This article is a report of Projet bibliographique of M1 at École Normale Supérieure. In this article we reviewed the historical developments in artificial gauge fields and spin-orbit couplings in cold atom systems. We resorted to origins of literatures to trace the ideas of the developments. For pedagogical purposes, we tried to work out examples carefully and clearly, to verified the validity of various approximations and arguments in detail, and to give clear physical and mathematical pictures of the problems that we discussed. The first part of this article introduced the fundamental concepts of Berry phase and Jaynes-Cummings model. The second part reviewed two schemes to generate artificial gauge fields with N-pod scheme in cold atom systems. The first one is based on dressed-atom picture which provide a method to generate non-Abelian gauge fields with dark states. The second one is about rotating scheme which is achieved earlier historically. Non-Abelian gauge field inevitably leads to spin-orbit coupling. We reviewed some developments in achieve spin-orbital coupling theoretically and experimentally. The fourth part was devoted to recently developed idea of optical flux lattice that provides a possibility to reach the strongly correlated regime in cold atom systems. We developed a geometrical interpretation based on Cooper's theory. Some useful formulae and their proofs were listed in the Appendix.
This article is a report of Projet bibliographique of M1 at École Normale Supérieure. In this article we reviewed the historical developments in artificial gauge fields and spin-orbit couplings in cold atom systems. We resorted to origins of literatures to trace the ideas of the developments. For pedagogical purposes, we tried to work out examples carefully and clearly, to verified the validity of various approximations and arguments in detail, and to give clear physical and mathematical pictures of the problems that we discussed. The first part of this article introduced the fundamental concepts of Berry phase and Jaynes-Cummings model. The second part reviewed two schemes to generate artificial gauge fields with N-pod scheme in cold atom systems. The first one is based on dressed-atom picture which provide a method to generate non-Abelian gauge fields with dark states. The second one is about rotating scheme which is achieved earlier historically. Non-Abelian gauge field inevitably leads to spin-orbit coupling. We reviewed some developments in achieve spin-orbital coupling theoretically and experimentally. The fourth part was devoted to recently developed idea of optical flux lattice that provides a possibility to reach the strongly correlated regime in cold atom systems. We developed a geometrical interpretation based on Cooper's theory. Some useful formulae and their proofs were listed in the Appendix.
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.
We start by reviewing the concept of gauge invariance in quantum mechanics, for Abelian and Non-Ableian cases. Then we idescribe how the various gauge potential and field can be associated with the geometrical phase acquired by a quantum mechanical wave function while adiabatically evolving in a parameter space. Subsequently we show how this concept is exploited to generate light induced gauge field for neutral ultra cold bosonic atoms. As an example of such light induced Abelian and Non Abelian gauge field for ultra cold atoms we disucss ultra cold atoms in a rotating trap and creation of synthetic spin orbit coupling for ultra cold atomic systems using Raman lasers.
When a gauge potential is constructed from the Berry phase, there is an associated scalar potential coming with the gauge potential. In this paper, we investigate the impact of this scalar potential on cold atoms by constructing an artificial gauge potential in a cold-atomic system. We interpret the scalar potential as the coupling between the atom and the mass of the non-Abelian part of the gauge potential. We demonstrate that the gauge potential can produce spin-orbit coupling and that the scalar potential will suppress the spin Hall currents which are generated by spin-orbit coupling. We also discuss the observation of these phenomena in a real experiment.
When a neutral atom moves in a properly designed laser field, its center-of-mass motion may mimic the dynamics of a charged particle in a magnetic field, with the emergence of a Lorentz-like force. In this Colloquium the physical principles at the basis of this artificial (synthetic) magnetism are presented. The corresponding Aharonov-Bohm phase is related to the Berry's phase that emerges when the atom adiabatically follows one of the dressed states of the atom-laser interaction. Some manifestations of artificial magnetism for a cold quantum gas, in particular, in terms of vortex nucleation are discussed. The analysis is then generalized to the simulation of non-Abelian gauge potentials and some striking consequences are presented, such as the emergence of an effective spin-orbit coupling. Both the cases of bulk gases and discrete systems, where atoms are trapped in an optical lattice, are addressed.
The Higgs-like excitations, which distinguish from the Higgs amplitude mode in many-body system, are single-particle excitations in system with non-Abelian gauge potential. We investigate the Higgs-like excitations of cold atom system in artificial non-Abelian gauge potential. We demonstrate that when a non-Abelian gauge potential is reduced to Abelian potential, its Abelian part constructs spin-orbit coupling, and its non-Abelian part emerges Higgs-like excitations. The Higgs-like excitations induce a mass of the non-Abelian gauge field, which offsets the defect of massless of the gauge theories. We show that the mass of gauge field can affect the spin Hall currents which are produced by the spin-orbit coupling. We also discuss the observation of these phenomena in real experiment.
Gauge fields play a prominent role in modern physics from electromagnetic theory to standard model of particle physics. These are cutting edge mathematical tools to understand fundamental forces and interaction between sub-atomic particles. Though gauge fields are considered natural tools of study in high-energy physics, they can be extended to low-energy condensed matter physics to study the behaviour of quantum particles in special environments like electrons in a magnetic field. Apart from naturally occurring gauge fields, controllable artificial gauge fields can be used to tailor physical effects on quantum systems like neutral atoms, qubits, photons, anions etc. This thesis reports the study of laser-induced artificial non-Abelian gauge fields interacting with an ultracold atomic wave packet in free space. In sharp contrast to the Abelian gauge fields, spatially uniform non-Abelian gauge fields can induce non-inertial motion of a particle. This motion is locked to the pseudo-spin state of the system and induces spin-orbit coupling. These non-Abelian gauge fields are experimentally realized by adiabatic evolution of the state of the atoms in the degenerate subspace of the coupling Hamiltonians. We produce SU(2) and SU(3) non-Abelian gauge fields using doubly degenerate dark state subspace of tripod and triply degenerate dark state subspace of double-tripod atom-light coupling schemes, respectively. Using the SU(2) gauge field in one dimension, we demonstrate the function of an atomtronic Datta-Das transistor (DDT) where spin polarized atoms enter the gate region and exit with their spin orientation controlled by the gate parameter analogous to the gate voltage in conventional field effect transistor. We demonstrate the geometric nature of the spin rotation in our DDT by showing insensitivity of the result to the input velocity and velocity dispersion of the atomic cloud. We have also implemented a Ramsey interferometric sequence to extract the phase of the output state of the DDT showing that the spin rotation by the DDT is coherent. In two dimensions, the dynamics of the wave packet in SU(2) gauge field show Zitterbewegung oscillations revealing spin Hall effect characteristics of the system. Whereas the dynamics of the wave packet in SU(3) gauge field show oscillations with multiple frequencies emulating color-orbit coupling. These frequencies correspond to the energy separation between the eigenstates of the Hamiltonian. Using the SU(3) Hamiltonian, we show two paths of reaching the same final state from the initial state demonstrating the existence of additional ladder operators in the SU(3) system in addition to the well-known raising and lowering operators of angular momentum represented by SU(2) symmetry. With this study, we realize a system that can be used to study the rich physics of SU(3) Hamiltonians on table top experiments.