Kartashov Yaroslav V, Malomed Boris A, Shnir Yasha, Torner Lluis
ICFO-Institut de Ciencies Fotoniques, and Universitat Politecnica de Catalunya, Mediterranean Technology Park, E-08860 Castelldefels (Barcelona), Spain and Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, 142190 Moscow, Russia.
Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, 69978 Tel Aviv, Israel.
Phys Rev Lett. 2014 Dec 31;113(26):264101. doi: 10.1103/PhysRevLett.113.264101. Epub 2014 Dec 24.
Toroidal modes in the form of so-called Hopfions, with two independent winding numbers, a hidden one (twist s), which characterizes a circular vortex thread embedded into a three-dimensional soliton, and the vorticity around the vertical axis (m), appear in many fields, including field theory, ferromagnetics, and semi- and superconductors. Such topological states are normally generated in multicomponent systems, or as trapped quasilinear modes in toroidal potentials. We uncover that stable solitons with this structure can be created, without any linear potential, in the single-component setting with the strength of repulsive nonlinearity growing fast enough from the center to the periphery, for both steep and smooth modulation profiles. Toroidal modes with s=1 and vorticity m=0, 1, 2 are produced. They are stable for m≤1, and do not exist for s>1. An approximate analytical solution is obtained for the twisted ring with s=1, m=0. Under the application of an external torque, it rotates like a solid ring. The setting can be implemented in a Bose-Einstein condensate (BEC) by means of the Feshbach resonance controlled by inhomogeneous magnetic fields.
以所谓霍普夫离子形式存在的环形模式，具有两个独立的缠绕数，一个隐藏的缠绕数（扭曲数s），它表征嵌入三维孤子中的圆形涡旋线，以及围绕垂直轴的涡度（m），出现在许多领域，包括场论、铁磁学以及半导体和超导体中。这种拓扑态通常在多组分系统中产生，或者作为环形势中的捕获准线性模式产生。我们发现，在单组分环境中，对于陡峭和平滑的调制轮廓，在排斥性非线性强度从中心到周边足够快地增长的情况下，可以在没有任何线性势的情况下创建具有这种结构的稳定孤子。产生了s = 1且涡度m = 0、1、2的环形模式。它们对于m≤1是稳定的，而对于s>1则不存在。得到了s = 1、m = 0的扭曲环的近似解析解。在外部扭矩的作用下，它像实心环一样旋转。通过由非均匀磁场控制的费什巴赫共振，可以在玻色-爱因斯坦凝聚体（BEC）中实现这种设置。
