We have studied theoretically the Bose-Einstein condensation (BEC) of two-dimensional excitons in a ring with small random width variation. We derive a nonlinear Gross-Pitaevkii equation (GPE) for such a condensate. Our numerical solution of the ground state of the GPE displays a necklace-like structure in the presence of small random variation of ring width. This is a consequence of the interplay between random potential and the nonlinear repulsive term of the GPE. Our result suggests that the formation of ring and necklace-like structures observed recently in the photoluminescence of quantum wells may be stable even in the BEC phase.
We have studied theoretically the Bose-Einstein condensation (BEC) of two-dimensional excitons in a ring with small random width variation. We derive a nonlinear Gross-Pitaevkii equation (GPE) for such a condensate. Our numerical solution of the ground state of the GPE displays a necklace-like structure in the presence of small random variation of ring width. This is a consequence of the interplay between random potential and the nonlinear repulsive term of the GPE. Our result suggests that the formation of ring and necklace-like structures observed recently in the photoluminescence of quantum wells may be stable even in the BEC phase.
We have studied theoretically the Bose-Einstein condensation (BEC) of two-dimensional excitons in a ring with a small random variation of the effective exciton potential along the circumference. We derive a nonlinear Gross-Pitaevskii equation (GPE) for such a condensate, which is valid even in the presence of a weak magnetic field. For several types of random potentials, our numerical solution of the ground state of the GPE displays a necklace-like structure. This is a consequence of the interplay between the random potential and a strong nonlinear repulsive term in the GPE.
A Bose-Einstein Condensate (BEC) made of alkali-metal atoms at ultra-low temperatures is well described by the three-dimensional cubic nonlinear Schrödinger equation, the so-called Gross-Pitaevskii equation (GPE). Here we consider an attractive BEC in a ring and by solving the GPE we predict the existence of bright solitons with single and multi-peaks, showing that they have a limited domain of dynamical stability. We also discuss finite-temperature effects on the transition from the uniform phase to the localized phase.
By means of variational solution and direct numerical simulation of the Gross-Pitaevskii equation (GPE), we have studied the stability of matter-wave solitons in two-dimensional (2D) Bose-Einstein condensations (BECs), with 2D linear and nonlinear optical lattices (OLs). Using the static variational approach and Vakhitov-Kolokolov criterion necessary for stability, we obtain the stability condition for solitons in different combinations of OL's parameters. We show that the 2D linear and nonlinear optical lattices allow us to stabilize 2D solitons for both attractive and repulsive interactions. We also study the time-evolution problems of 2D BECs, using the time-dependent variational approach and numerical solution of GPE for 2D linear and nonlinear OLs. Very good agreement between the results corresponding to both treatments is observed.
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.
This work is motivated by the active surge in research on ring Bose–Einstein condensates of exciton polaritons, including those with artificially created defects (barriers). Currently, such systems are primarily described within a one-dimensional approach, taking into account that the order parameter depends only on the azimuthal angle. In this work, the profiles of a polariton ring Bose–Einstein condensate in the stationary case have been numerically simulated using the two-dimensional Gross–Pitaevskii equation. The radial profiles of the ground-state wavefunction are obtained for ring confinement potentials of various depths and widths, which have been used to calculate effectively one-dimensional interaction constants for various particle densities in the condensate. The results have been compared with analytical expressions for the linear limit and for a ring of a large radius, and the validity range for these simplified analytical descriptions have been determined.
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.