中国物理B ›› 2012, Vol. 21 ›› Issue (9): 90304-090304.doi: 10.1088/1674-1056/21/9/090304

• GENERAL • 上一篇    下一篇

Topological aspect of vortex lines in two-dimensional Gross–Pitaevskii theory

赵力a, 杨捷a, 谢群英a b, 田苗c   

  1. a Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, China;
    b School of Information Science and Engineering, Lanzhou University, Lanzhou 730000, China;
    c School of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University, Lanzhou 730000, China
  • 收稿日期:2011-08-25 修回日期:2012-04-14 出版日期:2012-08-01 发布日期:2012-08-01
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant Nos. 10905026 and 10905027) and the Program of Science and Technology Development of Lanzhou, China (Grant No. 2010-1-129).

Topological aspect of vortex lines in two-dimensional Gross–Pitaevskii theory

Zhao Li (赵力)a, Yang Jie (杨捷)a, Xie Qun-Ying (谢群英)a b, Tian Miao (田苗)c   

  1. a Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, China;
    b School of Information Science and Engineering, Lanzhou University, Lanzhou 730000, China;
    c School of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University, Lanzhou 730000, China
  • Received:2011-08-25 Revised:2012-04-14 Online:2012-08-01 Published:2012-08-01
  • Contact: Zhao Li E-mail:lizhao@lzu.edu.cn
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant Nos. 10905026 and 10905027) and the Program of Science and Technology Development of Lanzhou, China (Grant No. 2010-1-129).

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摘要： Using the φ-mapping topological theory, we study the topological structure of vortex lines in a two-dimensional generalized Gross-Pitaevskii theory in (3+1)-dimensional space-time. We obtain the reduced dynamic equation in the framework of two-dimensional Gross-Pitaevskii theory, from which a conserved dynamic quantity is derived on the stable vortex lines. Such equations can also be used to discuss Bose-Einstein condensates in heterogeneous and highly nonlinear systems. We obtain an exact dynamic equation with a topological term, which is ignored in traditional hydrodynamic equations. The explicit expression of vorticity as a function of the order parameter is derived, where the δ function indicates that the vortices can only be generated from the zero points of Φ and are quantized in terms of the Hopf indices and Brouwer degrees. The φ-mapping topological current theory also provides a reasonable way to study the bifurcation theory of vortex lines in the two-dimensional Gross-Pitaevskii theory.

关键词: Gross-Pitaevskii equation, Bose-Einstein condensate, vortex line, bifurcation theory

Abstract: Using the φ-mapping topological theory, we study the topological structure of vortex lines in a two-dimensional generalized Gross-Pitaevskii theory in (3+1)-dimensional space-time. We obtain the reduced dynamic equation in the framework of two-dimensional Gross-Pitaevskii theory, from which a conserved dynamic quantity is derived on the stable vortex lines. Such equations can also be used to discuss Bose-Einstein condensates in heterogeneous and highly nonlinear systems. We obtain an exact dynamic equation with a topological term, which is ignored in traditional hydrodynamic equations. The explicit expression of vorticity as a function of the order parameter is derived, where the δ function indicates that the vortices can only be generated from the zero points of Φ and are quantized in terms of the Hopf indices and Brouwer degrees. The φ-mapping topological current theory also provides a reasonable way to study the bifurcation theory of vortex lines in the two-dimensional Gross-Pitaevskii theory.

Key words: Gross-Pitaevskii equation, Bose-Einstein condensate, vortex line, bifurcation theory

中图分类号:  (Tunneling, Josephson effect, Bose-Einstein condensates in periodic potentials, solitons, vortices, and topological excitations)

  • 03.75.Lm
47.32.C- (Vortex dynamics) 02.40.Pc (General topology)