We investigate the formation, stability, and dynamics of solitons in a one-dimensional binary Bose-Einstein condensate under the action of the spin-orbit-coupling and Lee-Huang-Yang (LHY) correction to the underlying system of the Gross-Pitaevskii equations. We identify the semi-dipole (SD) family of solitons and thoroughly analyze its properties. The numerical analysis reveals intricate bifurcations, including transitions from real to complex-valued stationary wave functions of the SD solitons and norm-dependent dynamical instabilities. Stability maps in the plane of the solitons' norm and interaction strength exhibit areas of monostability, oscillatory behavior, and soliton splitting. Solitons with complex stationary wave functions emerge as ground states in broad parameter areas due to the effects of the LHY terms. The other soliton species, in the form of mixed modes (MMs), do not feature the compexification bifurcation. In the LHY-dominated regime, the SD and MM solitons exhibit identical values of the energy for the same norm. The results deepen the understanding of nonlinear matter-wave states and reveal multistable ones in quantum gases.
We investigate the formation, stability, and dynamics of solitons in a one-dimensional binary Bose-Einstein condensate under the action of the spin-orbit-coupling and Lee-Huang-Yang (LHY) correction to the underlying system of the Gross-Pitaevskii equations. We identify the semi-dipole (SD) family of solitons and thoroughly analyze its properties. The numerical analysis reveals intricate bifurcations, including transitions from real to complex-valued stationary wave functions of the SD solitons and norm-dependent dynamical instabilities. Stability maps in the plane of the solitons' norm and interaction strength exhibit areas of monostability, oscillatory behavior, and soliton splitting. Solitons with complex stationary wave functions emerge as ground states in broad parameter areas due to the effects of the LHY terms. The other soliton species, in the form of mixed modes (MMs), do not feature the compexification bifurcation. In the LHY-dominated regime, the SD and MM solitons exhibit identical values of the energy for the same norm. The results deepen the understanding of nonlinear matter-wave states and reveal multistable ones in quantum gases.
We study two-dimensional (2D) matter-wave solitons in spinor Bose–Einstein condensates under the action of the spin–orbit coupling and opposite signs of the self- and cross-interactions. Stable 2D two-component solitons of the mixed-mode type are found if the cross-interaction between the components is attractive, while the self-interaction is repulsive in each component. Stable solitons of the semi-vortex type are formed in the opposite case, under the action of competing self-attraction and cross-repulsion. The solitons exist with the total norm taking values below a collapse threshold. Further, in the case of the repulsive self-interaction and inter-component attraction, stable 2D self-trapped modes, which may be considered as quantum droplets (QDs), are created if the beyond-mean-field Lee–Huang–Yang terms are added to the self-repulsion in the underlying system of coupled Gross–Pitaevskii equations. Stable QDs of the mixed-mode type, of a large size with an anisotropic density profile, exist with arbitrarily large values of the norm, as the Lee–Huang–Yang terms eliminate the collapse. The effect of the spin–orbit coupling term on characteristics of the QDs is systematically studied. We also address the existence and stability of QDs in the case of SOC with mixed Rashba and Dresselhaus terms, which makes the density profile of the QD more isotropic. Thus, QDs in the spin-orbit-coupled binary Bose–Einstein condensate are for the first time studied in the present work.
We present the numerical results of the structure and dynamics of the self-bound ground state arising solely because of the presence of beyond-mean-field quantum fluctuation in spin-orbit- (SO-) coupled binary Bose-Einstein condensates in one dimension. We obtain an analytical soliton solution for nonzero SO coupling that matches quite well with the numerical results. Further, we investigate the dynamical stability of these solitons by adopting three protocols, such as (i) adding initial velocity to each component, (ii) quenching the SO- and Rabi-coupling parameters at initial and finite time, and (iii) allowing collision between the two spin components by giving equal and opposite direction velocity to them. For small velocity perturbation, the soliton has the presence of the breathing oscillation, while large velocity perturbations transform the soliton into moving solitons. The maximum breathing frequency exhibits power-law dependence on the Rabi-coupling frequency with an exponent $\ensuremath{\sim}0.16$. The quenching of SO and Rabi couplings results in several interesting dynamical features, such as dynamical phase transition from plane to stripe phase, dynamical flipping of the spin state, and secondary solitonic modes. In the absence of the SO and Rabi couplings, depending upon the velocity of the up- and down-spin components, the collision between them is either elastic or inelastic, which is consistent with the earlier numerical and experimental observations. However, in the presence of coupling parameters, the collision appears to be inelastic and quasielastic in nature.
The interplay of quantum fluctuation through the Lee-Huang-Yang (LHY) correction and spin-orbit coupling can modify the stability of the ultradilute Bose-Bose mixtures and result in rich nonlinear physics. Here, under the action of spin-orbit coupling, LHY correction, and rotation, the ground state and vortex excitation of spin-orbit coupled Bose-Einstein condensates (BECs) with LHY correction trapped in a rotating potential are studied. Stable isotropic and anisotropic ground states with zero momentum (in the zero-momentum phase) and nonzero momentum (in the plane wave phase) are obtained. It is shown that anisotropic ground states are persistent near the phase boundary. Particularly, unpolarized and polarized vortex states are predicted. The phase transition condition, the effective mass and quadrupole surface mode describing the anisotropic (deformation) of the condensates, and the critical rotation frequency for vortex excitation are provided analytically. Owing to the action of spin-orbit coupling, the effective mass of the condensates is spatially dependent and asymmetric in two phases, resulting in distinct quadrupole modes, deformations, and vortex excitations in the two phases, especially when the LHY correction is present. The LHY correction promotes the system in the zero-momentum phase with unpolarized vortex states, while the spin-orbit coupling promotes the emergence of a plane wave phase with polarized vortex states. The phase transition and vortex excitation can be well manipulated by spin-orbit coupling, Raman coupling, LHY correction, and external potential.
We present an analysis of two-dimensional (2D) matter-wave solitons, governed by the pseudospinor system of Gross-Pitaevskii equations with self- and cross attraction, which includes the spin-orbit coupling (SOC) in the general Rashba-Dresselhaus form, and, separately, the Rashba coupling and the Zeeman splitting. Families of semivortex (SV) and mixed-mode (MM) solitons are constructed, which exist and are stable in free space, as the SOC terms prevent the onset of the critical collapse and create the otherwise missing ground states in the form of the solitons. The Dresselhaus SOC produces a destructive effect on the vortex solitons, while the Zeeman term tends to convert the MM states into the SV ones, which eventually suffer delocalization. Existence domains and stability boundaries are identified for the soliton families. For physically relevant parameters of the SOC system, the number of atoms in the 2D solitons is limited by ∼1.5×10^{4}. The results are obtained by means of combined analytical and numerical methods.
We consider possibilities to control dynamics of solitons of two types, maintained by the combination of cubic attraction and spin-orbit coupling (SOC) in a two-component system, namely, semi-dipoles (SDs) and mixed modes (MMs), by making the relative strength of the cross-attraction, γ , a function of time periodically oscillating around the critical value, γ = 1 , which is an SD/MM stability boundary in the static system. The structure of SDs is represented by the combination of a fundamental soliton in one component and localized dipole mode in the other, while MMs combine fundamental and dipole terms in each component. Systematic numerical analysis reveals a finite bistability region for the SDs and MMs around γ = 1 , which does not exist in the absence of the periodic temporal modulation (“management”), as well as emergence of specific instability troughs and stability tongues for the solitons of both types, which may be explained as manifestations of resonances between the time-periodic modulation and intrinsic modes of the solitons. The system can be implemented in Bose-Einstein condensates (BECs), and emulated in nonlinear optical waveguides.
We report families of two-dimensional (2D) composite solitons in spinor dipolar Bose-Einstein condensates, with two localized components linearly mixed by the spin-orbit coupling (SOC), and the intrinsic nonlinearity represented by the dipole-dipole interaction (DDI) between atomic magnetic moments polarized in plane by an external magnetic field. Recently, stable solitons were predicted in the form of semivortices (composites built of coupled fundamental and vortical components) in the 2D system combining the SOC and contact attractive interactions. Replacing the latter by the anisotropic long-range DDI, we demonstrate that, for a fixed norm of the soliton, the system supports a continuous family of stable spatially asymmetric vortex solitons (AVSs), parameterized by an offset of the pivot of the vortical component relative to its fundamental counterpart. The offset is limited by a certain maximum value, while the energy of the AVS practically does not depend on the offset. At small values of the norm, the vortex solitons are subject to a weak oscillatory instability. In the present system, with the Galilean invariance broken by the SOC, the composite solitons are set in motion by a kick the strength of which exceeds a certain depinning value. The kicked solitons feature a negative effective mass, drifting along a spiral trajectory opposite to the direction of the kick. A critical angular velocity, up to which the semivortices may follow rotation of the polarizing magnetic field, is found too.