Stable High-Order Vortices in Spin-Orbit-Coupled Spin-1 Bose-Einstein Condensates

TL;DR

The paper addresses how to realize stable high-order vortex states in spin-1 Bose-Einstein condensates with spin-orbit coupling by exploiting ground-state phase transitions. It combines an exact linear solution at $\alpha=\beta$ with nonlinear simulations including density and spin interactions, complemented by stability analyses. The results show that excited states can be driven to GS by tuning $\beta$ (and $\alpha$), enabling arbitrary winding numbers as GS; repulsive spin interactions preserve the linear predictions, while attractive interactions yield mixed states near transitions, all of which are linearly and dynamically stable. Importantly, GS transitions also occur for $\alpha\neq\beta$, making these states more accessible experimentally and shedding light on topological phenomena in SOC BECs.

Abstract

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.

Figures (8)

• Figure 1: (a) The corresponding chemical potential $\mu_n(\beta)$ plotted pursuant to Eq. \ref{['mu']}. The dots are values of $\beta_n$ defined by Eq. \ref{['beta']}. (b) Scatter points show the dependence of the $n$th critical point $\beta_n$ on $n$; the dashed line indicates the asymptotic behavior given by Eq. \ref{['asymptotic']}.
• Figure 2: Profiles of the radial wavefunctions $R_{\pm1,0}^{(n)}$, defined as per Eqs. \ref{['R']} and \ref{['R0']}, with quantum numbers (a) $n=0$, (b) $n=1$, (c) $n=2$ and (d) $n=3$.
• Figure 3: The map of values of the winding number (magnetic quantum number) $m$ corresponding to GS of the linear system in the ($\alpha$, $\beta$) parameter plane.
• Figure 4: (a-d) Angular momentum $M$, defined as per Eqs. \ref{['M']}, as produced by the imaginary-time simulations of the full (nonlinear) system (\ref{['main']}), for $\beta$ varying from $0$ to $1.8$. Other parameter in Eq. \ref{['main']} is $c_0 =1$.
• Figure 5: Distributions of the absolute values, $\left\vert \psi_{\pm1,0}\right\vert$, and phases, $\Theta_{\pm1,0}$, of wave functions of the three components in the GS of the vortex type, for $\beta =0.32,\mu=0.2812$ (panels $a_{1}-a_{3}$); $\beta =0.80,\mu=0.2794$ (panels $b_{1}-b_{3}$); $\beta =1.16,\mu=-0.7114$ (panels $c_{1}-c_{3}$) and $\beta =1.52,\mu=-1.7008$ (panels $d_{1}-d_{3}$). Other parameters in Eq. \ref{['main']} are $c_0=c_2 =1$.