Abstract The ground-state phases of rotating spin-1 Bose-Einstein condensates in three different box traps are studied numerically. In circular or square box trap, influenced by the combined effects of spin-exchange interaction and rotation, traditional skyrmion vortex for a rotating spin-1 BEC in harmonic trap are replaced by the homogeneous coreless vortices layered with fourfold rotational symmetry, which are usually generated from the trap boundary every four. In a triangular box trap, one skyrmion vortex structure is firstly found in the trap center, while other vortices are all generated from the boundary and arrange parallel to the trap edge in the form of vortex chains .
Abstract The ground-state phases of rotating spin-1 Bose-Einstein condensates in three different box traps are studied numerically. In circular or square box trap, influenced by the combined effects of spin-exchange interaction and rotation, traditional skyrmion vortex for a rotating spin-1 BEC in harmonic trap are replaced by the homogeneous coreless vortices layered with fourfold rotational symmetry, which are usually generated from the trap boundary every four. In a triangular box trap, one skyrmion vortex structure is firstly found in the trap center, while other vortices are all generated from the boundary and arrange parallel to the trap edge in the form of vortex chains .
Realization of spinor Bose-Einstein condensate in an optical trap has made it possible to create a variety of topological nontrivial structures, due to the vector character of the order parameter. Recently, artificial spin-orbit coupling in the spinor Bose-Einstein condensate, owing to coupling between the spin and the center-of-mass motion of the atom, provides an unprecedented opportunity to search for novel quantum states. As is well known, the potential well in the Bose-Einstein condensate is adjustable. The toroidal trap is an important model potential because of its simplicity and richness in physics. In particular, the spinor Bose-Einstein condensate under the toroidal trap has brought an ideal platform for studying fascinating properties of a superfluid, such as persistent flow and symmetry-breaking localization. For the case of the spin-orbit-coupled Bose-Einstein condensate, the previous studies of the toroidal trap mainly focused on the two-component or antiferromagnetic case. However, in the presence of a toroidal trap, there remains an open question whether the combined effects of the spin-orbit coupling and rotation can produce previously unknown types of topological excitations in the ferromagnetic Bose-Einstein condensate. In this work, by using quasi two-dimensional Gross-Pitaevskii equations, we study the ground state structure of spin-orbit coupled rotating ferromagnetic Bose-Einstein condensate in the toroidal trap. We concentrate on the effects of the spin-orbit coupling and the rotation on the ground states. The numerical results show that in the presence of a toroidal trap, the ground state structure is displayed as half-skyrmion chain with circular distribution. Adjusting the strength of spin-orbit coupling not only changes the number of half-skyrmion in the system, but also controls the symmetry of half-skyrmion with circular distribution. As the rotation frequency increases, the system undergoes the transitions from the plane wave to the half-skyrmion chain with circular distribution, and eventually developing the half-skyrmion phase of triangular lattice. Next, we examine the effect of spin-independent interaction on spin-orbit coupled rotating spinor Bose-Einstein condensate. As the spin-independent interaction increases, the topological defects in the condensate increase due to the variation of the local magnetic order. We also discuss the influence of well shape on the ground state structure. These topological structures can be detected via the time-of-flight absorption imaging technique. The spin-orbit coupled spinor Bose-Einstein condensate in the toroidal trap is an important quantum platform, which not only opens up a new avenue for exploring the exotic topological structures, but also is crucial for realizing the transitions among different ground states. This work paves the way for futureexploring the topological defects and the corresponding dynamical stability in quantum system subjected to the toroidal trap.
We study numerically the vortex-lattice formation in a rapidly rotating uniform quasi-two-dimensional Bose-Einstein condensate (BEC) in a box trap. We consider two types of boxes: square and circle. In a square-shaped 2D box trap, when the number of generated vortices is the square of an integer, the vortices are found to be arranged in a perfect square lattice, although deviations near the center are found when the number of generated vortices is arbitrary. In case of a circular box trap, the generated vortices in the rapidly rotating BEC lie on concentric closed orbits. Near the center, these orbits have the shape of polygons, whereas near the periphery the orbits are circles. The circular box trap is equivalent to the rotating cylindrical bucket used in early experiment(s) with liquid He II. The number of generated vortices in both cases is in qualitative agreement with Feynman's universal estimate. The numerical simulation for this study is performed by a solution of the underlying mean-field Gross-Pitaevskii (GP) equation in the rotating frame, where the wave function for the generated vortex lattice is a stationary state. Consequently, the imaginary-time propagation method can be used for a solution of the GP equation, known to lead to an accurate numerical solution. We also demonstrated the dynamical stability of the vortex lattices in real-time propagation upon a small change of the angular frequency of rotation, using the converged imaginary-time wave function as the initial state.
The nucleation of vortices and the resulting structures of vortex arrays in dilute, trapped, zero-temperature Bose-Einstein condensates are investigated numerically. Vortices are generated by rotating a three-dimensional, anisotropic harmonic atom trap. The condensate ground state is obtained by propagating the Gross-Pitaevskii equation in imaginary time. Vortices first appear at a rotation frequency significantly larger than the critical frequency for vortex stabilization, consistent with a critical velocity mechanism for vortex nucleation. At higher frequencies, the structures of the vortex arrays are strongly influenced by trap geometry.
This thesis contains theoretical studies of weakly interacting, trapped Bose-Einstein condensed gases. The main emphasis is on the rotational properties of these systems. A typical Bose-Einstein condensate reacts to external rotation by forming singly quantized vortices. By the word ”typical” we refer to a condensate with repulsive interparticle interactions and with a frozen spin degree of freedom. However, there are some atomic species for which the low-energy interactions are attractive. Moreover, the tuning of different condensate properties is relatively easy. For example, there are ways to experimentally change the effective interaction from repulsive to attractive. For an attractively interacting condensate the vortex is not the only possible rotational state but the sample may also experience rotation through its center of mass. In the first part of the thesis, we analyze some properties of a rotating attractive Bose gas at the zero temperature limit. First, we note that in order to maintain energetical stability, the gas has to be rotated in an external trap which is steeper than a harmonic trap. We study the system in an anharmonic quadratic + quartic potential. The rotating ground state phase-space is determined with analytical and numerical methods. If the interactions are strong enough, the rotating ground state appears to be a state with the rotating center of mass. On the other hand, at the limit of weak interactions the system experiences rotation by forming a multiply quantized vortex. We also determine the spectrum of elementary excitations for these quantum many-particle states. We find that the broken symmetry of the center of mass rotating state induces an additional zero-energy (Goldstone) mode and that the small oscillations of the ground state exhibit the decoupling between the center of mass and relative motions. If the spin of the atoms is allowed to change freely, a rich variety of topological excitations in condensates can be studied. External rotation is one of the tools for examining the excitations in spinor condensates. In a spinless scalar condensate the appearance of a quantized vortex is manifested by the vortex line in which the condensate density vanishes. In the plane perpendicular to the vortex line, one speaks of the vortex core for the density minimum. We study the properties of slowly rotated F = 1 spinor condensates and show that for ferromagnetic interactions the ground state is a coreless vortex. While the individual spin components may carry ordinary vortices the overall density of the system is coreless. We also study a method to create a point-like monopole structure in a ferromagnetic condensate by suitably imprinting vortices and solitons for different spin components of the condensate.
Abstract We consider the ground-state properties of a rotating spin-orbit-coupled Bose—Einstein condensate under extreme elongation in a harmonic plus quartic potential. The effects of spin-orbit coupling and rotation on the ground-state vortex structures are investigated. In the absence of spin-orbit coupling, new nucleated vortices gradually form vortex lines and annular vortex structures with the increase of the rotation frequency. In the presence of spin-orbit coupling, part of the vortices arrange in a line and form a stable vortex chain, and the remanent vortices coexist in pairs aside such vortex chain. More specially, the remanent vortices of each component repel each other and form vortex pair for isotropic spin-orbit coupling, while attract each other and locate in the same positions for anisotropic spin-orbit coupling.
Compared with the scalar Bose-Einstein condensate, the spinor Bose-Einstein condensate, in which internal degrees of freedom are essentially free, has aroused the great interest in the study of topological excitations. In particular, the spinor Bose-Einstein condensate with rotation provides a new opportunity for studying novel quantum states including a coreless vortex and vortex lattice. To date, in the presence of rotation, a great many of studies on the topological excitations have focused on the Bose-Einstein condensate system with the uniform Zeeman field or without external magnetic field. However, the ground state structure of a rotating Bose-Einstein condensate in the presence of in-plane gradient-magnetic-field remains an open question. In this work, by using the imaginary-time propagation method, we study the ground state structure of a rotating Bose-Einstein condensate with in-plane quadrupole field. We first examine the effect of in-plane quadrupole field on trapped spinor Bose-Einstein condensate. The numerical results show that Mermin-Ho vortex can be induced only by the cooperation between quadrupole field and rotation. When magnetic field gradient is increased, the vortices around Mermin-Ho vortex display the symmetrical arrangement. For an even larger magnetic field gradient strength, the system only presents the Mermin-Ho vortex because the in-plane quadrupole field can prevent the vortices around Mermin-Ho vortex from occurring. Next, we examine the effect of the rotation on trapped spinor Bose-Einstein condensate. A phase transition from a polar-core vortex to a Mermin-Ho vortex is found through applying a rotational potential, which is caused by the cooperation between the in-plane quadrupole field and the rotation. We further study the combined effects of spin exchange interaction and density-density interaction. The results confirm that in the presence of the quadrupole field both spin exchange interaction and density-density interaction, acting as controllable parameters, can control the number of the vortices around Mermin-Ho vortex. The corresponding number of the vortices shows step behavior with increasing the ratio between spin exchange interaction and density-density interaction, which behaves as hexagon, pentagon, square and triangle. It is found that two types of topology structures, i.e., the hyperbolic meron and half-skyrmion, can occur in the present system. These vortex structures can be realized via time-of-flight absorption imaging technique. Our results not only provide an opportunity to investigate the exotic vortex structures and the corresponding phase transitions in a controlled platform, but also lay the foundation for the study of topological defect subjected to gauge field and dipolar interaction in future.