The ring dark solitons in spin-1 ${}^{23}$Na and ${}^{87}$Rb Bose-Einstein condensates are studied numerically in the framework of the time-dependent Gross-Pitaevskii equations. By simulating the phase engineering technique in real experiments, we explore the roles of the parameters characterizing the far-off resonant laser pulse which can be used to generate the ring dark solitons. The variations of these parameters have a dramatic effect on the lifetime and the decay profiles of the ring dark solitons. If only one ring dark soliton is generated in one component of the condensate, then ring dark solitons in other components are inclined to be induced, resulting in a coexistence state composed of interdependent ring dark solitons coming from different components of the condensate. Ring dark solitons in this coexistence state exhibit dynamical oscillations for hundreds of milliseconds. By studying the lifetime and decaying profiles of ring dark solitons, we explore the similarities and differences of ${}^{23}$Na and ${}^{87}$Rb condensates. In addition, taking into account the fact that the center of the ring may not coincide with that of the trap, we study the dynamics and decaying profiles of the off-centered ring dark solitons in the presence of the symmetry-breaking effect.
The ring dark solitons in spin-1 23Na and 87Rb Bose-Einstein condensates are studied numerically in the framework of the time-dependent Gross-Pitaevskii equations. By simulating the phase engineering technique in real experiments, we explore the roles of the parameters characterizing the far-off resonant laser pulse which can be used to generate the ring dark solitons. The variations of these parameters have a dramatic effect on the lifetime and the decay profiles of the ring dark solitons. If only one ring dark soliton is generated in one component of the condensate, then ring dark solitons in other components are inclined to be induced, resulting in a coexistence state composed of interdependent ring dark solitons coming from different components of the condensate. Ring dark solitons in this coexistence state exhibit dynamical oscillations for hundreds of milliseconds. By studying the lifetime and decaying profiles of ring dark solitons, we explore the similarities and differences of 23Na and 87Rb condensates. In addition, taking into account the fact that the center of the ring may not coincide with that of the trap, we study the dynamics and decaying profiles of the off-centered ring dark solitons in the presence of the symmetry-breaking effect.
We investigate the dynamics and pattern formation of two ring dark solitons in a two-dimensional binary Bose-Einstein condensate with tunable intercomponent interaction via numerical simulation of the time-dependent Gross-Pitaevskii equation. Both the black and gray ring dark solitons are considered for cases where the modulation frequency of the intercomponent interaction is resonant or nonresonant with the one of the trapping potential. Our results show that in the presence of periodic modulation of the intercomponent interaction not only are the lifetimes of the ring dark solitons largely extended but also their decaying dynamics are dramatically affected. Before snaking instability sets in, new ring dark solitons are formed, and both the numbers and depths of the ring dark solitons exhibit collective oscillations. With the development of instability, the system exhibits different decaying processes, and a variety of decay profiles, such as vortex necklace, distorted octagon, vortex-antivortex ring, and cross, are formed, showing a strong dependence on the modulation frequency of the intercomponent interaction and the initial depth of the ring dark soliton.
The successive dynamical evolution of a Bose-Einstein condensate confined in a cylindrical well is numerically studied in the framework of the time-dependent Gross-Pitaevskii equation. Interference in the nonlinear matter wave leads to concentric density rings. The phase distribution exhibits a discontinuous sequence of plateaulike belts. Abrupt jumps in the phase between adjacent belts imply large radial superfluid velocity at the borderline. This, however, does not mean large particle current because the corresponding superfluid density is nearly zero. The density zeros along with the large gradient are identified as ring dark solitons, which have a brief lifetime before evolving into other soliton states.
We consider the dynamics and formation of vortices from ring dark solitons in a two-dimensional Bose-Einstein condensate with the Rashba spin-orbit coupling based on the time-dependent coupled Gross-Pitaevskii equation. Compared with previous results, the system exhibits complex dynamical behaviors in the presence of the spin-orbit coupling. With the modulation of the spin-orbit coupling, not only the lifetime of ring dark solitons is greatly prolonged, but also their attenuation kinetics is significantly affected. For two shallow ring dark solitons with the equal strength of the spin-orbit coupling, the radius of ring dark solitons increases to a maximum value over time and then shrinks into a minimum value. Due to the effect of the snake instability, ring dark solitons split into a series of ring-like clusters of vortex pairs, which perform complex oscillations. This indicates that the system is strongly dependent on the presence of the spin-orbit coupling. Furthermore, the effect of different initial modulation depths on the dynamics of ring dark solitons is investigated.
We systematically construct stationary soliton states in a one-component, two-dimensional, repulsive, Gross-Pitaevskii equation with a ring-shaped targetlike trap similar to the potential used to confine a Bose-Einstein condensate in a recent experiment [R. Mathew, A. Kumar, S. Eckel, F. Jendrzejewski, G. K. Campbell, M. Edwards, and E. Tiesinga, Phys. Rev. A 92, 033602 (2015)]. In addition to the ground-state configuration, we identify a wide variety of excited states involving phase jumps (and associated dark solitons) inside the ring. These configurations are obtained from a systematic bifurcation analysis starting from the linear, small atom density, limit. We study the stability and, when unstable, the dynamics of the most basic configurations. Often these lead to vortical dynamics inside the ring persisting over long time scales in our numerical experiments. To illustrate the relevance of the identified states, we showcase how such dark-soliton configurations (even the unstable ones) can be created in laboratory condensates by using phase-imprinting techniques.
We present an overview of our recent theoretical studies on the quantum phenomena of the spin-1 Bose-Einstein condensates, including the phase diagram, soliton solutions and the formation of the topological spin textures. A brief exploration of the effects of spin-orbit coupling on the ground-state properties is given. We put forward proposals by using the transmission spectra of an optical cavity to probe the quantum ground states: the ferromagnetic and polar phases. Quasi-one-dimension solitons and ring dark solitons are studied. It is predicted that characteristics of the magnetic solitons in optical lattice can be tuned by controlling the long-range light-induced and static magnetic dipoledipole interactions; solutions of single-component magnetic and single-, two-, three-components polar solitons are found; ring dark solitons in spin-1 condensates are predicted to live longer lifetimes than that in their scalar counterparts. In the formation of spin textures, we have considered the theoretical model of a rapidly quenched and fast rotating trapped spin-1 Bose-Einstein condensate, whose dynamics can be studied by solving the stochastic projected Gross-Pitaevskii equations. Spontaneous generation of nontrivial topological defects, such as the hexagonal lattice skyrmions and square lattice of half-quantized vortices was predicted. In particular, crystallization of merons (half skyrmions) can be generated in the presence of spin-orbit coupling.
Soliton is an exotic topological excitation, and it widely exists in various nonlinear systems, such as nonlinear optics, Bose-Einstein condensates, classical and quantum fluids, plasma, magnetic materials, etc. A stable soliton can propagate with constant amplitude and velocity, and maintain its shape. Two-dimensional and three-dimensional solitons are usually hard to stabilize, and how to realize stable two-dimensional or three-dimensional solitons has aroused the great interest of the researchers. Ring dark soliton is a kind of two-dimensional soliton, which was first theoretically predicted and experimentally realized in nonlinear optical systems. Compared with the usual two-dimensional solitons, ring dark solitons have good stability and rich dynamical behaviors. Owing to their highly controllable capability, Bose-Einstein condensates provide a new platform for studying the ring dark solitons. Based on the recent progress in Bose-Einstein condensates and solitons, this paper reviews the research on the analytic solutions, stability, as well as the decay dynamics of ring dark solitons in Bose-Einstein condensates. A transform method is introduced, which generalizes the analytic solutions of ring dark solitons from a homogeneous system with time-independent nonlinearity to a harmonically trapped inhomogeneous system with time-dependent nonlinearity. The stability phase diagram of the ring dark soliton under deformation perturbations is discussed by numerically solving the Gross-Pitaevskii equations in the mean-field theory. A method of enhancing the stability of ring dark solitons by periodically modulating the nonlinear coefficients is introduced. It is also shown that the periodically modulated nonlinear coefficient can be experimentally realized by the Feshbach resonance technology. In addition, we discuss the dynamics of the decay of ring dark solitons. It is found that the ring dark soliton can decay into various vortex clusters composed of vortices and antivortices. This opens a new avenue to the investigation of vortex dynamics and quantum turbulence. It is also found that the ring dark solitons combined with periodic modulated nonlinearity can give rise to the pattern formation, which is an interesting nonlinear phenomenon widely explored in all the fields of nature. Finally, some possible research subjects about ring dark solitons in future research are also discussed.