Quasi-one-dimensional soliton in a self-repulsive spin-orbit-coupled dipolar spin-half and spin-one condensates

TL;DR

This work identifies and characterizes a broad family of quasi-1D solitons in SO-coupled, self-repulsive dipolar BECs for pseudo spin-half and spin-one spinor condensates. By analyzing both the analytical linear problem and nonlinear stationary states solved via imaginary-time propagation, it shows how SO coupling and dipolar interactions produce dark-bright, bright-bright, and multi-component solitons, some with spatially periodic density stripes characteristic of supersolids. The study reveals that large SO coupling induces stripe modulations with period $\pi/\gamma$ in component densities while preserving a uniform total density, and demonstrates dynamical stability of all identified solitons through long-time real-time propagation under perturbations. These results deepen understanding of soliton formation in complex spinor BECs and suggest avenues for experimental realization of multi-component, stripe-ordered solitons in SO-coupled dipolar systems.

Abstract

We study the formation of solitons in a uniform quasi-one-dimensional (quasi-1D) spin-orbit (SO) coupled self-repulsive pseudo spin-half and spin-one dipolar Bose-Einstein condensates (BEC), using the mean-field Gross-Pitaevskii equation. The dipolar atoms are taken to be polarized along the quasi-1D $x$ direction. In the pseudo spin-half case, for small SO-coupling, one can have dark-bright and bright-bright solitons. For large SO coupling, the dark-bright and bright-bright solitons may acquire a spatially-periodic modulation in density; for certain values of contact interaction paramerers there is only the normal bright-bright soliton without spatially-periodic modulation in density. In the spin-one anti-ferromagnetic case, for small SO coupling, one can have bright-bright-bright, dark-bright-dark, and bright-dark-bright solitons; and for large SO coupling, the dark-bright-dark and bright-dark-bright solitons are found to have spatially-periodic modulation in density. In the spin-one ferromagnetic case, for both small and large SO coupling, we find only bright-bright-bright solitons. All these solitons, specially those with a dark-soliton component, are dynamically stable as demonstrated by real-time propagation using the converged stationary solution obtained by imaginary-time propagation as the initial state.

Figures (6)

• Figure 1: (Color online) Density of components $j=1,2$ and total density of a quasi-1D SO-coupled pseudo spin-half dipolar (a) dark-bright, (b) a phase-separated bright-bright, and (c) a bright-bright soliton for SO-coupling strength $\gamma =0.1$. In (c) we also show the total density of the solitons in plots (a) (t-1a) and (b) (t-1b). The plots (d)-(f) display a dark-bright, a phase-separated bright-bright, and a bright-bright soliton, for $\gamma =1$, with the first two having a spatially-periodic modulation in density. The energy ($E$), magnetization (mag), and $\gamma$ are given in the inset of all plots of this paper. The parameters are $c_0=c_2=0.5, d=2$. All quantities in this and following figures are dimensionless and the total density is always normalized to unity, viz. Eq. (\ref{['noma']}).
• Figure 2: Density of components $j=1,2$ and total density of a quasi-1D SO-coupled pseudo spin-half dipolar (a) dark-bright, and (b) bright-bright soliton for SO-coupling strength $\gamma =0.1$. The plots (c)-(e) display a dark-bright, a phase-separated bright-bright, and a bright-bright soliton, respectively for $\gamma =1$. The plot (f) presents the phase of the wave functions of the dark-bright soliton of plot (a). The parameters are $c_0=-0.5, c_{2}=0.5, d=2$.
• Figure 3: Density of components $j=1,2$ and total density of a quasi-1D SO-coupled pseudo spin-half dipolar bright-bright soliton for (a) $\gamma =0.1$ and (b) $\gamma=1$. The parameters are $c_0=0.5, c_{2}=-0.5, d=2$.
• Figure 4: Density of components $j=1,2,3$ and total density of a quasi-1D SO-coupled spin-one dipolar (a) dark-bright-dark, (b) partially phase-separated bright-bright-bright and (c) bright-dark-bright solitons for SO-coupling strength $\gamma =0.1$ for an anti-ferromagnetic spin-one BEC. The plots (d)-(e) display a dark-bright-dark and a bright-dark-bright soliton, both with a spatially-periodic modulation in density, respectively for $\gamma =1$. The plot (f) depicts the phase of the component wave functions of the soliton of plot (a). The parameters are $c_0=0.5, c_2=0.5, d=2$.
• Figure 5: Density of components $j=1,2,3$ and total density of a quasi-1D SO-coupled spin-one dipolar (a) bright-bright-bright, (b) a phase-separated bright-bright-bright solitons for SO-coupling strength $\gamma =0.1$ for a ferromagnetic spin-one BEC. The plot (c) displays a bright-bright-bright soliton for $\gamma =1$. The parameters are $c_0=0.5, c_2=-0.1, d=2$.