Few and many-body systems usually feature interesting and novel behaviors compared with their counterparts in three dimensions. On one hand, low dimensional physics presents challenges due to strong interactions and divergences in the perturbation theory; On the other hand, there exist powerful theoretical tools such as the renormalization group and the Bethe ansatz. In this thesis, I discuss two examples: three interacting bosons in two dimensions and interacting bosons/fermions in one dimension. In both examples, there are intraspecies repulsion as well as interspecies attraction, producing a rich spectrum of phenomena. In the former example, a universal curve of three-body binding energies versus scattering lengths is obtained efficiently by evolving a matrix renormalization group equation. In the latter example, exact solutions for the BCS-BEC crossover are obtained and the unexpected robust features in their excitation spectra are explained by a comprehensive semiclassical analysis.
Few and many-body systems usually feature interesting and novel behaviors compared with their counterparts in three dimensions. On one hand, low dimensional physics presents challenges due to strong interactions and divergences in the perturbation theory; On the other hand, there exist powerful theoretical tools such as the renormalization group and the Bethe ansatz. In this thesis, I discuss two examples: three interacting bosons in two dimensions and interacting bosons/fermions in one dimension. In both examples, there are intraspecies repulsion as well as interspecies attraction, producing a rich spectrum of phenomena. In the former example, a universal curve of three-body binding energies versus scattering lengths is obtained efficiently by evolving a matrix renormalization group equation. In the latter example, exact solutions for the BCS-BEC crossover are obtained and the unexpected robust features in their excitation spectra are explained by a comprehensive semiclassical analysis.
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 presence of few-body bound states is an important issue in the study of interacting bosonic systems due to the instability it introduces into the condensation. The problem is more interesting in two dimensions because the system can host only a finite number of bound states, in contrast to the Efimov effect in three dimensions. The authors considered a two-component bosonic system in two dimensions, with repulsive intra- and attractive inter-species interactions. They found that there is only one three-particle bound state and the binding energies for various scattering lengths fall on an universal curve, where the microscopic details are of no relevance. Their conclusion is pertinent for various systems such as excitons in quantum well structures and bosonic dipoles in bilayer geometries. They developed an efficient renormalization group type method, capable of handling both short- and long- range physics and free of numerical instabilities. This method also makes use of the Hopf coordinates and running basis to reduce the proliferation of hyperspherical harmonics appearing in alternative approaches, yielding significant saving in numerical implementations.
Understanding quantum many-body systems with interactions is an important problem common to various fields of physics, such as atomic physics, condensed matter physics, nuclear physics, and particle physics.Among others, ultracold atoms have provided ideal grounds to study strongly interacting systems and to explore exotic quantum states because of their high controllability.We can widely control the strengths of the interactions from weak to strong as well as the geometry of the systems.In order to deepen our understanding of quantum manybody systems with interactions, we in this thesis investigate two types of one-dimensional (1D) systems realizable with ultracold atoms; bosons and spinless fermions near two-body resonances and bosons near a three-body resonance.One-dimensional systems near two-body resonances have been intensively studied because, in the homogeneous cases, they are exactly solvable systems.The energy spectra and thermodynamic quantities exactly obtained by the Bethe ansatz have greatly contributed to deepening our comprehension of many-body systems in 1D.On the other hand, the exact computations of correlation functions, which encode information about excitations of the systems, are in general much more complicated even in the integrable systems.Motivated by this background, we derive a series of exact relations for correlation functions in bosons and fermions near two-body resonances.These relations called universal relations originate from the universal properties of the resonant systems: Their properties depend on the interactions only through the scattering lengths characterizing the low-energy scattering.The universal relations are strong constraints on the systems because the relations hold for any particle number, scattering length, temperature, and with or without a trapping potential.The relations include the asymptotic behaviors of correlation functions in high-energy regime as well as the energy relations, in which the energies are expressed in term of the momentum distributions.The universal relations involve two-and three-body contacts, which are the integrals of local pair and triad correlations, respectively.Firstly we derive, in both bosons and fermions, universal relations for static correlation functions such as static structure factors and momentum distributions within the first-quantized formalisms.The power laws of the static correlation functions for large momentum and the energy relations are obtained.The coefficients in these power-law behaviors are proportional to the two-body contact.We clarify that the three-body contact makes no contribution to the bosonic energy relation, but it plays a crucial role in the fermionic one.Secondly, we derive universal relations for dynamic correlation functions for bosons.Using the operator product expansion (OPE) in the field theoretical formalism, we obtain the largeenergy-momentum behaviors of the dynamic structure factor, the dynamic current correlation, and the single-particle spectral density as well as the quasiparticle energy and the scattering rate in high-energy regime.While the behaviors of the dynamic structure factor and the current correlation are proportional to the two-body contact, that of the single-particle spectral density is proportional to the number density.Because of the interrelation between 1D bosons and fermions near two-body resonances, the results of the dynamic structure factor and the current i
The author explores some of the inherent simplifications of "quantum lattice physics." He distinguishes between fermions and bosons and analyzes the n-body problem for each, with n=1,2,3… typically a small number. With delta-function (zero-range) interactions, the three-body problem on a lattice is manageable, and some results can even be extrapolated to $n\ensuremath{\ge}4$. Such calculations are not limited to one dimension (where the well-known Bethe ansatz solves a number of n-body problems). On the contrary, studies cited are mainly in three dimensions and actually simplify with increasing dimensionality. For example, it is found that bound states of $n\ensuremath{\ge}3$ particles in $d\ensuremath{\ge}3$ dimensions are formed discontinuously as the strength of two-body attractive forces is increased, and are therefore always in the easily analyzed "strong coupling limit." In the Appendix, an exactly solved example from the theory of itinerant-electron magnetism illustrates how a rigorous solution to the few-body problem is capable of yielding information concerning the N-body problem.
In this dissertation, we analyze both many- and few-body systems under external confinement with tunable interactions. First, we develop a density-renormalization ap-proach for describing two-component fermionic systems with short-range interactions. This renormalized zero-range interaction eliminates the instabilities produced by a bare Fermi pseudopotential and provides a simple description of the interactions from the weakly interacting BCS region up to unitarity. In the second part of the thesis, we focus on few-body systems in the BCS-BEC crossover. To obtain the solutions, we implement two different numerical techniques: a correlated-Gaussian-basis-set expansion and a fixed-node diffusion Monte Carlo tech-nique. We also develop an innovative numerical technique for obtaining solutions to the four-body problem in the hyperspherical representation. Our solutions provide an accurate description of few-body trapped systems. The analysis of two-, three-, and four-body systems, for instance, provides a few-body per-spective on the BCS-BEC crossover problem. The analysis of the spectrum of such systems allows us to visualize important pathways for molecule formation. We then use
Recent experiments have been moving ever closer to the goal of creating cold quantum-degenerate Bose gases in one and two dimensional systems. In this thesis we investigate such systems by adapting the mean-field theories used so successfully to describe Bose-Einstein condensates in the three dimensional case. We find that the key to describing the low dimensional systems lies in an accurate description of the interactions within those systems. In three dimensions the interactions may be well approximated by a simple form in which atoms are assumed to interact in a vacuum. Many-body effects on the scattering then lead to only a small perturbation. However, in one and two dimensions we find that the inclusion of many-body effects on the scattering is crucial, even for a first order calculation. We use a many-body T-matrix to describe the interactions, and develop a simple approximation in the zero-temperature limit that involves the off-shell two-body T-matrix evaluated at a shifted interaction energy. We therefore derive expressions for the off-shell two-body T-matrix in one, two, and three dimensions, and so find expressions for the coupling parameters which describe interactions in one and two dimensional Bose gases. These lead to interaction terms in the one and two dimensional Gross-Pitaevskii equations which differ significantly from the three dimensional case. We present numerical solutions of the Gross-Pitaevskii equation in both one and two dimensions, and in one dimension we find that the results are in agreement with exact results in the extreme limits of the interaction strength. We also investigate higher order theories in the one-dimensional case, with a view to describing the cross-over from a weakly-interacting Bose-Einstein condensate to a quasicondensate as the interactions become stronger. Finally, we study the Bogoliubov-de Gennes excitation spectrum for a one dimensional condensate containing solitons. We show that it has an anomalous mode with a negative energy, and that this represents the collective motion of the soliton relative to the background condensate. (author)