We systematically investigate the weakly trapped spin-1 Bose-Einstein condensates with spin-orbit coupling in an external Zeeman field. We find that the mean-field ground state favors either a magnetized standing-wave phase or plane-wave phase when the strength of the Zeeman field is below a critical value related to the strength of spin-orbit coupling. The Zeeman field can induce a phase transition between standing-wave and plane-wave phases, and we determine the phase boundary analytically and numerically. The magnetization of these two phases responds to the external magnetic field in a very unique manner: the linear Zeeman effect magnetizes the standing-wave phase along the direction of the magnetic field, but the quadratic Zeeman effect demagnetizes the plane-wave phase. When the strength of Zeeman field surpasses the critical value, the system is completely polarized to a ferromagnetic state or polar state with zero momentum.
We systematically investigate the weakly trapped spin-1 Bose-Einstein condensates with spin-orbit coupling in an external Zeeman field. We find that the mean-field ground state favors either a magnetized standing-wave phase or plane-wave phase when the strength of the Zeeman field is below a critical value related to the strength of spin-orbit coupling. The Zeeman field can induce a phase transition between standing-wave and plane-wave phases, and we determine the phase boundary analytically and numerically. The magnetization of these two phases responds to the external magnetic field in a very unique manner: the linear Zeeman effect magnetizes the standing-wave phase along the direction of the magnetic field, but the quadratic Zeeman effect demagnetizes the plane-wave phase. When the strength of Zeeman field surpasses the critical value, the system is completely polarized to a ferromagnetic state or polar state with zero momentum.
We study the dynamics of a quasi-two-dimensional (2D) spin-orbit-coupled ferromagnetic Bose-Einstein condensate (BEC) under a linear Zeeman magnetic field disturbed by a moving obstacle. The Bogoliubov excitation spectrums and corresponding critical excitations in different situations are analyzed. The structure of the coreless vortex or antivortex generated by the moving obstacle has been investigated. When the Zeeman field is parallel to the 2D system plane, the vortex cores for the three components of a (an) vortex (antivortex) could be arranged into a line which is vertical to the Zeeman field, and their order would be reversed as the spin-orbit coupling increases. When the Zeeman field is perpendicular to the plane, a skyrmionlike vortex ground state could be induced via spin-orbit interaction even by a static obstacle and thus becomes unique in contrast with the scalar BECs or spinor BECs without spin-orbit coupling. This topological structure is also found to be dynamically stable if the obstacle is moving at a relatively small velocity.
We predict a dynamical resonant effect, which is driven by externally applied linear and quadratic Zeeman fields, in a spin-orbit-coupled spin-1 Bose-Einstein condensate. The Bose-Einstein condensate is assumed to be initialized in some superposed state of Zeeman sublevels and subject to a sudden shift of the trapping potential. It is shown that the time-averaged center-of-mass oscillation and the spin polarizations of the Bose-Einstein condensate exhibit remarkable resonant peaks when the Zeeman fields are tuned to certain strengths. The underlying physics behind this resonance can be traced back to the out-of-phase interference of the dynamical phases carried by different spin-orbit states. By analyzing the single-particle spectrum, the resonant condition is summarized as a simple algebraic relation, connecting the strengths of the linear and quadratic Zeeman fields. This property is potentially applicable in quantum information and quantum precision measurement.
We study the ground-state phases of two-dimensional spin-orbit coupled spin-1 Bose-Einstein condensate loaded in a plane quadrupole field. In the absence of rotation, for the fixed spin-orbit coupling strength, the ordinary stripe phase is found when the strength of the magnetic field gradient is small. As the strength of magnetic field gradient enhances, the system realizes the phases with three layer vortices along the radial direction. The number of vortices in the second layer is successively increased and the vortices in the outermost layer disappear when the strength of magnetic field gradient surpass the critical value. For the large strength of magnetic field gradient, the system only has the inner layer vortices. The magnetic field inhibits the region of vortices. For the fixed magnetic field gradient strength, the vortices of the system elongate along the radial direction and form a series of vortex lines, the number of the vortex line increases as the strength of spin-orbit coupling enhances. By adding the rotation, for the fixed strengths of spin-orbit coupling and magnetic field gradient, the number of second layer vortices also successively increases as the rotational frequency increases. The number of vortices in the certain layer of the ground-state density can be regularly changed under the effects of the magnetic field and spin-orbit coupling.
A weakly interacting, spin-orbit-coupled, ultracold, dilute Bose gas on a two-dimensional square lattice with an external Zeeman field is studied. We explore the plane- and stripe-wave phases of the system involving nonzero condensate momenta, which occur when the Zeeman field is below a critical value. Their excitation spectra are found using Bogoliubov theory and by two different routes. The validity of each method to obtain the excitation spectrum is discussed, and it is found that projection on the lowest single-particle band is an excellent approximation in the plane-wave phase, while it is a poor approximation in the stripe-wave phase. While the plane-wave phase has a phonon minimum at its single condensate momentum, revealing a nonzero sound velocity of the excitations, the stripe-wave phase has quadratic minima at its two condensate momenta showing zero sound velocity of the excitations. We discuss how the presence of more than one condensate momentum is essential for these differences between the two phases. Additionally, it is emphasized that the zero sound velocity in the stripe-wave phase is a lattice effect, since continuum studies of the same phase have shown nonzero sound velocity.
We investigate the transitions of ground states induced by zero momentum (ZM) coupling in pseudospin-1/2 Rashba spin-orbit coupled Bose-Einstein condensates confined in a harmonic trap. In a weak harmonic trap, the condensate presents a plane wave (PW) state, a stripe state or a spin polarized ZM state, and the particle distribution of the stripe state is weighted equally at two points in the momentum space without ZM coupling. The presence of ZM coupling induces an imbalanced particle distribution in the momentum space, and leads to the decrease of the amplitude of the stripe state. When its strength exceeds a critical value, the system experiences the transition from stripe phase to PW phase. The boundary of these two phases is shifted and a new phase diagram spanned by the ZM coupling and the interatomic interactions is obtained. The presence of ZM coupling can also achieve the transition from ZM phase to PW phase. In a strong harmonic trap, the condensate exhibits a vortex lattice state without ZM coupling. For the positive effective Rabi frequency of ZM coupling, the condensate is driven from a vortex lattice state to a vortex-free lattice state and finally to a PW state with the increase of coupling strength. In addition, for the negative effective Rabi frequency, the condensate is driven from a vortex lattice state to a stripe state, and finally to a PW state. The stripe state found in the strong harmonic trap is different from that in previous works because of its nonzero superfluid velocity along the stripes. We also discuss the influences of the ZM coupling on the spin textures, and indicate that the spin textures are squeezed transversely by the ZM coupling.
Abstract We explore the existence and stability domains of stationary and moving bright solitons in spin–orbit-coupled spin-1 Bose–Einstein condensates in an external Zeeman field. Two families of bell-shape bright solitons (plane-wave (PW) phases) and two families of stripe-shape bright solitons (standing-wave (SW) phases) are obtained. We find a relation between the existence of these bright solitons and the single-particle energy spectrum, as well as the requirements of their existence for atomic interactions. The stability of four families of bright solitons is systematically analyzed using the linear stability analysis method. For two families of PW bright solitons, they are unstable only when the strength of the Zeeman field is above a critical value. The critical value is affected by the spin–orbit coupling and hardly affected by atomic interactions. When the velocities of solitons are zero (nonzero), the critical values of these two families are equal (unequal). For two families of SW bright solitons, their stability domains are identical in the absence of the Zeeman field, and they are unstable when the ferromagnetic interaction is strong. In the presence of the Zeeman field, their stability domains are complementary and complex, and one family of SW bright solitons can exist stably in the area with stronger ferromagnetic interaction. Furthermore, we find the collisions of stable bright solitons with different velocities can generate intriguing dynamics.