Interacting Fermi gases are one of the chief paradigms of condensed matter physics. They have been studied since the beginning of the development of quantum mechanics, but continue to produce surprises today. Recent experimental developments in the field of ultracold atomic gases, as well as conventional solid state materials, have produced new and exotic forms of Fermi gases, the theoretical understanding of which is still in its infancy. This Thesis aims to provide updated tools and additional insights into some of these systems, through the application of both numerical and analytical techniques. The first Part of this Thesis is concerned with the development of improved numerical tools for the study of interacting Fermi gases. These tools take the form of accurate model potentials for the dipolar and contact interactions, as found in various ultracold atomic gas experiments, and a new form of Jastrow correlation factor that interpolates between the radial symmetry of the inter-electron Coulomb potential at short inter-particle distances, and the symmetry of the numerical simulation cell at large separation. These methods are designed primarily for use in quantum Monte Carlo numerical calculations, and provide high accuracy along with considerable acceleration of simulations. The second Part shifts focus to an analytical analysis of spin-imbalanced Fermi gases with an attractive contact interaction. The spin-imbalanced Fermi gas is shown to be unstable to the formation of multi-particle instabilities, generalisations of a Cooper pair containing more than two fermions, and then a theory of superconductivity is built from these instabilities. This multi-particle superconductivity is shown to be energetically favourable over conventional superconducting phases in spin-imbalanced Fermi gases, and its unusual experimental consequences are discussed.
Interacting Fermi gases are one of the chief paradigms of condensed matter physics. They have been studied since the beginning of the development of quantum mechanics, but continue to produce surprises today. Recent experimental developments in the field of ultracold atomic gases, as well as conventional solid state materials, have produced new and exotic forms of Fermi gases, the theoretical understanding of which is still in its infancy. This Thesis aims to provide updated tools and additional insights into some of these systems, through the application of both numerical and analytical techniques. The first Part of this Thesis is concerned with the development of improved numerical tools for the study of interacting Fermi gases. These tools take the form of accurate model potentials for the dipolar and contact interactions, as found in various ultracold atomic gas experiments, and a new form of Jastrow correlation factor that interpolates between the radial symmetry of the inter-electron Coulomb potential at short inter-particle distances, and the symmetry of the numerical simulation cell at large separation. These methods are designed primarily for use in quantum Monte Carlo numerical calculations, and provide high accuracy along with considerable acceleration of simulations. The second Part shifts focus to an analytical analysis of spin-imbalanced Fermi gases with an attractive contact interaction. The spin-imbalanced Fermi gas is shown to be unstable to the formation of multi-particle instabilities, generalisations of a Cooper pair containing more than two fermions, and then a theory of superconductivity is built from these instabilities. This multi-particle superconductivity is shown to be energetically favourable over conventional superconducting phases in spin-imbalanced Fermi gases, and its unusual experimental consequences are discussed.
We consider imbalanced Fermi gases with strong attractive interactions, for which Cooper-pair formation plays an important role. The two-component mix-tures consist either of identical fermionic atoms in two different hyperfine states, or of two different atomic species both occupying only a single hyperfine state. In both cases, the number of atoms for each component is allowed to be differ-ent, which leads to a spin imbalance, or spin polarization. Two different atomic species also lead to a mass imbalance. Imbalanced Fermi gases are relevant to condensed-matter physics, nuclear physics and astroparticle physics. They have been studied intensively in recent years, following their experimental realization in ultracold atomic Fermi gases. The experimental control in such a system allows for a systematic study of the equation of state and the phase diagram as a function of temperature, spin polarization and interaction strength. In this re-view, we discuss the progress in understanding strongly-interacting imbalanced Fermi gases, where a main goal is to describe the results of the highly controlled experiments. We start by discussing Feshbach resonances, after which we treat the imbalanced Fermi gas in mean-field theory to give an introduction to the relevant physics. We encounter several unusual superfluid phases, including phase-separation, gapless Sarma superfluidity, and supersolidity. To obtain a more quantitative description of the experiments, we review also more sophisti-cated techniques, such as diagrammatic methods and the renormalization-group theory. We end the review by discussing two theoretical approaches to treat the inhomogeneous imbalanced Fermi gas, namely the Landau-Ginzburg theory and the Bogoliubov-de Gennes approach. Key words:
Advances in experiments on dilute ultracold atomic gases have given us access to highly tunable quantum systems. In particular, there have been substantial improvements in achieving different kinds of interaction between atoms. As a result, utracold atomic gases oer an ideal platform to simulate many-body phenomena in condensed matter physics, and engineer other novel phenomena that are a result of the exotic interactions produced between atoms. In this dissertation, I present a series of studies that explore the physics of dilute ultracold atomic gases in different settings. In each setting, I explore a different form of the inter-particle interaction. Motivated by experiments which induce artificial spin-orbit coupling for cold fermions, I explore this system in my first project. In this project, I propose a method to perform universal quantum computation using the excitations of interacting spin-orbit coupled fermions, in which effective p-wave interactions lead to the formation of a topological superfluid. Motivated by experiments which explore the physics of exotic interactions between atoms trapped inside optical cavities, I explore this system in a second project. I calculate the phase diagram of lattice bosons trapped in an optical cavity, where the cavity modes mediates effective global range checkerboard interactions between the atoms. I compare this phase diagram with one that was recently measured experimentally. In two other projects, I explore quantum simulation of condensed matter phenomena due to spin-dependent interactions between particles. I propose a method to produce tunable spin-dependent interactions between atoms, using an optical Feshbach resonance. In one project, I use these spin-dependent interactions in an ultracold Bose-Fermi system, and propose a method to produce the Kondo model. I propose an experiment to directly observe the Kondo effect in this system. In another project, I propose using lattice bosons with a large hyperfine spin, which have Feshbach-induced spin-dependent interactions, to produce a quantum dimer model. I propose an experiment to detect the ground state in this system. In a final project, I develop tools to simulate the dynamics of fermionic superfluids in which fermions interact via a short-range interaction.
Fermionic matter is ubiquitous in nature, from the electrons in metals and semiconductors or the neutrons in the inner crust of neutron stars, to gases of fermionic atoms, like 40K or 6Li that can be created and studied under laboratory conditions. It is especially interesting to study these systems at very low temperatures, where we enter the world of quantum mechanical phenomena. Due to the Fermi-Dirac statistics, a dilute system of spin-polarised fermions exhibits no interactions and can be viewed as an ideal Fermi gas. However, interactions play a crucial role for fermions of several spin species. This thesis addresses several questions concerning interacting Fermi gases, in particular the transition between the normal and the superfluid phase and dynamical properties at higher temperatures. First we will look at the unitary Fermi gas: a two-component system of fermions interacting with divergent scattering length. This system is particularly interesting as it exhibits universal behaviour. Due to the strong interactions perturbation theory is inapplicable and no exact theoretical description is available. I will describe the Determinant Diagrammatic Monte Carlo algorithm with which the unitary Fermi gas can be studied from first principles. This algorithm fails in the presence of a spin imbalance (unequal number of particles in the two components) due to a sign problem. I will show how to apply reweighting techniques to generalise the algorithm to the imbalanced case, and present results for the critical temperature and other thermodynamic observables at the critical point, namely the chemical potential, the energy per particle and the contact density. These are the first numerical results for the imbalanced unitary Fermi gas at finite temperature. I will also show how temperatures beyond the critical point can be accessed and present results for the equation of state and the temperature dependence of the contact density. At sufficiently high temperatures a semiclassical description captures all relevant physical features of the system. The dynamics of an interacting Fermi gas can then be studied via a numerical simulation of the Boltzmann equation. I will describe such a numerical setup and apply it to study the collision of two spin-polarised fermionic clouds. When the two components are separated in an elongated harmonic trap and then released, they collide and for sufficiently strong interactions can bounce off each other several times. I will discuss the different types of the qualitative behaviour, show how they can be interpreted in terms of the equilibrium properties of the system, and explain how they relate to the coupling between different excitation modes. I will also demonstrate how transport coefficients, for instance the spin drag, can be extracted from the numerical data.
Rapid progress in the field of ultracold atoms allows the study of many new and old models of quantum many-body physics. In this doctoral dissertation we theoretically explore exotic phases of ultracold quantum gases, with a special focus spin-imbalanced attractive Fermi gases in lower dimensional situations. Chapter 2 reviews the mean-field theory approach to pairing in twocomponent Fermi gases. Applications of this theory are illustrated in Chapter 3, where we discuss mostly well-known results of mean-field theory applied to imbalanced Fermi gases. Adapted from the author's prior publications, Chapters 4, 5 use the theory developed in Chapters 2, 3. In Chapter 6 we discuss the physics of Fermi gases, squeezed into one spatial dimension. In this and Chapter 7, we go beyond mean-field theory, approaching the problem through the Bethe ansatz, exact solutions to few-body problems and Fermi-Bose mappings ("fermionization"). We also show results from a joint effort with the experimental group of Randy Hulet at Rice University to experimentally realize and probe a strongly interacting one dimensional paired Fermi gas. In Chapter 8, after a brief introduction to rapidly rotating two dimensional Bose gases, we introduce a new protocol to create few atom fractional quantum Hall states. Finally, in Chapter 9 we study the effects of two-body losses on lattice Bose gases with hardcore interactions in one and two spatial dimensions.
Ultracold atomic gases provide an ideal system with which to study fundamental many-body physics.Exhibiting universal interactions in clean and controllable environments, long-used simple models as well as more exotic models can now be realized.The interplay between theory and experiment is therefore very active, and, in this thesis, I will detail several works, both exact analytic results and numerical calculations, which have impacts on current experiments.I begin with an introduction to the field including a brief discussion of experiments, the microscopic model of two species of interacting fermions, the BCS-BEC crossover and an overview of the various phases of atomic Fermi gases.I then describe the various results of my theoretical investigations, which are divided into five chapters.First, I describe radio frequency (RF) spectroscopy experiments and how they probe the single-particle spectral function.This leads to my results on an exact feature of the spectral lineshape, a universal large-momentum structure which exists for all states of interacting Fermi systems and has been verified in recent angle-resolved RF experiments.Second, I focus on gases which have a normal Fermi liquid ground state and show that their lineshape exhibits a characteristic jump discontinuity.I illustrate this Fermi surface singularity and the previously mentioned universal large momentum tail with explicit calculations.Third, I turn to the low energy structure of the single-particle spectral function in the superfluid state.I argue that sharp low energy quasiparticle excitations exist across the BCS-BEC crossover using a general argument that includes the interaction of fermions with the low-energy collective mode.This is illustrated with an explicit calculation within an approximation scheme.Fourth, I address the trap-induced inhomogeneity and use a Bogoliubov-deGennes analysis to test if a simple local density approximation (LDA) can provide an adequate description of a spin imbalanced Fermi system.Finally, I discuss dynamics of atomic gases motivated by a recent experiment in which two spin-polarized clouds are allowed to collide with each other.Despite the underlying attractive interactions, the clouds are seen to bounce off of each other.Using a hydrodynamic description, I argue that the short and intermediate time dynamics reflects a metastable state in which the effective interaction is repulsive.It is my hope that the work presented in this thesis can be used to gain key insights which inspire future endeavors.6.2 Time dependence of the center-of-mass positions zσ of the two spin clouds at unitarity for an initial separation z↑ (t = 0) = R z , z↓ (t = 0) = -R z using the upper branch energy functional and damping γ = 1.Red (upper) and blue (lower) curves denote the different spin species. . . . . . . . . . . . . . . .71 6.3 The upper branch energy functional obtained from lowest order constraint variational method.(a), energy as a function of the polarization P for various values of k F a s as indicated in the figure.(b), energy as a function of the interaction parameter k F a s for various values of polarization P as indicated in the figure.(c), phase diagram of the interacting two-component Fermi gases.The two vertical red dashed lines indicates the separation between three phases.For 1/(k F a s ) > 1.124, the system is in the Fermi liquid (FL) phase.For 1.124 > 1/(k F a s ) > 0.91, the system is in the partially polarized phase (PP) and for 1/(k F a s ) < 0.91, the system is fully polarized. . . . . .74 6.4 (Color online) Time dependence of the center-of-mass positions zσ of the two spin clouds for k F (0)a s = 2. . . . . . . . . . . . . . . . . . . . . . . . . . .75 6.5 (Color online) Time dependence of the center-of-mass positions zσ of the two spin clouds for k F (0)a s = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . .
The quantum many-body problem is at the heart of some of the most formidable open problems in modern physics, such as high-Tc superconductivity or the behaviour of neutron stars. Ultracold atomic systems can now be used to simulate model Hamiltonians of condensed matter or nuclear physics, in very well-controlled environnement. In this thesis, we have developed a general method to probe the thermodynamics of homogeneous quantum systems using trapped atomic gases. These measurements are directly compared to the predictions of theories of the quantum many-body problem. We have applied this technique to the spin-1/2 Fermi gas and the Bose with short-range interactions. Using fermionic 6Li, we explored a part of the wide parameter space by changing the interaction strength, the spin-population imbalance or the temperature of the gas. This system exhibits remarkably rich physics, such as normal/superfluid phase transitions (that can be of thermal or quantum character) or a Fermi liquid-type behaviour of the normal phase. We have also used this method to probe a Bose gas of 7Li atoms close to a Feshbach resonance. We have measured the Equation of State of the Bose gas as a function of interactions at very low temperature. For the first time, we measured quantitatively the Lee-Huang-Yang beyond mean-field correction to the ground-state energy of the system, first predicted in 1957. We compared the experimental results to Quantum Monte-Carlo calculations. We have extended this study using out-of-equilibrium measurements of the Bose gas in the strongly interacting regime, which gives a first hint on the properties of the hypothetical unitary Bose gas.