| With the experimental realization of Bose-Einstein condensations in toroidal trap,the study of this system has become one of the hotspots in ultracold atoms.Combining the special geometric structure of toroidal trap with the advantages of precise and controllable parameters of Bose-Einstein condensates,provides an excellent quantum simulation platform to study superfluid,hysteresis and atomic devices.However,the electroneutrality of atoms in a Bose-Einstein condensate limits the ability to simulate charge-related physical phenomena in condensed matter physics.The proposal of artificial gauge field theory and the experimental realization in Bose-Einstein condensates system address this limitation.Through the interaction of light and atoms,atoms have properties similar to the behavior of electrons.In artificial gauge field theory,spin-orbit coupling is the most important mechanism,in which the spin and linear momentum of particles are directly coupled to each other,which is crucial for spin Hall effect,topological insulators,and spintronic devices.In this paper,a Bose-Einstein condensate system with spin-orbit coupling in toroidal trap is mainly studied,and the influence of different spin-orbit couplings on the ground state phase diagram and dynamic properties of the system is emphatically analyzed.The two-dimensional weakly interacting Bose-Einstein condensate is processed based on the mean-field theory;the imaginary-time evolution is performed using the split-step Fourier method to obtain ground states of the system numerically;the numerical results are analyzed,and then an effective quasi-one-dimensional model is used to model the system,and the system can be solved analytically to reveal its physical nature.The main results of this paper are as follows:Firstly,a Bose-Einstein condensate in toroidal trap is studied when two-component Rashba spin-orbit coupling and Dresselhaus spin-orbit coupling coexisted.The ground state of condensates shows a stripe phase with an inclined distribution along the ring,which reveals that the coexistence of these two kinds of couplings breaks the rotational symmetry of the system.The ground state exhibits interesting periodic motions in both radial and azimuth directions.Secondly,three-component Rashba spin-orbit coupling in a toroidal Bose-Einstein condensate is studied.The ground state of the system presents an even-petal phase spatially separated in azimuth and a persistent flow phase separated in radial direction,which proves that the system has rotational symmetry.The two-dimensional numerical simulation results and the analytical results of quasi-one-dimensional model are further verified by analyzing their symmetry.The number of petals in petal phase and the number of phase windings of persistent flow phase can be selected by controlling the strength of spin-orbit coupling.Finally,Raman-induced spin-orbit coupling and spin-tensor-momentum coupling are introduced into the toroidal Bose-Einstein condensate,and its density is found to occupy a unique two-lobed structure;the introduction of the quadratic Zeeman shift produces more interesting stripe phases.The results of two-dimensional numerical simulation are consistent with the results of dispersion relation corresponding to the Hamiltonian,and the physics behind it is explained by analyzing symmetries of the two spin-orbit couplings. |
