Structured light fields embody strong spatial variations of polarization,phase,and amplitude.Understanding,characterization,and exploitation of such fields can be achieved through their topological properties.Three-di...Structured light fields embody strong spatial variations of polarization,phase,and amplitude.Understanding,characterization,and exploitation of such fields can be achieved through their topological properties.Three-dimensional(3D)topological solitons,such as hopfions,are 3D localized continuous field configurations with nontrivial particle-like structures that exhibit a host of important topologically protected properties.Here,we propose and demonstrate photonic counterparts of hopfions with exact characteristics of Hopf fibration,Hopf index,and Hopf mapping from real-space vector beams to homotopic hyperspheres representing polarization states.We experimentally generate photonic hopfions with on-demand high-order Hopf indices and independently controlled topological textures,including Néel-,Bloch-,and antiskyrmionic types.We also demonstrate a robust free-space transport of photonic hopfions,thus showing the potential of hopfions for developing optical topological informatics and communications.展开更多
Hopfions are three-dimensional(3D)topological states discovered in field theory,magnetics,and hydrodynamics that resemble particle-like objects in physical space.Hopfions inherit the topological features of the Hopf f...Hopfions are three-dimensional(3D)topological states discovered in field theory,magnetics,and hydrodynamics that resemble particle-like objects in physical space.Hopfions inherit the topological features of the Hopf fibration,a homotopic mapping from unit sphere in 4D space to unit sphere in 3D space.Here we design and demonstrate dynamic scalar optical hopfions in the shape of a toroidal vortex and expressed as an approximate solution to Maxwell’s equations.Equiphase lines correspond to disjoint and interlinked loops forming complete ring tori in 3D space.The Hopf invariant,product of two winding numbers,is determined by the topological charge of the poloidal spatiotemporal vortices and toroidal spatial vortices in toroidal coordinates.Optical hopfions provide a photonic testbed for studying topological states and may be utilized as high-dimensional information carriers.展开更多
We investigate a kind of solitons in the two-component Bose-Einstein condensates with axisymmetric configurations in the R2 × S1 space. The corresponding topological structure is referred to as Hopfion. The spin ...We investigate a kind of solitons in the two-component Bose-Einstein condensates with axisymmetric configurations in the R2 × S1 space. The corresponding topological structure is referred to as Hopfion. The spin texture differs from the conventional three-dimensional （3D） skyrmion and knot, which is characterized by two homotopy invariants. The stability of the Hopfion is verified numerically by evolving the Gross-Pitaevskii equations in imaginary time.展开更多
基金the National Natural Science Foundation of China(Grant Nos.62075050,11934013,and 61975047)the High-Level Talents Project of Heilongjiang Province(Grant No.2020GSP12)the European Research Council iCOMM project(Grant No.789340).
文摘Structured light fields embody strong spatial variations of polarization,phase,and amplitude.Understanding,characterization,and exploitation of such fields can be achieved through their topological properties.Three-dimensional(3D)topological solitons,such as hopfions,are 3D localized continuous field configurations with nontrivial particle-like structures that exhibit a host of important topologically protected properties.Here,we propose and demonstrate photonic counterparts of hopfions with exact characteristics of Hopf fibration,Hopf index,and Hopf mapping from real-space vector beams to homotopic hyperspheres representing polarization states.We experimentally generate photonic hopfions with on-demand high-order Hopf indices and independently controlled topological textures,including Néel-,Bloch-,and antiskyrmionic types.We also demonstrate a robust free-space transport of photonic hopfions,thus showing the potential of hopfions for developing optical topological informatics and communications.
基金We acknowledge the support from the National Natural Science Foundation of China(NSFC)(92050202(Q.Z.),61875245(C.W.)),Shanghai Science and Technology Committee(19060502500(Q.Z.)),and Wuhan Science and Technology Bureau(2020010601012169(C.W.)).
文摘Hopfions are three-dimensional(3D)topological states discovered in field theory,magnetics,and hydrodynamics that resemble particle-like objects in physical space.Hopfions inherit the topological features of the Hopf fibration,a homotopic mapping from unit sphere in 4D space to unit sphere in 3D space.Here we design and demonstrate dynamic scalar optical hopfions in the shape of a toroidal vortex and expressed as an approximate solution to Maxwell’s equations.Equiphase lines correspond to disjoint and interlinked loops forming complete ring tori in 3D space.The Hopf invariant,product of two winding numbers,is determined by the topological charge of the poloidal spatiotemporal vortices and toroidal spatial vortices in toroidal coordinates.Optical hopfions provide a photonic testbed for studying topological states and may be utilized as high-dimensional information carriers.
基金supported by the National Natural Science Foundation of China(Grant No.11374036)the National Basic Research Program of China(Grant No.2012CB821403)
文摘We investigate a kind of solitons in the two-component Bose-Einstein condensates with axisymmetric configurations in the R2 × S1 space. The corresponding topological structure is referred to as Hopfion. The spin texture differs from the conventional three-dimensional （3D） skyrmion and knot, which is characterized by two homotopy invariants. The stability of the Hopfion is verified numerically by evolving the Gross-Pitaevskii equations in imaginary time.
