In this paper, we consider the ground state properties of a spin-orbit coupled binary Bose-Einstein condensates with dipole-dipole interaction.Our results are obtained by numerically solving the Gross-Pitaevskii equation in the mean field.It is shown that the density structures are greatly affected by spin-orbit coupled strength and dipole strength.With the increase of spin-orbit coupled strength, we find that a stripe phase and phase separation of two components appear, and the number of stripes increases.In addition, concentric circularity stripes with spatial symmetry appear.We also verify that the phase separation can be broken as the dipole strength increases.However, the stripe phase is still kept in this process.As a result, there are no regions of zero density inside the cloud and the nonfragmented density BEC is exhibited.Also, the critical dipole strength decreases with increasing the SOC strength for the non-fragmented state.This result reflects a competition in which the spin-orbit coupling dominates the dipole-dipole interaction.
In this paper, we consider the ground state properties of a spin-orbit coupled binary Bose-Einstein condensates with dipole-dipole interaction.Our results are obtained by numerically solving the Gross-Pitaevskii equation in the mean field.It is shown that the density structures are greatly affected by spin-orbit coupled strength and dipole strength.With the increase of spin-orbit coupled strength, we find that a stripe phase and phase separation of two components appear, and the number of stripes increases.In addition, concentric circularity stripes with spatial symmetry appear.We also verify that the phase separation can be broken as the dipole strength increases.However, the stripe phase is still kept in this process.As a result, there are no regions of zero density inside the cloud and the nonfragmented density BEC is exhibited.Also, the critical dipole strength decreases with increasing the SOC strength for the non-fragmented state.This result reflects a competition in which the spin-orbit coupling dominates the dipole-dipole interaction.
In this paper, we study the spin dynamics of spin-orbit-coupled (SOC) spin-2 Bose–Einstein condensates (BECs) with dipole–dipole interaction (DDI). Numerical results are obtained by solving the nonlocal Gross–Pitaevskii equation. Our discussions are divided into two parts. In the absence of DDI, the spin dynamics presents periodic oscillation. Increasing the magnetization makes the amplitude increase, and increasing SOC strength accelerates the spin dynamics and thus changes the period. Nevertheless, in the presence of DDI, the periodicity oscillation of spin dynamics is broken. The thermalization time decreases with the increase of magnetization for fixed SOC strength. In addition, the thermalization time becomes zero with strong SOC strength when magnetization is fixed. We also show the formation of domain in density profile when DDI is considered. Moreover, the components m = 2 and m = −2 exchange their location with a little increase of SOC strength. This is a specific reflection of the competition between the SOC and DDI.
In this paper, we study the ground state of two-component spin–orbit-coupled (SOC) Bose–Einstein condensates with mass imbalance. Our results are based on the framework of mean-field Gross–Pitaevskii theory. The effects of unequal atomic mass and SOC strength are studied. Different density structures such as heliciform stripes, linear stripes and string vortex chains are found. Different density structures such as heliciform stripes, linear stripes and string vortex chains are found. With the increase of SOC strength, the azimuthal phase separation is kept while radial phase separation is broken. In addition, increasing the mass ratio is unfavorable to vortex formation, whereas more vortices can be generated by increasing the SOC strength. We also discuss the physical quantities such as angular momentum per atom and spin polarisation, for a larger mass ratio, showing that angular momentum gets a little bigger as the SOC strength increases and the first-order phase transition does not exist.
In this paper, we study the vortex formation of spin-orbit coupled (SOC) spin-1 dipolar Bose–Einstein condensates by numerically solving the Gross–Pitaevskii equation. We focus the effects of dipole–dipole interaction (DDI) on vortex structures and dynamics behavior. We find that the symmetry of density profiles in vortex lattice formation process and time evolution of the angular momentum is broken in the presence of DDI. This is due to the symmetry breaking of anisotropy DDI. Also, the total angular momentum decreases with increasing the relative dipole strength. However, it is shown that increasing SOC strength can improve the rotation efficiency. This result reflects the SOC dominates the DDI.
The spinor Bose-Einstein condensate (BEC) provides an ideal platform for observing and manipulating topological structures, which arise from the spin degrees of freedom and the superfluid nature of the gas. Artificial helicoidal spin-orbit coupling (SOC) in the spinor BEC, owing to the spatially varying gauge potential and the more flexible adjustability, provides possibly an unprecedented opportunity to search for novel quantum states. The previous studies of the BEC with helicoidal SOC mainly focus on the two-component case. However, there are few reports on the studies of helicoidal SOC in three-component BEC. Especially considering one-dimensional three-component BEC, whether the helicoidal SOC can generate previously unknown types of topological excitations and phase diagrams is still an unsolved problem. In this work, by solving quasi one-dimensional Gross-Pitaevskii equations, we study the ground state structure of one-dimensional helicoidal spin-orbit coupled three-component BEC. The numerical results show that the helicoidal SOC can induce a phase separation among the components in ferromagnetic BEC. Through numerical calculations of the system, a phase diagram is obtained as a function of the helicoidal SOC strength and gauge potential, which shows the critical conditions for phase separation and phase miscibility in ferromagnetic BEC. Meanwhile, we also study the influences of the helicoidal SOC and the gauge potential on the antiferromagnetic BEC ground state. The numerical results show that the helicoidal SOC is beneficial for the miscibility in antiferromagnetic BEC. When the helicoidal SOC strength or gauge potential increases, the ground state of antiferromagnetic BEC exhibits a stripe soliton structure. Adjusting the strength of helicoidal SOC or gauge potential can control the transitions between a plane-wave soliton and a stripe soliton. In addition, we show the changes of the particle number density maximum and the number of peaks of stripe solitons for adjusting the helicoidal SOC strength or gauge potential. Our results show that helicoidal spin-orbit coupled BEC not only provides a controlled platform for investigating the exotic topological structures, but also is crucial for the transitions between different ground states. This work paves the way for exploring the topological defect and the corresponding dynamical stability in quantum systems subjected to the helicoidal SOC in future.
In this paper, we investigate the spin–orbit coupling of F = 1 dipolar Bose–Einstein condensates in a quasi-one-dimensional trap by solving the mean field Gross–Pitaevskii equation. We focus on the two phases, ferromagnetic and antiferromagnetic. For the former, we find that the spin echo signal emerges gradually as the spin–orbit coupling strength increases, which favors miscibility in density profile. The dipolar interaction plays a tiny role in the spin dynamics process, the qualitative behavior remains almost unchanged. Nevertheless, for the antiferromagnetic phase, the physical picture greatly changes. The spin echo signal abruptly increases with the increase in spin–orbit coupling strength, and the density structure exhibits phase separation. With the emergence of dipolar interaction, the spin echo signal vanishes, and the evolution of the three components is ruleless. Meanwhile, the condensate deviates from the property of phase separation. Also, we analyze the effect of the number of atoms on spin dynamics. It is obvious that the reduction of atoms increases the number of spin echo signals for the ferromagnetic phase regardless of the nature of dipolar interaction. In addition, for the antiferromagnetic phase, such a decrease gives rise to the disappearance of the spin echo signal in the absence of dipolar interaction, and its influence is small when dipolar interaction is added.
This work concerns a theoretical study of a two-component ultra-cold bosonic gas of polarised magnetic dipolar atoms. The atoms interact by short-ranged contact interactions and long-ranged dipole-dipole interactions. The two components can correspond to two different spin-states of a single atomic species or two different atomic species, as realised in recent experiments. Here, we develop a general theory for the ground states and excitations of a two-component dipolar Bose-Einstein condensate. The ground states are described by dipolar Gross-Pitaevskii equations and Bogoliubov-de Gennes equations describe the excitations. We apply these theories to systems in 3D harmonic traps and a uniform planar potential. Specialised numerical methods are developed to provide accurate solutions of these equations, and to deal with the singular dipole-dipole interaction kernel. These methods involve spectral representations based on plane waves and Bessel functions. We also discuss the methods we used for obtaining ground state solutions of the nonlinear Gross-Pitaevskii energy functional. Dipole-dipole interactions can favour spatial structure, as demonstrated in the recent observation of supersolid states of matter in a single-component dipolar gas. In the two-component system these interactions can also favour a ground state with a periodic order in the pseudo-spin density. A two-component system can be either miscible (components overlap) or immiscible (components are separated). Our focus here is in the spin-stripe state, which occurs in a flattened trap geometry and involves an immiscible stripe pattern along one of the weakly trapped or untrapped directions. We examine the dynamical stability conditions of the uniform miscible state in planar confinement, and produce an associated stability phase diagram. We identify the various instabilities and analytical criteria for estimating these. We focus on the occurrence of, and conditions that favour, density-roton or spin-roton excitations, anticipating that the softening of these excitations will lead to structured ground states. To characterize these excitations we introduce density and spin dynamical structure factors, which can be measured in experiments using Bragg spectroscopy. Building on the stability phase diagram we compute ground states where the immiscible state is unstable to spin-rotons, finding triangular and stripe immiscible states with a length scale set by the confinement and interactions. We find that the transition to the stripe state is continuous and coincides with the spin-roton softening in the uniform miscible state. We study the excitations of the spin stripe state which exhibits three gapless excitation branches arising from the spontaneous broken symmetries. We identify a balanced parameter regime, where the excitations exhibit a nonsymmorphic symmetry (a symmetry is nonsymmorphic when it is a combination of point-group operations with nonprimitive lattice translations, i.e. not an integer multiple of lattice constants), which is revealed in the density and spin dynamical structure factors. We also consider the dynamics of the system following a quench from the uniform to immiscible stripe states. In the early time dynamics, a spin-stripe state is formed in a such way that the stripe order only extends over small domains. In subsequent dynamics, these domain sizes grow as the system phase-orders towards its equilibrium state. We identify the relevant topological defects such as disclinations, dislocations and grain boundaries. Using ensembles of large scale simulations we show that dynamical scaling holds for the growth of order.