Bose-Einstein condensates (BECs) in ultracold atomic gas have been studied intensively, since the state of atoms is well controlled by lasers and external electromagnetic fields.Synthetic gauge fields can be induced in a charge-neutral system by mechanical rotation of the system or by using lasers.In a BEC under a synthetic gauge fields, a quantized vortex or a vortex lattice, where multiple quantized vortices are periodically located, are observed.In binary BECs composed of bosonic atoms in two hyperfine states, parallel and antiparallel synthetic gauge fields are induced by the former (mechanical rotation) and the latter (laser) techniques, respectively.In the mean-field theory, vortex lattices appear with various configurations which depend on the ratio of the intercomponent interaction to the intracomponent one.While the ground-state phase diagrams for parallel and antiparallel fields are the same in the mean-field regime, two phase diagrams are significantly different in the quantum Hall regime, where a magnetic field is strong.In the first part of this thesis, we study the excitation spectra, correlation functions and the quantum depletion of vortex lattices by means of the effective-field theory and the Bogoliubov theory under the lowest-Landau-level (LLL) approximation.We obtain the effective-field theory of vortex lattices in binary BECs by introducing the missing elastic energy term from the symmetry consideration.By renormalizing the coupling constants, we find the effective-field theory for strong magnetic fields so that we can compare it with the Bogoliubov theory.We obtain the analytical expression of the dispersion relations for parallel and antiparallel fields by diagonalizing the effective Hamiltonian and the rescaling relations between them.From the rescaling relations, we clarify that the different excited states for parallel and antiparallel fields are indeed related to each other.By calculating the correlation function, we show that a one-particle density matrix exhibits a quasi-longrange order and that the fraction of depletion increases logarithmically with respect to the number of fluxes.By means of the Bogoliubov theory, we numerically obtain energy spectra with quadratic and linear dispersion relations for parallel and antiparallel fields and confirm the rescaling relations for all the phases.We find that the quantum depletion depends crucially on the direction of synthetic gauge fields and diverges logarithmically with respect to the number of fluxes.By calculating the correction to the ground-state energy due to zero-point fluctuations by the Bogoliubov theory, we show that the transition points shift appreciably due to quantum fluctuations.Furthermore, we show that the variations of an inner angle and an aspect ratio with respect to the ratio of interactions are altered from those obtained by the mean-field theory for rhombic and rectangular lattices, respectively.We find that the quantum fluctuation changes the ground-state phase diagram and leads to the distinction between the phase diagrams for parallel and antiparallel fields which are unique to binary BECs.Moreover, we relate the quantum depletion to shifts of the transition points since the phase diagram is largely modified when the quantum depletion which characterizes the quantum fluctuation is also large.In the second part of the present thesis, we study the intercomponent entanglement spectrum (ES) and entanglement entropy (EE) of vortex lattices in binary BECs by exploiting the formalisms developed in the first part.By means of the effective-field theory, we show that the ES exhibits a square-root dispersion relation for all the phases and that the rescaling relation holds between the ESs for parallel and antiparallel fields, similarly to the energy spectra.We confirm these predictions through a numerical calculation by List of PublicationsThis Thesis is based on the following publications. Collective modes of vortex lattices in two-component Bose-Einstein condensates under synthetic gauge fields
Bose-Einstein condensates (BECs) in ultracold atomic gas have been studied intensively, since the state of atoms is well controlled by lasers and external electromagnetic fields.Synthetic gauge fields can be induced in a charge-neutral system by mechanical rotation of the system or by using lasers.In a BEC under a synthetic gauge fields, a quantized vortex or a vortex lattice, where multiple quantized vortices are periodically located, are observed.In binary BECs composed of bosonic atoms in two hyperfine states, parallel and antiparallel synthetic gauge fields are induced by the former (mechanical rotation) and the latter (laser) techniques, respectively.In the mean-field theory, vortex lattices appear with various configurations which depend on the ratio of the intercomponent interaction to the intracomponent one.While the ground-state phase diagrams for parallel and antiparallel fields are the same in the mean-field regime, two phase diagrams are significantly different in the quantum Hall regime, where a magnetic field is strong.In the first part of this thesis, we study the excitation spectra, correlation functions and the quantum depletion of vortex lattices by means of the effective-field theory and the Bogoliubov theory under the lowest-Landau-level (LLL) approximation.We obtain the effective-field theory of vortex lattices in binary BECs by introducing the missing elastic energy term from the symmetry consideration.By renormalizing the coupling constants, we find the effective-field theory for strong magnetic fields so that we can compare it with the Bogoliubov theory.We obtain the analytical expression of the dispersion relations for parallel and antiparallel fields by diagonalizing the effective Hamiltonian and the rescaling relations between them.From the rescaling relations, we clarify that the different excited states for parallel and antiparallel fields are indeed related to each other.By calculating the correlation function, we show that a one-particle density matrix exhibits a quasi-longrange order and that the fraction of depletion increases logarithmically with respect to the number of fluxes.By means of the Bogoliubov theory, we numerically obtain energy spectra with quadratic and linear dispersion relations for parallel and antiparallel fields and confirm the rescaling relations for all the phases.We find that the quantum depletion depends crucially on the direction of synthetic gauge fields and diverges logarithmically with respect to the number of fluxes.By calculating the correction to the ground-state energy due to zero-point fluctuations by the Bogoliubov theory, we show that the transition points shift appreciably due to quantum fluctuations.Furthermore, we show that the variations of an inner angle and an aspect ratio with respect to the ratio of interactions are altered from those obtained by the mean-field theory for rhombic and rectangular lattices, respectively.We find that the quantum fluctuation changes the ground-state phase diagram and leads to the distinction between the phase diagrams for parallel and antiparallel fields which are unique to binary BECs.Moreover, we relate the quantum depletion to shifts of the transition points since the phase diagram is largely modified when the quantum depletion which characterizes the quantum fluctuation is also large.In the second part of the present thesis, we study the intercomponent entanglement spectrum (ES) and entanglement entropy (EE) of vortex lattices in binary BECs by exploiting the formalisms developed in the first part.By means of the effective-field theory, we show that the ES exhibits a square-root dispersion relation for all the phases and that the rescaling relation holds between the ESs for parallel and antiparallel fields, similarly to the energy spectra.We confirm these predictions through a numerical calculation by List of PublicationsThis Thesis is based on the following publications. Collective modes of vortex lattices in two-component Bose-Einstein condensates under synthetic gauge fields
Abstract We study binary Bose–Einstein condensates subject to synthetic magnetic fields in mutually parallel or antiparallel directions. Within the mean-field theory, the two types of fields have been shown to give the same vortex-lattice phase diagram. We develop an improved effective field theory to study properties of collective modes and ground-state intercomponent entanglement. Here, we point out the need to introduce renormalized coupling constants for coarse-grained densities. We show that the low-energy excitation spectra for the two types of fields are related to each other by suitable rescaling with the renormalized coupling constants. By calculating the entanglement entropy, we find that for an intercomponent repulsion (attraction), the two components are more strongly entangled in the case of parallel (antiparallel) fields, in qualitative agreement with recent studies for a quantum (spin) Hall regime. We also find that the entanglement spectrum exhibits an anomalous square-root dispersion relation, which leads to a subleading logarithmic term in the entanglement entropy. All of these are confirmed by numerical calculations based on the Bogoliubov theory with the lowest-Landau-level approximation. Finally, we investigate the effects of quantum fluctuations on the phase diagrams by calculating the correction to the ground-state energy due to zero-point fluctuations in the Bogoliubov theory. We find that the boundaries between rhombic-, square-, and rectangular-lattice phases shift appreciably with a decrease in the filling factor.
We study collective modes of vortex lattices in two-component Bose–Einstein condensates subject to synthetic magnetic fields in mutually parallel or antiparallel directions. By means of the Bogoliubov theory with the lowest-Landau-level approximation, we numerically calculate the excitation spectra for a rich variety of vortex lattices that appear commonly for parallel and antiparallel synthetic fields. We find that in all of these cases there appear two distinct modes with linear and quadratic dispersion relations at low energies, which exhibit anisotropy reflecting the symmetry of each lattice structure. Remarkably, the low-energy spectra for the two types of fields are found to be related to each other by simple rescaling when vortices in different components overlap owing to an intercomponent attraction. These results are consistent with an effective field theory analysis. However, the rescaling relations break down for interlaced vortex lattices appearing with an intercomponent repulsion, indicating a nontrivial effect of an intercomponent vortex displacement beyond the effective field theory. We also find that high-energy parts of the excitation bands exhibit line or point nodes as a consequence of a fractional translation symmetry present in some of the lattice structures.
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.
The dissertation explores the effects of synthetic spin-orbit coupling on the behaviour of quantum gases in several different contexts. We first study realistic methods to create vortices in spin-orbit-coupled (SOC) Bose-Einstein condensates (BEC). We propose two different methods to induce thermodynamically stable static vortex configurations: (1) to rotate both the Raman lasers and the anisotropic trap; and (2) to impose a synthetic Abelian field on top of synthetic spin-orbit interactions. We solve the Gross-Pitaevskii equation for several experimentally relevant regimes and find new interesting effects such as spatial separation of left- and right-moving spin-orbit-coupled condensates, and the appearance of unusual vortex arrangements. Next we consider cold atoms in an optical lattice with synthetic SOC in the Mott-insulator regime. We calculate the parameters of the corresponding tight-binding model and derive the low-energy spin Hamiltonian which is a combination of Heisenberg model, quantum compass model and Dzyaloshinskii-Moriya interaction. We find that the Hamiltonian supports a rich classical phase diagram with collinear, spiral and vortex phases. Next we study the time evolution of the magnetization in a Rashba spin-orbit-coupled Fermi gas, starting from a fully-polarized initial state. We model the dynamics using a Boltzmann equation, which we solve in the Hartree-Fock approximation. The resulting non-linear system of equations gives rise to three distinct dynamical regimes controlled by the ratio of interaction and spin-orbit-coupling strength λ: for small λ, the magnetization decays to zero. For intermediate λ, it displays undamped oscillations about zero and for large λ, a partially magnetized state is dynamically stabilized. Motivated by an interesting stripe phase which appears in BEC with SOC [Li et al., Phys. Rev. Lett. 108, 225301 (2011)], we study the finite-temperature phase diagram of a pseudospin-1/2 Bose gas with contact interactions. We show that strong inter-spin interactions can lead to the appearance of magnetically ordered phases at temperatures above the superfluid transition. For the case of inter-spin attraction, we also discuss the possibility of a bosonic analogue of the Cooper-paired phase, however this state is not energetically favourable. We extend our calculations to a spin-orbit-coupled Bose gas to investigate the possibility of stripe ordering in the normal phase. However, within our approximations, we do not find an instability towards stripe formation. Finally, we consider a two-dimensional Bose gas at zero temperature with an underlying quartic single-particle dispersion in one spatial direction. This Hamiltonian can be realized using the NIST scheme of spin-orbit coupling [Y.-J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature 471, 83 (2011)], or using the shaken lattice scheme of Parker et al. [C. V. Parker, L.-C. Ha and C. Chin, Nature Physics 9, 769 (2013)]. By numerically comparing energies of various trial wave-functions, we show that, at low densities, the ground state is strongly correlated, in contrast to a typical mean-field BEC. The trial wave-function with the lowest energy is of Jastrow-type and it describes a state with finite, but strongly reduced, condensate fraction.
Abstract We consider a two-dimensional weakly interacting ultracold Bose gas whose constituents are two-level atoms. We study the effects of a synthetic density-dependent gauge field that arises from laser–matter coupling in the adiabatic limit with a laser configuration such that the single-particle zeroth-order vector potential corresponds to a constant synthetic magnetic field. We find a new exotic type of current nonlinearity in the Gross–Pitaevskii equation which affects the dynamics of the order parameter of the condensate. We investigate the rotational properties of this system in the Thomas–Fermi limit, focusing in particular on the physical conditions that make the existence of a quantized vortex in the system energetically favourable with respect to the non-rotating solution. We point out that two different physical interpretations can be given to this new nonlinearity: firstly it can be seen as a local modification of the mean field coupling constant, whose value depends on the angular momentum of the condensate. Secondly, it can be interpreted as a density modulated angular velocity given to the cloud. Looking at the problem from both of these viewpoints, we show that the effect of the new nonlinearity is to induce a rotation to the condensate, where the transition from non-rotating to rotating states depends on the density of the cloud.
In this dissertation, we study rotational properties of ultracold bosonic quantum gases in two trapped configurations, a quasi-two-dimensional gas in a harmonic trap and a quasi-one-dimensional gas in a toroidal trap. First, we investigate the effects of correlations on the properties of the ground state of the rotating harmonically-trapped Bose gas by adding Bogoliubov fluctuations to the mean-field ground state of an N-particle single-vortex system. We demonstrate that the fluctuation-induced correlations lower the energy compared to that of the mean-field ground state, that the vortex core is pushed slightly away from the center of the trap, and that an unstable mode with negative energy (for rotations slower than a critical frequency) emerges in the energy spectrum, thus, pointing to a better state for slow rotation. We construct mean-field ground states of 0-, 1-, and 2-vortex states as a function of rotation rate and determine the critical frequencies for transitions between these states, as well as the critical frequency for appearance of a metastable state with an off-center vortex and its image vortex in the evanescent tail of the cloud. Then, we show how the configuration-space form of the above-mentioned Bogoliubov ground-state wave function of a bosonic condensate with a single vortex in a harmonic trap can be described in terms of bosonic Jastrow correlations. We then generalize this result to study the effects of such correlations on a mean-field vortex lattice state and show that the included correlations lower the energy below that of the mean-field state. Although the reduction is relatively small, it is a precursor of the more general expected effect of correlations in describing the melting of the vortex lattice at a high angular momentum per particle. Finally, we study the stability and dynamics of an ultracold bosonic gas trapped in a toroidal geometry and driven by rotation in the absence of dissipation. We first delineate, via the Bogoliubov mode expansion, the regions of stability and the nature of instabilities of the system for both repulsive and attractive interaction strengths. To study the response of the system to variations in the rotation rate, we introduce a “disorder” potential, breaking the rotational symmetry. We demonstrate the breakdown of adiabaticity as the rotation rate is slowly varied and find forced tunneling between the eigenstates of the system. The nonadiabaticity is signaled by the appearance of a swallow-tail loop in the lowest-energy level, a general sign of hysteresis. Then, we show that this system is in one-to-one correspondence with a trapped gas in a double-well potential and thus exhibits macroscopic quantum self-trapping. Finally, we show that self-trapping is a direct manifestation of the behavior of the lowest-energy level.