In this thesis, we present theoretical studies on three topics related to multi-component ultracold gases and gauge fields. The first topic that we discuss is artificial gauge fields in ultracold gases. Recently, methods to create artificial gauge fields coupled to neutral ultracold systems using a light-induced Berry's connection have been rapidly developing. These methods are not only capable of creating Abelian gauge fields, such as a conventional magnetic field, but also non-Abelian gauge fields, which opens a way to explore and simulate a wide variety of physical models. In this thesis, we discuss various properties of bosons with Rashba-Dresselhaus spin-orbit coupling, which is a special type of non-Abelian gauge field. We investigate the stability of Bose-Einstein condensates with Rashba-Dresselhaus spin-orbit coupling, and show that the condensates are stable against quantum and thermal fluctuations. We also consider the renormalization of the bare interaction by calculating the t-matrix and its consequence on the ground state phase diagrams. The second topic discussed here is three-component ultracold fermionic systems. It is known that ferromagnetism and superfluidity can coexist at low enough temperature in three-component ultracold fermions. In this thesis, we elucidate how fermionic pairing and population imbalance enhance each other. We also describe a crossover from Bardeen-Cooper-Schrieffer state of fermionic pairing state to the limit of Bose-Einstein condensate of three weakly interacting species of molecules, as the interaction increases. Furthermore, we find an interesting similarity in the free energies between three-component ultracold fermions and quantum chromodynamics. The last topic discussed here is Niels Bohr's double-slit interference gedankenexperiment with charged particles, which argues that the consistency of elementary quantum mechanics requires that the electromagnetic field must be quantized. In the experiment a particle's path through the slits is determined by measuring the Coulomb field that it produces at large distances. Under these conditions the interference pattern must be suppressed; otherwise quantum mechanics is not consistent. The mechanism for the suppression of the interference pattern is that, as the particle's trajectory is bent in diffraction by the slits, it must radiate and the radiation must carry away phase information. Thus, the radiation field must be a quantized dynamical degree of freedom. We also consider the related setup in which one attempts to determine the path of a massive particle through an interferometer by measuring the Newtonian gravitational potential the particle produces. In this case, we show that the interference pattern would have to be finer than the Planck length and thus indiscernible. Therefore, unlike for the electromagnetic field, Bohr's argument does not imply that the gravitational field must be quantized.
In this thesis, we present theoretical studies on three topics related to multi-component ultracold gases and gauge fields. The first topic that we discuss is artificial gauge fields in ultracold gases. Recently, methods to create artificial gauge fields coupled to neutral ultracold systems using a light-induced Berry's connection have been rapidly developing. These methods are not only capable of creating Abelian gauge fields, such as a conventional magnetic field, but also non-Abelian gauge fields, which opens a way to explore and simulate a wide variety of physical models. In this thesis, we discuss various properties of bosons with Rashba-Dresselhaus spin-orbit coupling, which is a special type of non-Abelian gauge field. We investigate the stability of Bose-Einstein condensates with Rashba-Dresselhaus spin-orbit coupling, and show that the condensates are stable against quantum and thermal fluctuations. We also consider the renormalization of the bare interaction by calculating the t-matrix and its consequence on the ground state phase diagrams. The second topic discussed here is three-component ultracold fermionic systems. It is known that ferromagnetism and superfluidity can coexist at low enough temperature in three-component ultracold fermions. In this thesis, we elucidate how fermionic pairing and population imbalance enhance each other. We also describe a crossover from Bardeen-Cooper-Schrieffer state of fermionic pairing state to the limit of Bose-Einstein condensate of three weakly interacting species of molecules, as the interaction increases. Furthermore, we find an interesting similarity in the free energies between three-component ultracold fermions and quantum chromodynamics. The last topic discussed here is Niels Bohr's double-slit interference gedankenexperiment with charged particles, which argues that the consistency of elementary quantum mechanics requires that the electromagnetic field must be quantized. In the experiment a particle's path through the slits is determined by measuring the Coulomb field that it produces at large distances. Under these conditions the interference pattern must be suppressed; otherwise quantum mechanics is not consistent. The mechanism for the suppression of the interference pattern is that, as the particle's trajectory is bent in diffraction by the slits, it must radiate and the radiation must carry away phase information. Thus, the radiation field must be a quantized dynamical degree of freedom. We also consider the related setup in which one attempts to determine the path of a massive particle through an interferometer by measuring the Newtonian gravitational potential the particle produces. In this case, we show that the interference pattern would have to be finer than the Planck length and thus indiscernible. Therefore, unlike for the electromagnetic field, Bohr's argument does not imply that the gravitational field must be quantized.
Since the realization of Bose-Einstein condensation in 1995, experimentalists in ultracold atomic gas community have developed a set of entirely new methods to study quantum gases. In ultracold atomic gases, implementation of synthetic gauge fields, the spin-orbit coupling, optical lattices and Feshbach resonances, provides a clean and highly controllable platform to tune the parameters of the system, which are typically fixed for real materials. Motivated by the rapid developments of experimental progresses, I investigate the ultracold atomic gases with synthetic gauge fields. \n \nIn this thesis, a bosonic gas in optical lattices in the presence of an effective staggered magnetic flux will be studied, using a generalized Bose-Hubbard model. Phase transition from the Mott insulating phase to the superfluid phase will be discussed. Novel superfluid phases including the stripe phase and the plane wave phase will be discussed. Furthermore, the existence of a tricritical point in the phase diagram is predicted. \n \nTo study Fermi systems with synthetic gauge fields, a spin-1=2 Fermi gas with Rashba spin-orbit coupling is investigated by studying the structure of vortex line through the Bardeen-Cooper-Schrieffer to Bose-Einstein condensation (BCSBEC) crossover. Due to the spin-orbit coupling, the s-wave interacting Fermi gas behaves like an effective p-wave paired system. A discussion about the vortex line structure will be provided by calculating the spatial distribution of the gap function, the density, the particle current, and also the elementary excitations. It has been found that the most robust superfluid occurs around the unitary point, indicated by the smallest healing length and the largest critical current. \n \nFinally, motivated by the recent progress of p-wave resonantly Fermi gas in experiment, an extension of Nozières-Schmitt-Rink method to the p-wave case will be given. An investigation of the normal state properties will be given by calculating the equation of state and the p-wave contacts, which have been measured by the experiment. Furthermore, the superfluid transition temperature with realistic parameters appropriate for experiment is calculated and is found to be within experimental reach.
The present thesis studies a variety of cold atomic systems in artificial gauge fields. In the first part we focus on fractional quantum Hall effects, asking whether interesting topological states can be realized with cold atoms. We start by making a close connection to solid-state systems and first consider fermionic atoms with dipolar interactions. Assuming the system to be in the Laughlin state, we evaluate the energy gap in the thermodynamic limit as a measure for the robustness of the state. We show that it can be increased by additionally applying a non-Abelian gauge field squeezing the Landau levels. We then switch to bosonic systems with repulsive contact interactions. Artificial magnetic fields for cold bosons have extensively been discussed before in the context of rotating Bose gases. We follow a different approach where the gauge field is due to an atom-laser coupling. Thus, transitions between different dressed states have to be included. They are shown to break the cylindrical symmetry of the system. Modifying the Laughlin state and the Moore-Read state accordingly, we determine the parameter regimes where they are good representations for the ground state of the system obtained via exact diagonalization. One of the most interesting feature of fractional quantum Hall states is the anyonic behavior of their excitations. We therefore also study quasiholes in the Laughlin state and the modified Laughlin state. They are shown to posses anyonic properties, which become manifest even in small systems. Moreover, the dynamics of a single quasihole causes visible traces in the density of the system which allow to clearly distinguish the Laughlin regime from less correlated phases. In the latter, a sequence of collapses and revivals of the quasihole can be observed, which is absent in the Laughlin regime. Extending our study to bosonic systems with a pesudospin-1/2 degree of freedom, we discuss the formation of strongly correlated spin singlets. Strikingly, at filling v=4/3, the system is described by a state with non-Abelian excitations, which is constructed as the zero-energy ground state of repulsive three-body contact interactions. Systems with internal degrees of freedom also allow for implementing artificial spin-orbit coupling. It is shown to give rise to a variety of incompressible states. In the second part of the thesis, we concentrate on condensed system. Bose-Einstein condensates with spin-orbit coupling are shown to have a degeneracy on the mean-field level, which is lifted by quantum and thermal fluctuations. The system becomes experimentally feasible in three dimensions, where the condensate depletion remains finite, and thus allow for an experimental observation of this order-by-disorder mechanism. Finally, we study the influence of Abelian and non-Abelian gauge fields on the quantum phase transitions of bosons in a square optical lattice. Re-entrant superfluid phases and superfluids at finite momenta are interesting properties featured by such systems.
Gauge theories are ubiquitous in physics. Many intriguing phenomena in condensed matter physics owe to the action of the electromagnetic field, which is an Abelian gauge theory. The numerical treatment of many-body systems is inherently complex due to the exponentially growing size of the Hilbert space. While in one dimension an area law guarantees that numerical methods on classical computers can deal with strongly correlated systems, in higher dimensions the quantum simulation comes as the panacea for the many-body problem. The present thesis comprises the elaboration of experimentally feasible methods for the quantum simulation of dynamical Abelian gauge fields with ultra-cold gases of neutral atoms and the theoretical analysis of the related model Hamiltonians. As neutral atoms do not interact with external vector potentials like charged particles would do, the gauge fields have to be artificially engineered. The elements of a gauge theory that need to be replicated on a quantum simulator vary depending on the subject of investigation. The key ingredient at the root of many condensed matter phenomena, from the quantum Hall effect to superconductivity and chiral topological insulators, is the Berry phase. Whilst artificial static gauge fields have been widely explored, much remains to do regarding the realization of artificial dynamical gauge fields. In Chapter 3 we present a method based on the amplitude modulation of a one-dimensional optical lattice, which allows for an unprecedented degree of control over a wide range of parameters. The method also comprises the generation of a density-dependent complex phase, fundamental to the creation of anyonic pseudo-particles. The anyons are amenable of observation through interferometric measurement, realizable with the same experimental set-up. With regard to gauge theories, the Berry phase is just the visible tip of the iceberg. Below the waterline, there is more to consider in order to comprehensively reproduce a gauge theory, like the electric and magnetic fields in quantum electrodynamics. Moreover, a full account for the inherent symmetry is crucial to investigate phenomena proper of non-Abelian gauge theories in the context of high-energy physics, such as confinement. For this collection of topics, one can turn to lattice gauge theories. In Chapter 5, we consider a class of lattice gauge theories particularly suitable for quantum simulation, the Quantum Link Model. The study of the Abelian U(1) Quantum Link Model on a ladder geometry reveals a highly non-trivial phase diagram, featuring a symmetry-protected topological phase. In both Chapters, innovative solutions for the experimental realization of the model Hamiltonians are designed and proposed. To gain numerical access to the ground-state properties and the dynamics of the systems investigated we make use of state-of-the-art numerical methods based on Tensor Networks. The elements of the numerical analysis carried out throughout this thesis are presented in Chapter 6. In the last part we offer an outlook on research perspectives related to the topics discussed in the thesis.
In this Thesis we discussed ultracold Fermi gas with an s-wave interaction and synthetic spin-orbit coupling under a variety of conditions. We considered the system in both three and two spatial dimensions, with equal-Rashba-Dresselhaus type or Rashba-only type of spin-orbit-coupling, and with or without an artificial Zeeman field. We found competing effects on Fermionic superfluidity from spin-orbit coupling and Zeeman fields, and topologically non-trivial states in the presence of both fields. We gave an outlook on the many-body physics in the last.
The study of systems formed from ultracold atomic gases has emerged to become one of the most active research fields within the condensed matter landscape. These highly controllable macroscopic systems amalgamate ideas from many sub disciplines of physics, including the study of low temperatures, quantum optics and quantum information theory as well as the seemingly disparate field of high energy physics. The central concept of this thesis is gauge theories as applied to systems of bosonic atoms, which at temperatures close to absolute zero form Bose-Einstein condensates. To simulate the mathematical structure of a gauge theory, the geometric (Berry) phase formalism is adopted. This is in turn accomplished by considering the adiabatic following of the eigenstates of the light-matter coupling for an ensemble of atoms forming a Bose-Einstein condensate. These concepts are then applied to show how one can generate a spin-orbit coupling in a one-dimensional condensate, which additionally features a random mass term that allows us to study the physics of Anderson localization in an intriguing “quasi” relativistic regime. One of the features of light induced gauge potentials is that they are static; in the sense that there is no feedback between the light-matter interaction and the matter field. In the second part of this thesis it is demonstrated how such a feedback mechanism can be induced by the appropriate modification of the light-matter interaction. The consequences this has for the condensate are then described at the mean-field level, including the expected experimental signatures of the resulting ‘interacting’ gauge theory, in terms of the expansion of the condensate and also the structure of the solitons of this nonlinear system. Finally, this nonlinear model is applied to a double well system, from which the associated Bose-Hubbard model is derived and analysed; and the nonlinear Josephson problem studied.
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