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.
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 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 study of ultracold trapped fermionic gases is a rapidly exploding subject [1-10] which is defining new directions in condensed matter and atomic physics. It has also captured the attention of physicists who study color superconducting quark matter as well as nuclear matter [11-13]. Indeed, it is hard, in recent times, to find a subfield of physics which appeals this broadly to the research community. As we come to understand the gases, experimentalists will move to address fermionic atoms in optical lattices; this will provide important insight to condensed matter physicists as analogue systems for “solving ” intractable many body problems. What makes these gases (and lattices) so important is their remarkable tunability and controllability. Using a Feshbach resonance, one can tune the attractive two-body interaction from weak to strong, and thereby make a smooth crossover from a BCS superfluid to a Bose-Einstein condensation (BEC) [14, 15]. Not only does this allow high transition temperatures Tc (relative to the Fermi energy EF) but it may also provide insights into the high temperature cuprate superconductivity [16-19]. Furthermore, one can tune the population of the two spin states, essentially at will, in this way, allowing exploration [20-22] of exotic polarized phases such as the FFLO [23-25] superfluids, of interest to condensed matter, nuclear and particle physicists. One will be able to tune the lattice parameters such as bandwidth, on-site repulsion, even random disorder, etc and thereby study the famous, and as yet, unsolved Hubbard model Hamiltonian. This paper will concentrate on those issues relating to the effects which are of current interest to experimentalists. In particular, we will study BCS-BEC crossover in atomic Fermi gases, looking at a wide range of different experiments. Our group has been principally interested in the effects of finite temperature [26, 17, 18] and in this review, we will discuss how temperature T enters into the standard crossover theory, and how temperature can be measured, and how temc○
We study ultracold gases of alkali-metal atoms in the quantum degenerate regime. The interatomic interactions in these type of systems can be tuned using resonances induced by magnetic or electric fields. The tunability of the interactions, together with the possibility of confining the atoms with several kinds of external potentials, allows for a completely novel approach to study basic problems in many-body physics, and moreover, allows to enter regimes which have never been accessible in condensed matter or nuclear physics. For example, this has led to the experimental demonstration of an intimate relation between two types of superfluidity: the crossover from Bose-Einstein condensation of tightly bound molecules to the superfluid behavior related to weakly bound BCS-like pairs. Another experimental landmark was reached when the existence of universal Efimov three-body bound states was proven in experiments with ultracold bosonic cesium atoms. In this thesis, we study several aspects of these strongly interacting and ultracold atomic gases. We develop an analytical model that encapsulates all of the relevant scattering physics in atomic systems where open-channel shape resonances and closedchannel Feshbach resonances give rise to complicated and non-trivial scattering properties. This model provides lots of physical insight and is shown to describe important quantities, such as the molecular energies and scattering phase shifts, with a high level of accuracy. The model is compared to full numerical coupled-channel calculations in two atomic systems: rubidium and lithium. We study the BCS-BEC crossover using a many-body description of the ultracold gas that includes the non-trivial energy dependence of the scattering model. We show that it gives rise to superfluid behavior associated with the formation of BCS-like pairs while the low-energy interactions are repulsive in character. The energy dependence of the interactions is crucial, as it gives rise to attractive interactions at the Fermi energy, necessary for the formation of Cooper pairs. We demonstrate new ways of controlling the interatomic interactions using a combination of magnetic and electric fields. This leads to experimental control of, for instance, the three-body parameter in the context of Efimov physics and of non-universal behavior in the BCS-BEC crossover in fermionic gases. Using a four-body method based on first principles, we solve the molecule-molecule scattering problem to calculate several important properties of bosonic and fermionic Summary 145 molecules that consist of light and heavy atoms. These type of molecules are of current experimental interest, and we predict several exciting relations between three- and four-body observables in these type of systems.
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.
This thesis covers two fields of research carried out in the context of ultracold Fermi gases: Non-equilibrium physics in the BEC-BCS crossover and the use of machine learning methods in physics. \n\t \nThe BEC-BCS crossover is of particular interest for the investigation of low-temperature superconductivity and related many-body phenomena. We realise the BEC-BCS crossover experimentally by cooling a mixture of two hyperfine states of fermionic lithium atoms far below degeneracy. To tune the interaction between the constituents, we make use of a broad magnetic Feshbach resonance between the states. We achieve temperatures far in the degenerate regime that can be tuned by a trap release heating procedure. Being able to tune temperature and interaction allows us to access the whole BEC-BCS crossover regime and investigate related phenomena. \n\t \nOne key quantity across the crossover is the critical temperature of the thermal to superfluid phase transition that signals condensation of pairs. The critical temperature can be calculated in the limits of weak interactions, however, for strong interactions theoretical treatment is difficult. In this thesis, we use neural networks to explore the critical temperature in the whole crossover regime with unprecedented accuracy. We show that a supervisedly trained neural network can not only detect the critical temperature, but also distinguish between Cooper pairs and Feshbach molecules, which was not possible before. Moreover, we show that an unsupervisedly trained autoencoder network is able to determine the critical temperature without any external input other than time-of-flight pictures. \n\t \nAnother important quantity in the crossover regime is the superfluid gap parameter. We measure the gap by exciting the Higgs mode through radio-frequency modulation of the atom population. In the BCS limit, the frequency of the Higgs mode is given by twice the gap frequency. By observing the frequency dependent response of the system we can therefore measure the gap parameter. Towards strong interactions, the observed feature broadens, signalling an instability of the mode due to the gradual violation of the required particle-hole symmetry. \n\t \nExploring the non-equilibrium behaviour of a Fermi gas in the BEC-BCS crossover might hold the key to understand the formation dynamics of pairs and how to induce superconductivity in a system that does not exhibit superconductivity in its equilibrium state. In this work, we present two ways to expose the system to a sudden change of interaction, resulting in a non-equilibrium response. Firstly, we use a fast radio-frequency transfer of one of the hyperfine states into a different, third state. We observe dynamics on several timescales in the momentum distribution and condensate fraction. Secondly, we develop a new coil system that is able to change the magnetic field at the atom position by tens of Gauss within a few microseconds. We detail the construction, testing and alignment process and present investigations into the long-time response of the system. \n\t \nMachine learning procedures can not only be used for data analysis, but also to improve upon the control over experimental systems. In this work we present a novel approach to improve the detection and correction of experimental errors exponentially. We use two machine learning approaches, principal component analysis and small, fully connected neural networks to create an empirical model of the experimental setup. Using this model allows us to reduce the required amount of data for compensation of noise sources exponentially. We demonstrate both approaches in the context of micromotion compensation on a single trapped ytterbium ion in a radio-frequency Paul trap.
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