Using exact diagonalization for a small system of cold bosonic atoms, we analyze the emergence of strongly correlated states in the presence of an artificial magnetic field. This gauge field is generated by a laser beam that couples two internal atomic states, and it is related to Berry's geometrical phase that emerges when an atom follows adiabatically one of the two eigenstates of the atom-laser coupling. Our approach allows us to go beyond the adiabatic approximation, and to characterize the generalized Laughlin wave functions that appear in the strong magnetic-field limit.
Using exact diagonalization for a small system of cold bosonic atoms, we analyze the emergence of strongly correlated states in the presence of an artificial magnetic field. This gauge field is generated by a laser beam that couples two internal atomic states, and it is related to Berry's geometrical phase that emerges when an atom follows adiabatically one of the two eigenstates of the atom-laser coupling. Our approach allows us to go beyond the adiabatic approximation, and to characterize the generalized Laughlin wave functions that appear in the strong magnetic-field limit.
We use the exact diagonalization method to analyze the possibility of generating strongly correlated states in two-dimensional clouds of ultracold bosonic atoms that are subjected to a geometric gauge field that was created by coupling two internal atomic states to a laser beam. On tuning the gauge field strength, the system undergoes stepwise transitions between different ground states (GSs), which we describe by using analytical trial wave functions, including the Pfaffian (Pf), the Laughlin and a Laughlin quasiparticle many-body state. Whereas for an infinitely strong laser field, the internal degree of freedom of the atoms can adiabatically follow their center-of-mass movement, a finite laser intensity gives rise to non-adiabatic transitions between the internal states, which are shown to break the cylindrical symmetry of the Hamiltonian. We study the influence of the asymmetry on the GS properties of the system. The main effect is to reduce the overlap of the numerical solutions with the analytical trial expressions by occupying states with higher angular momentum. Thus, we propose generalized wave functions arising from the Laughlin and Pf wave functions by including components where extra Jastrow factors appear while preserving important features of these states. We analyze quasihole excitations over the Laughlin and generalized Laughlin states and show that they possess effective fractional charge and obey anyonic statistics. Finally, we discuss the observability of the Laughlin state for increasing numbers of particles.
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 propose a scheme to generate an Abelian gauge field in an atomic gas using two crossed laser beams. If the internal atomic state follows adiabatically the eigenstates of the atom-laser interaction, Berry's phase gives rise to a vector potential that can nucleate vortices in a Bose gas. The present scheme operates even for a large detuning with respect to the atomic resonance, making it applicable to alkali-metal atoms without significant heating due to spontaneous emission. We test the validity of the adiabatic approximation by integrating the set of coupled Gross-Pitaevskii equations associated with the various internal atomic states, and we show that the steady state of the interacting gas indeed exhibits a vortex lattice, as expected from the adiabatic gauge field.
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.
Synthetic gauge fields for ultracold neutral atoms-engineered using the interaction between laser fields and the atoms' internal 'spin' degrees of freedom-provide promising techniques for generating the large (synthetic) magnetic fields required to reach the fractional quantum Hall (FQH) limit in quantum gases, bosonic or fermionic alike.Because neutral atoms can move in a nearly disorder-free environment and they have extremely simple contact interactions, the resulting FQH states would be revealed in their most essential form.Moreover, bosonic FQH states represent a new frontier and have never been seen in any setting.Going beyond electromagnetismʼs conventional scalar gauge field, it is possible to create more general non-Abelian gauge potentials.When these are spatially uniform, they are equivalent to spin-orbit coupling familiar in material systems, and can lead to cold atom analogs of topological insulators and topological superconductors.In this tutorial, we introduce basic concepts underlying these gauge fields, making connections to the Aharonov-Bohm phase and geometric phase.We focus on the system of neutral atoms 'dressed' by multiple laser beams, where the eigenstates of the resulting Hamiltonian are known as dressed states.Synthetic gauge potentials arise from the unitary transformation required to express these dressed states in terms of the laser-free eigenstates.We discuss stability of laser-dressed atoms corresponding to the adiabatic condition and the probability of non-adiabatic transitions.Adopting both the semiclassical and quantum mechanical approaches, we demonstrate they agree in the suitable limit.We also analyze using both the conventional adiabatic picture and exact picture, where the kinetic energy is neglected in the former and retained in the latter picture.
Cold atomic gases are perfect laboratories for realization of quantum simulators. In order to simulate solid state systems in the presence of magnetic fields special effort has to be made because atoms are charge neutral. There are different methods for realization of artificial magnetic fields, that is the creation of specific conditions so that the motion of neutral particles mimics the dynamics of charged particles in an effective magnetic field. Here, we consider adiabatic motion of atoms in the presence of an evanescent wave. Theoretical description of the adiabatic motion involves artificial vector and scalar potentials related to the Berry phases. Due to the large gradient of the evanescent field amplitude, the potentials can be strong enough to induce measurable effects in cold atomic gases. We show that the resulting artificial magnetic field is able to induce vortices in a Bose-Einstein condensate trapped close to a surface of a prism where the evanescent wave is created. We also analyze motion of an atomic cloud released from a magneto-optical trap that falls down on the surface of the prism. The artificial magnetic field is able to reflect falling atoms that can be observed experimentally.