of the dynamics of ultracold gases has received little attention with QMC techniques, due to the intrinsic computational difficulty of the many-body problem, so that general hydrodynamic equations are often used for studying the propagation of smoothly varying perturbations. In this Thesis we use QMC techniques for studying the problem of ferromagnetism in repulsive or effectively repulsive ultracold gases without a lattice and the problem of the Bardeen-Cooper-Schrieffer to Bose-Einstein-Condensation (BCS-BEC) crossover in two dimensions. We use DFT in the Local Density Approximation (LDA) for calculating the density profiles of ultracold Fermi gases in harmonic magneto-optical traps, starting from QMC equations of state. We study the propagation of first and second sound in ultracold Fermi gases in cylindrical geometry, using the hydrodynamic equations of superfluids.
In this study we investigate the properties of trapped atoms subjected to rapid rotations. The study is divided into two distinct parts, one for fermions, another for bosons. In the case of the degenerate Fermi gas we explore the density structure of non-interacting cold atoms when they are rotated rapidly. On the other hand, for rapidly rotating two component Bose condensate, we search for new lattice structures in the presence of contact and dipolar interactions. First, the density structure of Fermi gases in a rotating trap is investigated. We focus on the anisotropic trap case, in which two distinct regimes, two and one dimensional regimes, depending on rotation frequency and anisotropy are observed. Two regimes can be illustrated by a simple description of maximum number of states between two Landau levels, which is strongly related to the dimensionality of the system. The regimes are separated from each other by a minimum point in this description. For small anisotropy values the density profiles show a step structure where each step is demonstrated by an elliptical plateau. Each plateau represents a Landau level with a constant density. The local density approximation describes the two dimensional regime with a perfect similarity in the structure of fermion density. The case for one dimensional regime is a little different from the two dimensional case. For large anisotropy values the Friedel oscillation is the dominant aspect of the density profiles. The density profiles show gaussian structure along the direction of strong trapping, and a semicircular form with prominent oscillations along the weak confining direction. Again, the system is nicely described by local density approximation in this regime. A smooth crossover between two regimes is observed, with a switching from a step structure profile to a soft edge transition with Friedel oscillations. At finite temperatures, the step structures are smeared out in two dimension. In one dimensional regime the Friedel oscillations are cleaned as soon as the temperature is turned on. The second part of the study is devoted to the investigation of different lattice structures in two component Bose condensates subjected to very fast rotation, this time in the presence of interactions. We explore the existence of new vortex lattice structures for dipolar two component condensates scanning a wide range of interaction strengths. We introduce a phase diagram as a function of intra and inter-component interactions showing different type of vortex lattice structures. New types of lattice structures, overlapped square and overlapped rectangular, emerge as a result of dipolar interactions and s-wave interaction for a two component condensate. The region where the attractive inter-component interactions dominate the repulsive interactions, the overlapped lattices are formed. The intra-component interactions, which defines the behavior of each component inside, result in different type of lattices by changing the strength of interactions. Two different limits of phase diagram reproduce the results of ordinary two component and dipolar one component Bose condensates. The results of calculation are in agreement with the results of previous studies for two regimes.