In this Thesis, we report a detailed study of the ground-state properties of a set of quantum few- and many-body systems in one and two dimensions with different types of interactions by using Quantum Monte Carlo methods. Nevertheless, the main focus of this work is the study of the ground-state properties of an ultracold Bose system with dipole-dipole interaction between the particles. We consider the cases where the bosons are confined to a bilayer and multilayer geometries, that consist of equally spaced two-dimensional layers. These layers can be experimentally realized by imposing tight confinement in one direction. We specifically address the study of new quantum phases, their properties, and transitions between them. One expects these systems to have a rich collection of few- and many-body phases because the dipole-dipole interaction is anisotropic and quasi long-range.
In this Thesis, we report a detailed study of the ground-state properties of a set of quantum few- and many-body systems in one and two dimensions with different types of interactions by using Quantum Monte Carlo methods. Nevertheless, the main focus of this work is the study of the ground-state properties of an ultracold Bose system with dipole-dipole interaction between the particles. We consider the cases where the bosons are confined to a bilayer and multilayer geometries, that consist of equally spaced two-dimensional layers. These layers can be experimentally realized by imposing tight confinement in one direction. We specifically address the study of new quantum phases, their properties, and transitions between them. One expects these systems to have a rich collection of few- and many-body phases because the dipole-dipole interaction is anisotropic and quasi long-range.
In this Thesis, we report a detailed study of the ground-state properties of a set of quantum few- and many-body systems by using Quantum Monte Carlo methods. First, we introduced the Variational Monte Carlo and Diffusion Monte Carlo methods, which are the methods used in this Thesis to obtain the properties of the systems. The first systems we studied consist of few-body clusters in one-dimensional Bose-Bose and Bose-Fermi mixtures. Each mixture is formed by two different species with attractive interspecies and repulsive intraspecies contact interactions. For each mixture, we focused on the study of the dimer, tetramer, and hexamer clusters. We calculated their binding energies and unbinding thresholds. Combining these results with a three-body theory, we extracted the three-dimer scattering length close to the dimer-dimer zero crossing. For both mixtures, the three-dimer interaction turns out to be repulsive. Our results constitute a concrete proposal for obtaining a one-dimensional gas with a pure three-body repulsion. The next system analyzed consists of few-body clusters in a two-dimensional Bose-Bose mixture using two types of interactions. The first case corresponds to a bilayer of dipoles aligned perpendicularly to the planes and, in the second, we model the interactions by finite-range Gaussian potentials. We find that all the considered clusters are bound states and that their energies are universal functions of the scattering lengths, for sufficiently large attraction-to-repulsion ratios. Studying the hexamer energy close to the corresponding threshold, we discovered an effective three-dimer repulsion, which can stabilize interesting many-body phases. Once the existence of bound states in the dipolar bilayer has been demonstrated, we investigated whether halos can occur in this system. A halo state is a quantum bound state whose size is much larger than the range of the attractive interaction between the atoms that form it, showing universal ratios between energy and size. For clusters composed from three up to six dipoles, we find two very distinct halo structures. For large interlayer separation, the halo structure is roughly symmetric. However, for the deepest bound clusters and as the clusters approach the threshold, we discover an unusual shape of the halo states, highly anisotropic. Importantly, our results prove the existence of stable halo states composed of up to six particles. To the best of our knowledge, this is the first time that halo states with such a large number of particles have been predicted and observed in a numerical simulation. The next system we studied is a two-dimensional many-body dipolar fluid confined to a bilayer geometry. We calculated the ground-state phase diagram as a function of the density and the separation between layers. Our simulations show that the system undergoes a phase transition from a gas to a stable liquid as the interlayer distance increases. The liquid phase is stable in a wide range of densities and interlayer values. In the final part of this Thesis, we studied a system of dipolar bosons confined to a multilayer geometry formed by equally spaced two-dimensional layers. We calculated the ground-state phase diagram as a function of the density, the separation between layers, and the number of layers. The key result of our study in the dipolar multilayer is the existence of three phases: atomic gas, solid, and gas of chains, in a wide range of the system parameters. Remarkably, we find that the density of the solid phase decreases several orders of magnitude as the number of layers in the system increases. The results reported in this Thesis show that a dipolar system in a bilayer and multilayer geometries offer stable and highly controllable setups for observing interesting phases of quantum matter, such as halo states, and ultra-dilute liquids and solids. En esta Tesis, presentamos un estudio detallado de las propiedades del estado fundamental de un conjunto de sistemas cuánticos de pocos y muchos cuerpos mediante el uso de los métodos de Monte Carlo Cuántico. Primero, introducimos los métodos de Monte Carlo Variacional y Monte Carlo Difusivo que usamos en esta Tesis para obtener las propiedades de los sistemas. Los primeros sistemas que estudiamos son agregados de pocos cuerpos en mezclas unidimensionales de Bose-Bose y Bose-Fermi. Cada una de las mezclas está formada por dos especies con interacciones atractivas para interespecies y repulsivas para intraespecies. Para cada una de las mezclas nos centramos en el estudio de dímeros, tetrámeros y hexámeros. Calculamos las energías de ligadura y los valores umbrales de ruptura de los agregados. Combinando estos resultados con una teoría de tres cuerpos, extrajimos la longitud de dispersión de tres dímeros cerca del punto de ruptura dímero-dímero. Para ambas mezclas la interacción de tres dímeros resulta ser repulsiva. El siguiente sistema analizado son agregados de pocos cuerpos en una mezcla bidimensional de Bose-Bose con dos tipos de interacciones. El primer caso corresponde a una bicapa de dipolos con momentos dipolares orientados perpendicularmente a los planos y, en el segundo, modelamos las interacciones con potenciales gaussianos de rango finito. Encontramos que para relaciones de atracción-repulsión suficientemente grandes todos los agregados considerados son estados ligados y sus energías son funciones universales de las longitudes de dispersión. Estudiando la energía del hexámero cerca del punto umbral correspondiente, descubrimos una repulsión efectiva de tres dímeros, que puede estabilizar fases interesantes de muchos cuerpos. Después de demostrar la existencia de los estados ligados en la bicapa dipolar, investigamos si pueden ocurrir estados de halo en este sistema. Un estado de halo es un estado ligado cuántico cuyo tamaño es mucho mayor que el rango de la interacción atractiva entre los átomos que lo forman. Para agregados compuestos de tres hasta seis dipolos encontramos dos estructuras de halo muy distintas. Para separaciones grandes entre las capas, la estructura de halo es aproximadamente simétrica. Sin embargo, para los estados más ligados y, a medida que los agregados se acercan al punto umbral, descubrimos una estructura de halo inusual, altamente anisotrópica. Nuestros resultados demuestran la existencia de estados de halo estables compuestos de hasta seis dipolos. Hasta donde sabemos, esta es la primera vez que estados de halo con un número tan grande de partículas se predicen y observan en una simulación numérica. Después, estudiamos un fluido bidimensional dipolar de muchos cuerpos confinado a una geometría de bicapa. Calculamos el diagrama de fases del estado fundamental como función de la densidad y de la separación entre las capas. Nuestras simulaciones muestran que en el sistema ocurre una transición de fase, de un gas a un líquido a medida que se incrementa la distancia entre las capas. El líquido es estable en una región amplia de densidades y de la distancia entre las capas. En la parte final de esta Tesis se estudia un sistema de bosones dipolares confinados a una geometría multicapa formada por capas bidimensionales igualmente espaciadas. Calculamos el diagrama de fases del estado fundamental como función de la densidad, la separación entre las capas y el número de capas. Encontramos que existen tres fases: gas atómico, sólido y gas de cadenas, en una región amplia de los parámetros del sistema. La densidad del sólido disminuye varios órdenes de magnitud a medida que el número de capas en el sistema aumenta. Los resultados reportados en esta Tesis muestran que un sistema de dipolos confinados a una bicapa o multicapa ofrece configuraciones estables y altamente controlables para observar fases interesantes de materia cuántica, como estados
In this thesis, we study the physical properties of several ultracold few-boson systems depending on the interactions between their constituents. Nowadays, experimentally, it is possible to have great control with high precision over the geometry and the interactions between the particles, making them an excellent setup to test directly the principles of quantum mechanics. A very interesting point is to study the evolution of their properties with the number of particles. The theoretical study of these systems pretends to microscopically understand the current experimental results and give support to new experimental developments.\n\n\t\t\t\t The method that will be used is the exact diagonalization of the Hamiltonian of the system. As we will see, in spite of the attempts to improve it, the method is limited by the fact that, in practice, it is only useful to study few-particle systems. The method has several advantages. First of all, one has access to both the ground and the excited states. In second place, the method is variational and converges to the exact solution as long as the Hilbert space in which we diagonalize is enlarged.\n\t\t\t\t Moreover, since we have access to the states of the system, it is possible to calculate any observable quantity of interest.\n\n\t\t\t\t First, we will study a system of spinless bosons trapped in a two-dimensional harmonic potential. The effect of the trap is to keep the system bound. It will be seen how the presence of a repulsive interaction changes the energy spectrum and other properties of the system. For instance, the density profile, which is usually measurable, and also the two-body distribution function, which is intimately related to the existence of correlations.\n\n\t\t\t\t Afterwards, the focus will be on the particular case of having only two bosons in the system interacting through a strong repulsive force. Inspired by the one-dimensional case where the fermionization phenomenon takes place in the strongly-interacting limit, we will study whether in two dimensions there is a resembling reminiscent effect. In other words, we will analyze if there are properties of the two strongly-interacting bosons in two dimensions that are like the ones of two noninteracting fermions.\n\n\t\t\t\t After that, we will tackle the localization phenomenon in a one-dimensional system that is caused by an external speckle potential that introduces disorder in the system. We will show that the localization is a robust phenomenon against repulsive contact interactions.\n\n\t\t\t\t Finally, we will study the influence of the spin-orbit coupling in a system of bosons with two possible pseudospin components, associated, for instance, to two hyperfine levels, confined in a two-dimensional harmonic trap. We will present an exhaustive analysis of the combined effects of the interaction and the spin-orbit coupling in the spectrum and the properties of the system. In particular we show the existence of a crossover in the ground state of the system susceptible to be experimentally identified.
The ground-state phase properties of a two-dimensional Bose system with dipole-dipole interactions is studied by means of quantum Monte Carlo techniques. Limitations of mean-field theory in a two-dimensional geometry are discussed. A quantum phase transition from gas to solid is found. Crystal is tested for existence of a supersolid in the vicinity of the phase transition. Existence of mesoscopic analogue of the off-diagonal long-range order is shown in the one-body density matrix in a finite-size crystal. Non-zero superfluid fraction is found in a finite-size crystal, the signal being dramatically increased in presence of vacancies.
Strongly interacting ultracold dipolar molecules are a rich platform for quantum information and quantum simulation of phenomena relevant to condensed matter systems. This dissertation presents novel theoretical and computational studies of ultracold dipolar molecules relevant to these areas in three broad categories.First, the path integral ground state quantum Monte Carlo method is extended to simulations with both translational and rotational degrees of freedom, including the development of a novel estimator of the elementary excitation energies of a system with these degrees of freedom based on the well known Feynman-Bijl approximation. This is then used to study dipolar molecular ensembles confined to one-dimensional optical lattices. The ground state phase diagram is characterized with both a mean-field method and the aforementioned quantum Monte Carlo method. A quantum phase transition is found for very short optical lattice spacings or very large dipole moments. The prospects for experimentally realizing a system which exhibits this phase transition are discussed.Second, the path integral ground state quantum Monte Carlo method is utilized to study the ground state of dipole molecules confined to two-dimensional optical lattices. On square lattices a crossover from a weakly interacting disordered phase to a strongly interacting striped phase is found for short lattice spacings or large dipole moments. On triangular lattices a crossover between a disordered phase and a completely polarized phase is found, also for short lattice spacings or large dipole moments. The relevance of such simulations to self-assembled dipolar crystals in two dimensions is discussed and simulation results of these systems are presented. Simulations of polarized and harmonically confined molecular systems in two dimensions with translations and rotations are also presented and a realistic interaction potential using highly accurate ab initio quantum chemistry techniques is calculated for LiH--LiH interactions as a preliminary step toward developing a high quality trial wave function for the simulation of systems possessing both translational and rotational degrees of freedom.Finally, several novel applications for dipolar molecules confined to optical lattices are proposed. An experimental realization of a topological phase using 2Sigma ground state molecules with applied near resonant microwave fields and transverse magnetic fields is proposed. A method to create an array of dipole-coupled two-level molecules exhibiting superradiance and possessing novel spectroscopic properties using 1Sigma ground state molecules in the combination of a far off-resonant laser field and a moderate magnetic field is proposed. Additionally, a method to create entangled pairs of molecules using closed-shell dipolar molecules immersed in an intense far off-resonant laser field is reviewed and an extension in which parallel optical lattices are merged and molecules are allowed to collide in the presence of a far off-resonant laser field is proposed. Simulation results are presented indicating that a significant amount of entanglement may be generated in such a way in realistic molecular systems.
The research in this dissertation is devoted to few-body bound state physics in experimentally relevant systems of trapped atoms and molecules. First, the complexes of tubes containing dipoles are considered. The tubes are assumed to have zero width such that one-dimensional treatment can be applied. For this setup few-body bound structures are found for different polarization an- gles and dipole strengths by using stochastic variational methods. After that a similar analysis is provided for two-dimensional planes filled with dipolar par- ticles. At the end of the thesis, a system of particles in one-dimensional traps is analyzed assuming that two-body interactions are of zero range. It is shown that such a system is solvable in the limit of infinitely strong repulsion between particles.
This thesis presents a set of studies on atomic systems where quantum effects are particularly relevant. These studies have been developed by applying a variety of tools from many-body physics. First of all, we have studied the prospects for the existence of a superfluid transition in an ultracold gas of fermionic atoms, by generalizing the BCS theory of superconductivity to the case when the two species that pair have different densities and the ground state may spontaneously break one or more symmetries. In a second part, we have studied the dynamics of a Bose-Einstein condensate whose spin degree of freedom is free to evolve inside a quasi-onedimensional optical trap. We have used a mean-field formulation to address both the zero temperature case and the finite temperature one. Finally, we have performed a careful study of the ground state and the line tension of two-dimensional systems of helium-4. First, we have used Monte Carlo techniques, then with a Density Functional built on-purpose. Comments are welcome! KEYWORDS: Cold gases, Pairing, Spinor condensate, Helium, Two dimensions