The knot of spin texture is studied within the two-component Bose-Einstein condensates which are described by the nonlinear Gross-Pitaevskii equations. We start from the non-interacting equations including an axisymmetric harmonic trap to obtain an exact solution, which exhibits a non-trivial topological structure. The spin-texture is a knot with an integral Hopf invariant. The stability of the knot is verified by numerically evolving the nonlinear Gross-Pitaevskii equations along imaginary time.
The recent experimental creation of knotted solitons in spin-1 Bose-Einstein condensates opens an interesting avenue for exploring the stability and dynamics of knot solitons. Knotted and helical vortices can be accommodated in spin-orbit-coupled antiferromagnetic spin-1 Bose-Einstein condensates. We demonstrate the stability of these topological structures via numerical simulations of the corresponding Gross-Pitaevskii equation. By changing the strength of the spin-orbit coupling, we find that the ground state undergoes a change from a knotted spin texture to a helical spin texture. Our results reveal that these types of topological object are experimentally realizable in spin-1 Bose-Einstein condensates.
Hopfions are a class of three-dimensional (3D) solitons which are built as vortex tori carrying intrinsic twist of the toroidal core. They are characterized by two independent topological charges, \textit{viz}., vorticity
S and winding number
M of the intrinsic twist, whose product determines the \textit{Hopf number},
QH=MS, which is the basic characteristic of the hopfions. We construct hopfions as solutions of the 3D Gross-Pitaevskii equations (GPEs) for Bose-Einstein condensates in binary atomic gases. The GPE system includes the cubic mean-field self-attraction, competing with the quartic self-repulsive Lee-Huang-Yang (LHY) term, which represents effects of quantum fluctuations around the mean-field state, and a trapping toroidal potential (TP). A systematic numerical analysis demonstrates that families of the states with
S=1,M=0, i.e.,
QH=0, are stable, provided that the inner TP\ radius
R0 exceeds a critical value. Furthermore, true hopfions with
S=1,M=1∼7, which correspond, accordingly, to
QH=1∼7, also form partly stable families, including the case of the LHY\ superfluid, in which the nonlinearity is represented solely by the LHY term. On the other hand, the hopfion family is completely unstable in the absence of the LHY term, when only the mean-field nonlinearity is present. We illustrate the knot-like structure of the hopfions by means of an elementary geometric picture. For
QH=0, circles which represent the \textit{preimage} of the full state do not intersect. On the contrary, for
QH≥1 they intersect at points whose number is identical to
QH. The intersecting curves form multi-petal structures with the number of petals also equal to
QH.
Hopfions are a class of three-dimensional (3D) solitons which are built as vortex tori carrying intrinsic twist of the toroidal core. They are characterized by two independent topological charges, viz., vorticity S and winding number M of the intrinsic twist, whose product determines the Hopf number, QH=MS, which is the basic characteristic of the hopfions. We construct hopfion-like states (HLSs) as solutions of the 3D Gross−Pitaevskii equations (GPEs) for Bose−Einstein condensates in binary atomic gases. The GPE system includes the cubic mean-field self-attraction, competing with the quartic self-repulsive Lee−Huang−Yang (LHY) term, which represents effects of quantum fluctuations around the mean-field state, and a trapping toroidal potential (TP). Although our system exhibits two independent topological charges, it does not fully conform to the Hopf map structure, and therefore does not constitute hopfions in the complete sense. A systematic numerical analysis demonstrates that families of the states with S=1,M=0, i.e., QH=0, are stable, provided that the inner TP radius R0 exceeds a critical value. Furthermore, HLSs with S=1,M=1−7, which correspond, accordingly, to QH=1−7, also form partly stable families, including the case of the LHY superfluid, in which the nonlinearity is represented solely by the LHY term. On the other hand, the HLSs family is completely unstable in the absence of the LHY term, when only the mean-field nonlinearity is present. We illustrate the knot-like structure of the HLSs by means of an elementary geometric picture. For QH=0, circles which represent the preimage of the full state do not intersect. On the contrary, for QH≥1 they intersect at points whose number is identical to QH. The intersecting curves form multi-petal structures with the number of petals also equal to QH.
Searching for novel topological objects is always an intriguing task for scientists in various fields. We study a three-dimensional (3D) topological structure called a 3D dimeron in trapped two-component Bose-Einstein condensates. The 3D dimeron differs from the conventional 3D skyrmion for the condensates hosting two interlocked vortex rings. We demonstrate that the vortex rings are connected by a singular string and the complexity constitutes a vortex molecule. The stability of the 3D dimeron is examined in two different models using the imaginary time evolution method. We find that the stable 3D dimeron can be naturally generated from a vortex-free Gaussian wave packet incorporating a synthetic non-Abelian gauge potential into the condensates.