Ground states, solitons and spin textures in spin-1 Bose-Einstein condensates

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.     

Interaction of solitary spin waves

We present the results of a computer experiment devoted to the problem of the interaction of two magnetic solitary spin waves moving in the direction perpendicular to the axis of easy magnetization in an uniaxial ferromagnet. Such waves being particular solutions of the Landau-Lifshitz equations move like a domain wall under the influence of an external magnetic field. Our computer experiment shows that the two solitary spin waves during their interaction, behave as two solitons and thus the concerned Landau-Lifshitz equations allows N-soliton solutions.     

Dynamics and formation of vortices collapsed from ring dark solitons in a two-dimensional spin–orbit coupled Bose–Einstein condensate

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.     

Modulated spin waves and robust quasi-solitons in classical Heisenberg rings

We investigate the dynamical behavior of finite rings of classical spin vectors interacting via nearest-neighbor isotropic exchange in an external magnetic field. Our approach is to utilize the solutions of a continuum version of the discrete spin equations of motion (EOM) which we derive by assuming continuous modulations of spin wave solutions of the EOM for discrete spins. This continuum EOM reduces to the Landau-Lifshitz equation in a particular limiting regime. The usefulness of the continuum EOM is demonstrated by the fact that the time-evolved numerical solutions of the discrete spin EOM closely track the corresponding time-evolved solutions of the continuum equation. It is of special interest that our continuum EOM possesses soliton solutions, and we find that these characteristics are also exhibited by the corresponding solutions of the discrete EOM. The robustness of solitons is demonstrated by considering cases where initial states are truncated versions of soliton states and by numerical simulations of the discrete EOM equations when the spins are coupled to a heat bath at finite temperatures.     

Composite multihump vector solitons carrying topological charge

We propose composite solitons carrying topological charge: multicomponent two dimensional [ (2+1)D] vector (Manakov-like) solitons for which at least one component carries topological charge. These multimode solitons can have a single hump or exhibit a multihump structure. The "spin" carried by these multimode composite solitons suggests 3D soliton interactions in which the particlelike behavior includes spin, in addition to effective mass, linear, and angular momenta.     

Magnetized vector solitons in a spin-orbit coupled spin-1 Bose-Einstein condensate with Zeeman coupling

The property of matter-wave vector solitons in a spin-1 Bose-Einstein condensate with spin-orbit and Zeeman couplings is investigated by multiscale perturbation method. The excitation spectrum and the corresponding state vectors of the system are obtained analytically, and they can be adjusted by the parameters of the system. The bright and dark vector solitons are formulated by reducing the three-component coupled Gross-Pitaeviskii equations to a standard nonlinear Schrödinger equation, which has the solutions of the bright and dark solitons with positive or negative mass depending on the product of the effective dispersive and nonlinear coefficients. The moving vector solitons are demonstrated by adjusting specific momentum near the energy minimum. Finally, the magnetized features of the vector solitons are discussed by the spin polarization of the system.     

Dark solitons in ferromagnetic chains with first- and second-neighbor interactions

We present a study of the ferromagnetic spin chain with both first- and second-neighbor interactions. We obtain the condition for the appearance and stability of bright and dark solitons for arbitrary wave number in the Brillouin zone. The influence of the second-neighbor interaction and the anisotropy on the soliton properties is considered. The scattering of dark solitons from point defects in the discrete spin chain is investigated numerically.     

Dynamics of bright soliton in a spin–orbit coupled spin-1 Bose–Einstein condensate

We have investigated the dynamics of bright solitons in a spin–orbit coupled spin-1 Bose–Einstein condensate analytically and numerically. By using the hyperbolic sine function as the trial function to describe a plane wave bright soliton with a single finite momentum, we have derived the motion equations of soliton's spin and center of mass, and obtained its exact analytical solutions. Our results show that the spin–orbit coupling couples the soliton's spin with its center-of-mass motion, the spin oscillations induced by the exchange of atoms between components result in the periodical oscillation of center-of-mass, and the motion of center of mass of soliton can be viewed as a superposition of periodical and linear motions. Our analytical results have also been confirmed by the direct numerical simulations of Gross–Pitaevskii equations.     

Approximate solutions for half-dark solitons in spinor non-equilibrium Polariton condensates

In this work I generalize and apply an analytical approximation to analyze 1D states of non-equilibrium spinor polariton Bose–Einstein condensates (BEC). Solutions for the condensate wave functions carrying black solitons and half-dark solitons are presented. The derivation is based on the non-conservative Lagrangian formalism for complex Ginzburg–Landau type equations (cGLE), which provides ordinary differential equations for the parameters of the dark soliton solutions in their dynamic environment. Explicit expressions for the stationary dark soliton solution are stated. Subsequently the method is extended to spin sensitive polariton condensates, which yields ordinary differential equations for the parameters of half-dark solitons. Finally a stationary case with explicit expressions for half-dark solitons is presented.     

Bright soliton dynamics in Fourier synthesized optical lattice

We consider the dynamics of bright solitons in one dimensional Bose Einstein condensate with spin orbit coupling and trapped by an optical lattice potential. We mainly focus on the effects of the spin orbit coupling, Rabi coupling and detuning parameter on the soliton dynamics. The solution of the coupled Gross Pitaevskii equation is performed with the help of the variational and numerical methods for various parameters. The population imbalance and the center of mass motion show different dynamic behaviors depending on the parameters. We also obtain traveling bright solitons related with a given finite initial momentum.     

Kummer solitons in strongly nonlocal nonlinear media

We solve the three-dimensional (3D) time-dependent strongly nonlocal nonlinear Schrödinger equation (NNSE) in spherical coordinates, with the help of Kummer's functions. We obtain analytical solitary solutions, which we term the Kummer solitons. We compare analytical solutions with the numerical solutions of NNSE. We discuss higher-order Kummer spatial solitons, which can exist in various forms, such as the 3D vortex solitons and the multipole solitons.     

Dirac spinor solitons or bags

Explicit solutions of a non-linear Dirac equation in the four-dimensional Minkowski space have been found. They are continuous eigenfunctions of spin $\text{1}\text{2}$, energy and parity, and are confined to the interior of a sphere of definite radius a, with no tails outside the sphere. They are called solitons in the paper though the name of spinor droplets or bags may be more appropriate. Because of the Lorentz covariance of the Dirac equations, arbitrarily moving solitons can be obtained from the rest system solutions by applying proper Lorentz transformations.     

Non-linear spinor equations with localized solutions

A method is outlined for finding non-linear spinor field equations which have stationary, localized solutions (solitons or droplets) of definite spin and parity.This work has been sponsored in part by the National Science Foundation, GF 42060     

Quantum fluctuations in Sine-Gordon like magnetic chains: Corrections to soliton energies

The effect of quantum fluctuations on solitons in the easy-plane ferromagnetic chain is considered within the semiclassical approximation. In accordance with the low temperature ideal gas picture we treat the solitons as a Boltzmann gas and impose quantisation on the spin wave spectrum. We present a method which allows to calculate quantum corrections in a systematic perturbation expansion in 1/S, whereS is the spin length. We use this method to obtain the soliton energy to second order at zero temperature. Our results indicate that the semiclassical approach reasonably describes quantum effects on soliton properties.     

Gaussian solitons in nonlocal media: variational analysis

Exact solutions of Gaussian solitons in nonlinear media with a Gaussian nonlocal response are obtained. Using the variational approach, we obtain the approximate solutions of such solitons when the degree of the nonlocality is arbitrary. Specifically, we study the conditions for Gaussian solitons that propagate in weakly and highly nonlocal media. We also compare the variational result with the known exact solutions for weakly and highly nonlocal media.     

Two types of ground-state bright solitons in a coupled harmonically trapped pseudo-spin polarization Bose-Einstein condensate

We study two types of bright solitons in an attractive Bose-Einstein condensate with a spin-orbit interaction. By solving the coupled nonlinear Schr odinger equations with the variational method and the imaginary time evolution method,fundamental properties of solitons are carefully investigated in different parameter regimes. It is shown that the detuning between the Raman beam and energy states of the atoms dominates the ground state type and spin polarization strength.The soliton dynamics is also studied for various moving velocities for zero and nonzero detuning cases. We find that the shape of individual component solitons can be maintained when the moving speed of solitons is low and the detuning is small in the coupled harmonically trapped pseudo-spin polarization Bose-Einstein condensate.     

Tracking azimuthons in nonlocal nonlinear media

We study the formation of azimuthons, i.e., rotating spatial solitons, in media with nonlocal focusing nonlinearity. We show that whole families of these solutions can be found by considering internal modes of classical non-rotating stationary solutions, namely vortex solitons. This offers an exhaustive method to identify azimuthons in a given nonlocal medium. We demonstrate formation of azimuthons of different vorticities and explain their properties by considering the strongly nonlocal limit of accessible solitons.     

Soliton antiferromagnetic lattice in carbon nanotubes

Based on the Hubbard model for electrons of carbon nanotubes and the Anderson model in the theory of indirect interactions, an effective Hamiltonian describing the interaction of spin degrees of freedom in the longwave approximation and having the form of the Hamiltonian of the nonlinear sigma-model was proposed. The developed model was reduced to the classical nonlinear sigma-model. Periodic solutions to the equations of evolution of spin moments were obtained, which were interpreted as lattices composed of solitons of domain walls.