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arXiv:1708.02011 · Physical Review A

Dipolar bright solitons and solitary vortices in a radial lattice

Chunqing Huang, Lin Lyu, Hao Huang, Zhaopin Chen, Shenhe Fu, Haishu Tan, Boris A. Malomed, Yongyao Li

Published Nov 13, 2017• 19 citations
arXivHTMLPDF

Abstract

Stabilizing vortex solitons with high values of the topological charge $S$ is a challenging issue in optics, studies of Bose-Einstein condensates (BECs), and other fields. To develop an approach to the solution of this problem, we consider a two-dimensional dipolar BEC under the action of an axisymmetric radially periodic lattice potential, $V(r)\ensuremath{\sim}cos(2r+\ensuremath{\delta})$, with dipole moments polarized perpendicular to the system's plane, which gives rise to isotropic repulsive dipole-dipole interactions. Two radial lattices are considered, with $\ensuremath{\delta}=0$ and $\ensuremath{\pi}$, i.e., a potential maximum or minimum at $r=0$, respectively. Families of vortex gap soliton (GSs) with $S=1$ and $S\ensuremath{\ge}2$, the latter ones often being unstable in other settings, are completely stable in the present system (at least up to $S=11$), being trapped in different annular troughs of the radial potential. The vortex solitons with different $S$ may stably coexist in sufficiently far separated troughs. Fundamental GSs, with $S=0$, are found too. In the case of $\ensuremath{\delta}=0$, the fundamental solitons are ring-shaped modes, with a local minimum at $r=0.$ At $\ensuremath{\delta}=\ensuremath{\pi}$, they place a density peak at the center.

DOI
10.1103/physreva.96.053617

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Stabilizing vortex solitons with high values of the topological charge SSS is a challenging issue in optics, studies of Bose-Einstein condensates (BECs), and other fields. To develop an approach to the solution of this problem, we consider a two-dimensional dipolar BEC under the action of an axisymmetric radially periodic lattice potential, V(r)\ensuremath∼cos(2r+\ensuremathδ)V(r)\ensuremath{\sim}cos(2r+\ensuremath{\delta})V(r)\ensuremath∼cos(2r+\ensuremathδ), with dipole moments polarized perpendicular to the system's plane, which gives rise to isotropic repulsive dipole-dipole interactions. Two radial lattices are considered, with \ensuremathδ=0\ensuremath{\delta}=0\ensuremathδ=0 and \ensuremathπ\ensuremath{\pi}\ensuremathπ, i.e., a potential maximum or minimum at r=0r=0r=0, respectively. Families of vortex gap soliton (GSs) with S=1S=1S=1 and S\ensuremath≥2S\ensuremath{\ge}2S\ensuremath≥2, the latter ones often being unstable in other settings, are completely stable in the present system (at least up to S=11S=11S=11), being trapped in different annular troughs of the radial potential. The vortex solitons with different SSS may stably coexist in sufficiently far separated troughs. Fundamental GSs, with S=0S=0S=0, are found too. In the case of \ensuremathδ=0\ensuremath{\delta}=0\ensuremathδ=0, the fundamental solitons are ring-shaped modes, with a local minimum at r=0.r=0.r=0. At \ensuremathδ=\ensuremathπ\ensuremath{\delta}=\ensuremath{\pi}\ensuremathδ=\ensuremathπ, they place a density peak at the center.
We predict the existence of stable fundamental and vortical bright solitons in dipolar Bose-Einstein condensates (BECs) with repulsive dipole-dipole interactions (DDI). The condensate is trapped in the 2D plane with the help of the repulsive contact interactions whose local strength grows ∼r4\sim r^{4}∼r4 from the center to periphery, while dipoles are oriented perpendicular to the self-trapping plane. The confinement in the perpendicular direction is provided by the usual harmonic-oscillator potential. The objective is to extend the recently induced concept of the self-trapping of bright solitons and solitary vortices in the pseudopotential, which is induced by the repulsive local nonlinearity with the strength growing from the center to periphery, to the case when the trapping mechanism competes with the long-range repulsive DDI. Another objective is to extend the analysis for elliptic vortices and solitons in an anisotropic nonlinear pseudopotential. Using the variational approximation (VA) and numerical simulations, we construct families of self-trapped modes with vorticities ℓ=0\ell =0ℓ=0 (fundamental solitons), ℓ=1\ell =1ℓ=1, and ℓ=2\ell =2ℓ=2. The fundamental solitons and vortices with ℓ=1\ell =1ℓ=1 exist up to respective critical values of the eccentricity of the anisotropic pseudopotential, being stable in the entire existence regions. The vortices with ℓ=2\ell =2ℓ=2 are stable solely in the isotropic model.
We predict solitary vortices in quasiplanar condensates of dipolar atoms, polarized parallel to the confinement direction, with the effective sign of the dipole-dipole interaction inverted by means of a rapidly rotating field. Energy minima corresponding to vortex solitons with topological charges $\ensuremath{\ell}=1$ and 2 are predicted for moderately strong dipole-dipole interaction, using an axisymmetric Gaussian Ansatz. The stability of the solitons with $\ensuremath{\ell}=1$ is confirmed by full three-dimensional simulations, whereas their counterparts with $\ensuremath{\ell}=2$ are found to be unstable against splitting into a set of four fragments (quadrupole).
We analyse the existence, stability and dynamics of localized discrete modes with intrinsic vorticity S = 1 and S = 2 in the disc-shaped dipolar Bose-Einstein condensate loaded into a deep two-dimensional optical lattice. The condensate, which features the interplay of local contact and nonlocal dipole-dipole (DD) interactions between atoms, is modelled by the 2D discrete Gross-Pitaevskii equation which includes the long-range DD term. Various species of discrete vortex solitons, which are known in the model of the condensate with local interactions, are found to exist in the presence of the DD interaction too. In locally self-attractive condensates, the isotropic DD repulsion, which corresponds to the orientation of atomic dipoles perpendicular to the confinement plane, helps to extend the region of the vortex stability, while in the case of anisotropic DD interactions, corresponding to the in-plane orientation of the dipoles, vortices are unstable. In the former case, those vortices which are unstable may evolve into robust ring-shaped breathers. The attractive isotropic DD interaction can create localized vortices in the condensate with the local self-repulsion, but they all are unstable, evolving into single-peak asymmetric structures.
The three-dimensional study of the ring vortex solitons is conducted for both attractive and repulsive BECs subject to harmonic potential confinement. A family of stationary ring vortex solitons, which is defined by the radial excitation number and the winding number of the intrinsic vorticity, are obtained numerically for a given atomic interaction strength. We find that stabilities of the ground and radially excited states of the ring vortex soliton are dependent on the winding number differently. The ground state of the ring vortex soliton with the large winding number is unstable dynamically against random perturbation. The radially excited state of the ring vortex soliton with large winding number corresponds to the increased collapse threshold, and therefore can be made stable for sufficiently small atomic interaction strengths. The ground and radially excited states also demonstrate different dynamical evolutions under large atomic interaction strengths. The former exhibits simultaneously symmetrical splitting in the transverse plane, while the latter displays periodically expand-merge cycles in the longitudinal direction.
Solitons are self-trapped modes existing in various nonlinear systems. Creating stable solitons in two- and three-dimensional settings is a challenging goal in various branches of physics. Several methods have been developed theoretically and experimentally to achieve this, but few of them can support stable multi-dimensional solitons in free space. Recently, a new scheme using spin-orbit-coupling (SOC) has been proposed to create stable 2D solitons in Bose–Einstein condensates (BECs). This paper reviews recent theoretical progress on creating stable 2D solitons in spinor dipolar BEC with SOC, combined with long-range dipole-dipole interaction (DDI), Zeeman splitting (ZS) and contact nonlinearity, in free space. The continuous family of stable symmetric vortex solitons (SVS), asymmetric vortex solitons (AVS), as well as gap solitons (GS) is found via different settings. Their existence and stability conditions are summarized and discussed in detail. The mobility properties of these types of solitons are also addressed. For SVS, a potential method to manipulate its shape and mobility is investigated. These results are supposed to enrich our understanding of 2D solitons and help create multi-dimensional solitons in experiments.
A three-dimensional study of the ring vortex solitons is conducted for both attractive and repulsive Bose-Einstein condensates subject to harmonic potential confinement. A family of stationary ring vortex solitons, which is defined by the radial excitation number and the winding number of the intrinsic vorticity, are obtained numerically for a given atomic interaction strength. We find that stabilities of the ground and radially excited states of the ring vortex soliton are dependent on the winding number differently. The ground state of the ring vortex soliton with the large winding number is unstable dynamically against random perturbation. The radially excited state of the ring vortex soliton with large winding number corresponds to the increased collapse threshold and therefore can be made stable for sufficiently small atomic interaction strengths. The ground and radially excited states also demonstrate different dynamic evolutions under large atomic interaction strengths. The former exhibits simultaneous symmetric splitting in the transverse plane, while the latter displays periodic expand-merge cycles in the longitudinal direction.