A method to realize controllable inversion of energy levels in a one-dimensional spin-orbit (SO)-coupled two-component Bose-Einstein condensate under the action of a gradient magnetic field and harmonic-oscillator (HO) trapping potential is proposed. The linear version of the system is solved exactly. By adjusting the SO coupling strength and magnetic-field gradient, the energy-level inversion makes it possible to transform any excited state into the ground state. The full nonlinear system is solved numerically, and it is found that the results are consistent with the linear prediction in the case of the repulsive intercomponent interaction. On the other hand, the intercomponent attraction gives rise to states of superposition and edge types. Similar results are also reported for the system with the HO trap replaced by the box potential. These results suggest a possibility to realize any excited state and observe it in the experiment.
A method to realize controllable inversion of energy levels in a one-dimensional spin-orbit (SO)-coupled two-component Bose-Einstein condensate under the action of a gradient magnetic field and harmonic-oscillator (HO) trapping potential is proposed. The linear version of the system is solved exactly. By adjusting the SO coupling strength and magnetic-field gradient, the energy-level inversion makes it possible to transform any excited state into the ground state. The full nonlinear system is solved numerically, and it is found that the results are consistent with the linear prediction in the case of the repulsive intercomponent interaction. On the other hand, the intercomponent attraction gives rise to states of superposition and edge types. Similar results are also reported for the system with the HO trap replaced by the box potential. These results suggest a possibility to realize any excited state and observe it in the experiment.
We investigate vortex states in Bose-Einstein condensates under the combined action of the spin-orbit coupling (SOC), gradient magnetic field, and harmonic-oscillator trapping potential. The linear version of the system is solved exactly. Through the linear-spectrum analysis, we find that by varying the SOC strength and magnetic-field gradient one can perform energy-level inversion. With suitable parameters, initial higher-order vortex states can be made the ground state (GS). The nonlinear system is solved numerically, revealing that the results are consistent with the linear predictions in the case of repulsive intercomponent interactions. On the other hand, intercomponent attraction creates the GS in the form of mixed-mode states in a vicinity of the GS phase-transition points. The spin texture of both vortex- and mixed-mode GSs reveals that they feature the structure of 2D (baby) skyrmions.
We investigate phase transitions of the ground state (GS) of spin-1 Bose-Einstein condensates under the combined action of the spin-orbit coupling (SOC) and gradient magnetic field. Introducing appropriate raising and lowering operators, we exactly solve the linear system. Analyzing the obtained energy spectrum, we conclude that simultaneous variation of the magnetic-field gradient and SOC strength leads to the transition of excited states into the GS. As a result, any excited state can transition to the GS, at appropriate values of the system's parameters. The nonlinear system is solved numerically, showing that the GS phase transition, similar to the one in the linear system, still exists under the action of the repulsive nonlinearity. In the case of weak attraction, a mixed state appears near the GS transition point, while the GS transitions into an edge state under the action of strong attractive interaction.
Two-component Bose-Einstein condensate offers an ideal platform for investigating many intriguing topological defects due to the interplay between intraspecies and interspecies interactions. The recent realization of spin-orbit coupling in two-component Bose-Einstein condensate, owing to coupling between the spin and the centre-of-mass motion of the atom, provides possibly new opportunities to search for novel quantum states. In particular, the gradient magnetic field in the Bose-Einstein condensate has brought a new way to create topologically nontrivial structures including Dirac monopoles and quantum knots. Previous studies of the gradient magnetic field effect in the Bose-Einstein condensate mainly focused on the three-component case. However, it remains unclear how the gradient magnetic field affects the ground state configuration in the rotating two-component Bose-Einstein condensate with spin-orbit coupling. In this work, by using quasi two-dimensional Gross-Pitaevskii equations, we study the ground state structure of a rotating two-component Bose-Einstein condensate with spin-orbit coupling and gradient magnetic field. We concentrate on the effects of the spin-orbit coupling and the gradient magnetic field on the ground state. The numerical results show that increasing the strength of the spin-orbit coupling can induce a phase transition from skyrmion lattice to skyrmion chain in the presence of the gradient magnetic field. Unlike the study of skyrmion in rotating two-component Bose-Einstein condensate with only spin-orbit coupling, the skyrmion chain can occur under the isotropic spin-orbit coupling when the gradient magnetic field is considered. It is worth noting that the skyrmion chain here is arrayed along the diagonal direction. Next we examine the effect of the gradient magnetic field on spin-orbit coupled two-component Bose-Einstein condensate. For the case of weak spin-orbit coupling and the slow rotation, a phase transition from a single plane-wave to half-skyrmion is found through increasing magnetic field gradient strength. For the case of strong spin-orbit coupling and the fast rotation, the nature of the ground state is shown to support the formation of a hidden vortex as the gradient magnetic field is enhanced. These hidden vortices have no visible cores in density distributions but have phase singularities in phase distributions, which are arrayed along the diagonal direction. This result confirms a new method of creating the hidden vortices in the two-component Bose-Einstein condensate. These topological structures can be detected by using the time-of-flight absorption imaging technique. Our results illustrate that the gradient magnetic field not only provides an opportunity to explore the exotic topological structures in spin-orbit coupled spinor Bose-Einstein condensate, but also is crucial for realizing the phase transitions among different ground states. This work paves the way for the future exploring of topological defect and the corresponding dynamical stability in quantum systems subjected to a gradient magnetic field.
The present contribution explores phase transitions that occur in the ground state (GS) of spin-1 Bose-Einstein condensates (BECs) with spin-orbit coupling (SOC) under the action of gradient magnetic fields. By solving the corresponding linearized system in an exact fashion, we identify the conditions under which the GS phase transitions occur, thus transforming excited states into GS. The study of the full nonlinear system, including both density-density and spin-spin interactions, is numerically analyzed. For the case of repulsive spin-spin interactions, the results resemble the linear case, while attractive spin-spin interactions lead to the formation of mixed-states near the GS phase-transition points. Additionally, higher-order vortex solitons are found to be stable even in the nonlinear regime. These findings demonstrate that arbitrary winding numbers can be achieved as corresponding to stable GS and thus contributing to the understanding of topological properties in SOC BECs.
We study the collective dynamics of the spin-orbit coupled two pseudospin components of a Bose-Einstein condensate trapped in a quasi-one-dimensional harmonic potential, by using variational and directly numerical approach of binary mean-field Gross-Pitaevskii equations. The results show that, because of strong coupling of spin-orbit coupling (SOC), Rabi coupling, and atomic interaction, the collective dynamics of the system behave as complex characters. When the Rabi coupling is absent, the density profiles of the system preserve the Gauss type and the wave packets do harmonic oscillations. The amplitude of the collective oscillations increases with SOC. Furthermore, when the SOC strength increases, the dipole oscillations of the two pseudospin components undergo a transition from in-phase to out-of-phase oscillations. When the Rabi coupling present, there will exist a critical value of SOC strength (which depends on the Rabi coupling and atomic interaction). If the SOC strength is less than this critical value, the density profiles of the system can preserve the Gauss type and the wave packets do anharmonic (the frequency of dipole oscillations depends on SOC) oscillations synchronously (i.e., in-phase oscillations). However, if the SOC strength is larger than this critical value, the wave packets are dynamically fragmented and the stable dipole oscillations of the system can not exist. The collective dynamics of the system can be controlled by adjusting the atomic interaction, SOC, and Rabi-coupling strength.
Spin-orbit-coupled bosons can exhibit rich equilibrium phases at low temperature and in the presence of particle-particle interactions. In the case with a 1D synthetic spin-orbit interaction, it has been observed that the ground state of a Bose gas can be a normal phase, stripe phase, or magnetized phase in different parameter regimes. The magnetized states are doubly degenerate and consist of a many-particle two-state system. In this work, we investigate the nonequilibrium quantum dynamics by switching on a simple one-dimensional optical lattice potential as external perturbation to induce resonant couplings between the magnetized phases, and predict a quantum spin dynamics which cannot be obtained in the single-particle systems. In particular, due to particle-particle interactions, the transition of the Bose condensate from one magnetized phase to the other is forbidden when the external perturbation strength is less than a critical value, and a full transition can occur only when the perturbation exceeds such critical strength. This phenomenon manifests itself a dynamical phase transition, with the order parameter defined by the time-averaged magnetization over an oscillation period, and the critical point behavior being exactly solvable. The thermal fluctuations are also considered in detail. From numerical simulations and exact analytic studies we show that the predicted many-body effects can be well observed with the current experiments.