In this thesis we investigate strongly correlated phenomena in the field of ultracold atomic gases. Chapter 2 addresses a question of the insulating phases of cold spin-one bosonic particles with antiferromagnetic interactions, such as 23 Na, in optical lattices. Magnetic properties of the ground state in the insulating regime are studied using various techniques. Chapter 3 considers a one dimensional interacting Bose-Fermi mixture with equal masses of bosons and fermions, and with equal repulsive interactions between Bose-Fermi and Bose-Bose particles. Properties of such mixture are studied using exact Bethe-ansatz techniques. Chapter 4 deals with certain phenomena which appear in the experiments with imbalanced fermionic mixtures in strongly anisotropic traps. Chapter 5 gives a comprehensive review of interference phenomena, analyzing effects which contribute to the reduction of the interference fringe contrast in matter interferometers. iii Contents
In this thesis we investigate strongly correlated phenomena in the field of ultracold atomic gases. Chapter 2 addresses a question of the insulating phases of cold spin-one bosonic particles with antiferromagnetic interactions, such as 23 Na, in optical lattices. Magnetic properties of the ground state in the insulating regime are studied using various techniques. Chapter 3 considers a one dimensional interacting Bose-Fermi mixture with equal masses of bosons and fermions, and with equal repulsive interactions between Bose-Fermi and Bose-Bose particles. Properties of such mixture are studied using exact Bethe-ansatz techniques. Chapter 4 deals with certain phenomena which appear in the experiments with imbalanced fermionic mixtures in strongly anisotropic traps. Chapter 5 gives a comprehensive review of interference phenomena, analyzing effects which contribute to the reduction of the interference fringe contrast in matter interferometers. iii Contents
The recent advances in trapping, cooling, and manipulation of alkali atoms have opened the possibility to create and study novel states of matter. The quantum nature of matter becomes relevant at ultracold temperatures and emergent phenomena, such as Bose-Einstein condensation (BEC), are strongly affected by the interaction between atoms and their statistics. In this thesis we will address some of the physics in ultracold quantum gases, with Bose (Chapter 2 and Chapter 3) and Fermi statistics (Chapter 4), as well as ultracold Bose-Fermi mixtures (Chapter 3). We will discuss phenomena driven by nonlinear interactions, such as, localisation, macroscopic quantum self-trapping, intrinsic decoherence, Mott insulating symmetry states, formation of bro-ken symmetry states and the BCS-BEC crossover. In this thesis new major results can be summarised as follows: {u2022} The establishment of the relation between stationary states and decoherence origi-nated from many-body interactions in double well bosonic systems, Chapter 2. {u2022} The suppression or enhancement of localisation related phenomena (Superfluid and Mott-Insulator states or Macroscopic Quantum Self-trapping) in Bose-Fermi mixtures due to the presence of fermions and the interplay with many-body interactions in few site systems, Chapter 3. {u2022} The mapping of the BCS-BEC crossover problem to a magnetic impurity problem in the BCSside of a Feshbach resonance, and the possible origin of the pseudo gap in strongly interacting ultracold fermion systems, Chapter 4.
In this thesis, we discuss ultracold quantum gases both in continuum and optical lattices. For the continuum Fermi gases in BCS-BEC crossover, we present an effective field theory study on the recently discovered puzzling damping phenomena on the BCS side of the crossover. We find that in contrast to the previous proposed pair-breaking mechanism of damping, the damping process is due to the interaction between superfluid phonons and thermally excited fermionic quasi particles. Results from our effective field theory are compared quantitatively with experiments, showing a good agreement. For the ultracold fermionic atoms in optical lattices, we propose two novel quantum phases. Firstly, we show that a novel superconducting pairing occurs for spin-imbalanced Fermi gases with the spin up and down Fermi levels lying within the px- and s- orbital bands of a quasi-one-dimensional optical lattice. The pairs condense at a finite momentum equal to the sum of the two Fermi momenta of spin up and down fermions, and form a p-orbital condensate. The phase diagram shows that the p-orbital pair condensate occurs in a wide range of fillings. Secondly, we study instabilities of single-species fermionic atoms in the p-orbital bands in two-dimensional square optical lattices. From the nearly-perfect nesting Fermi surfaces, charge density wave and orbital density wave orderings with stripe or checkerboard patterns are found for attractive and repulsive interactions, respectively. The superconducting phase, usually expected of attractively interacting fermions, is strongly suppressed. We also use field theory to analyze the possible liquid crystal phases in our system. For bosons, we study ultracold bosonic atoms loaded in a one-dimensional optical lattice of two-fold p-orbital degeneracy at each site, and find an anti-ferro-orbital, a homogeneous px Mott insulator phase and two kinds of superfluid phases distinguished by the orbital ordering.
Recent theoretical and experimental progress on studying one-dimensional systems of bosonic, fermionic, and Bose-Fermi mixtures of a few ultracold atoms confined in traps is reviewed in the broad context of mesoscopic quantum physics. We pay special attention to limiting cases of very strong or very weak interactions and transitions between them. For bosonic mixtures, we describe the developments in systems of three and four atoms as well as different extensions to larger numbers of particles. We also briefly review progress in the case of spinor Bose gases of a few atoms. For fermionic mixtures, we discuss a special role of spin and present a detailed discussion of the two- and three-atom cases. We discuss the advantages and disadvantages of different computation methods applied to systems with intermediate interactions. In the case of very strong repulsion, close to the infinite limit, we discuss approaches based on effective spin chain descriptions. We also report on recent studies on higher-spin mixtures and inter-component attractive forces. For both statistics, we pay particular attention to impurity problems and mass imbalance cases. Finally, we describe the recent advances on trapped Bose-Fermi mixtures, which allow for a theoretical combination of previous concepts, well illustrating the importance of quantum statistics and inter-particle interactions. Lastly, we report on fundamental questions related to the subject which we believe will inspire further theoretical developments and experimental verification.
This thesis discusses two important topics in ultracold atomic gases: strong interactions in quantum gases, and quantum Hall physics in neutral atoms.First we give a brief introduction on basic scattering models in atomic physics, and an approach to adjust the interactions between atoms.We also include a list of experimental probes in cold atom physics.After these introductions, in Chapter 3, we report a few interesting problems in strongly interacting quantum gases.We introduce the BCS-BEC crossover model and relevant many-body techniques at the beginning, and discuss the details of several specific systems.We find the Fermi gases across narrow Feshbach resonances are strongly interacting at low temperature even when the magnetic field is several widths away from the resonance.We also discuss an approach to describe the metastable repulsive branch of Bose and Fermi gases across the resonance, and find a stable region of repulsive Bose gas close to unitarity.Some studies in two dimensional Fermi gases with spin imbalance are also included, and they are closely related to a number of recent experiments.In Chapter 4, we discuss quantum Hall physics in the context of neutral atomic gases.After illustrating how the Berry phase experienced by neutral atoms is equivalent to the magnetic field in electrons, we introduce the newly developed synthetic gauge field scheme in which a gauge potential is coupled to the neutral atoms.We give a detail introduction to this Raman coupling scheme developed by NIST group, and derive the theoretical model of the system.Then we make some predictions on the evolution of quantum Hall states when an extra anisotropy is applied from the external trap.Finally, we propose some experiments to verify our predictions.4.9 Phase diagram of the cloud in rotating anisotropic traps: two phases of the BEC and QH are identified. . . . . . . . . . . . . . . . . . . . . . . . . . . .4.10 Distortion and transition in vortex lattices of a rapidly rotating BEC in an anisotropic trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.11 Density profiles at cut x = 0 and y = 0 of the Laughlin wavefunctions at different positions in figure 4.9. . . . . . . . . . . . . . . . . . . . . . . . . .4.12 Distribution of zeros in the relative coordinate of the ground state of two body problem in rotating anisotropic traps. . . . . . . . . . . . . . . . . . .4.13 Variation in the distance of splitting zeros as a function of inverse interaction strength, for fixed anisotropy α = 0.96. . . . . . . . . . . . . . . . . . . . . .4.14 Density profiles after TOF expansion of the gas in a rotating isotropic trap.4.15 Density profiles after TOF expansion of anisotropy α = 0.94. . . . . . . . .4.16 Second order correlation in equation 4.64, in a rotating isotropic (left) and anisotropic (right) trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.17 A sketch of high-resolution in situ imaging technique in neutral atoms. . . .4.18 In situ images of the quantity |ψ(r/2, -r/2)| for an anisotropy α = 0.96. .x
This thesis explores certain extraordinary phenomena that occur when a gas of neutral atoms is cooled to the coldest temperatures in the universe --- much colder, in fact, than the electromagnetic radiation that permeates the vacuum of interstellar space. At those extreme temperatures, quantum effects dominate and the collective behaviour of the atoms can have unexpected consequences. For example, Bose-Einstein condensation may occur where the atoms lose their individual identities to coalesce into a macroscopic quantum particle. Although such ultracold atomic gases are interesting in their own right, much of the excitement generated in this field is due to the possibility that studying these gases could shed light on intractable problems in other areas of physics. This is predominantly due to the uniquely high degree of control over various physical parameters that ultracold atomic gases afford to experimentalists. Recent technological advances exploit this advantage to study quantum phenomena in a detail that would not be possible in other systems. For instance, atoms can be made to attract or repel each other, the strength of this interaction can be set to almost any value, and external potentials of various geometries and periodicities can be introduced. In this way, atoms can be used to model phenomena as diverse as the quark-gluon plasmas arising in high-energy particle physics, the colour superfluids conjectured to exist in the core of neutron stars, and the high-temperature superconductivity exhibited by electrons on the ion lattice of certain compounds. Indeed, ultracold atomic gases also have a demonstrated applicability to quantum information and computation. Due to a subtle interplay between electronic and nuclear spins known as the hyperfine interaction, atoms can have either an integer or half-integer total spin quantum number, making them either bosonic or fermionic at low temperatures, respectively. With the exception of chapter 7, the work presented here concerns fermionic atoms in periodic potential formed by interfering laser beams. Indeed, the standing light wave created by the interfering beams gives rise to a lattice potential because of the Stark effect which couples the electronic energy levels of the atoms to the spatially undulating electric field. Furthermore, fermionic atoms can be prepared in two different hyperfine states corresponding to the the \\spin-up" and \\spin-down" quantum states, and as such mimic electrons moving in the lattice structure of solids. This system is well described by the famous Hubbard model which we introduce in chapter 2 and, under certain conditions, undergoes a phase transition into the N eel state which believed to be a precursor to superconductivity in certain high-temperature superconductors. In chapter 3, we calculate precisely how the N eel state may be achieved in an ultracold fermionic atom gas. When the number of spin-up and spin-down atoms is unequal the system becomes spin-canted and exhibits both ferro- and antiferromagnetic characteristics, as we show in chapter 4. We also nd there are topological excitations present in the quantum spin texture known as merons which have never unambiguously been observed before. In order to form a Bose-Einstein condensate, fermionic atoms must rst form pairs, and can do so in two contrasting ways. The relationship between these two qualitatively di erent forms of pairing is described in chapter 5, and we examine how these two types of pairs transform into one another in an optical lattice in chapter 6. Finally, chapter 7 is a detailed eld-theoretic study of pairing as it occurs in an ultracold Bose gas. There, we nd there is an intriguing bosonic analogy of the two forms of fermion pairing and explore the properties of these pairs.
The study of out-of-equilibrium dynamics in ultra-cold gases is a new and exciting field, driven largely by the recent experimental advances in controlling and imaging cold clouds. The experimental and theoretical work thus far has been somewhat exploratory and largely numerical in nature, as the very paradigms for thinking about these systems are not well established. In this thesis I consider several different scenarios of ultra-cold bosonic and fermionic gases driven out of equilibrium and study their properties. In Chapter 1, I provide an overview of the phenomenology of ultra-cold gases, highlighting the timescales governing these systems and how the experimentalist can tune them. I discuss how cold gases can be cooled and trapped and discuss the basic physics behind optical lattices. I also discuss experimental probes of these gases, in particular the new high resolution imaging techniques developed recently at Chicago, Munich and Harvard. In Chapter 2, I discuss an early experiment (circa. 2008) which observed long lived spin dynamics in a thermal spin-1/2 Fermi gas. This experiment is an nice illustration of interesting physics resulting from the separation of timescales between spin and collisional dynamics. In my opinion, it is an excellent example of why cold gases are naturally suited to studying non-equilibrium dynamics. I simulate the experiment numerically using a collisionless Boltzmann equation and explain the observed spin dynamics both qualitatively and quantitively. In Chapter 3, I continue the discussion of spin waves in thermal gases by extending previous works on spin-1/2 gases to spin-1 Bose gases. In contrast to Chapter 2, the bulk of the work in this Chapter is analytic in nature. In particular, I find a spin wave instability in the thermal spin-1 Bose gas, which is the high temperature analog of the polar to ferromagnetic transition in a spin-1 Bose Einstein condensate. In Chapter 3, I turn my attention to bosonic systems and briefly review the the Bogoliubov mean-field theory. I calculate the momentum distribution and density-density correlation function of an interacting Bose gas within the Bogoliubov framework. Then I consider bosons in an optical lattice, and introduce the Bose Hubbard model. I calculate the mean-field phase diagram of the Bose Hubbard model and then consider fluctuations about the mean field, and derive the excitation spectrum of the lattice gas in the superfluid and insulating regimes. In Chapter 4, I ask what we learn by studying the dynamics of correlation functions following a sudden change in the interactions in a superfluid. Using the Bogoliubov theory developed in Chapter 3, I will show how the underlying excitation spectrum influences the long and short time behavior of the correlation functions. By considering a lattice dispersion, I study the analogous problem in a weak optical lattice and discuss how the lattice dispersion leads to additional features in the correlation functions. I will also discuss the timescale governing the revival of the condensate fraction in a quantum depleted gas. In Chapter 5, I derive equations of motion governing the dynamics of one and two body correlation functions in the single-band Bose Hubbard model, applicable to bosons in deep lattices. I then consider a simple quench from a Mott insulating initial state to a weakly interacting final state and produce analytic expressions describing the dynamics of correlations following such a quench. I discuss the timescale for the development of long range order following such a quench. I study the problem of chapter 4 using an equations of motion approach. This approach complements the Bogoliubov approach of Chapter 4. First, I derive exact expressions for a quench to a non-interacting state. I then consider how interactions redistribute quasi-momentum to first order in perturbation theory in different dimensions. In Chapter 6, I calculate the relevant timescales for local and global dynamics in trapped lattice Bose gases, a work done in collaboration with Dr. Kaden R. A Hazzard. Using a time-dependent Gutzwiller mean-field theory, I show that the timescale for local equilibration in these systems is fast in experimental terms. I then show that due to the spatial inhomogeneities inherent to cold gases, achieving global equilibrium can be quite complicated, sometimes taking longer than the lifetime of the experiment, an issue of practical importance to current day experiments. I continue this discussion in Chapter 7 which is a collaborative work with experimentalists David McKay and Prof. Brian DeMarco from the University of Toronto and the University of Illinois, Urbana Champaign. Using experimental and numerical methods, we show that the rapid timescales for local dynamics in interacting systems invalidates a frequently used cold atom technique for mapping out the momentum distribution of atoms in an optical lattice.