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A Simple and Efficient Numerical Method for Computing the Dynamics of Rotating Bose-Einstein Condensates Via Rotating Lagrangian Coordinates
SIAM Journal on Scientific Computing
  • Weizhu Bao
  • Daniel Marahrens
  • Qinglin Tang
  • Yanzhi Zhang, Missouri University of Science and Technology

We propose a simple, efficient, and accurate numerical method for simulating the dynamics of rotating Bose-Einstein condensates (BECs) in a rotational frame with or without longrange dipole-dipole interaction (DDI). We begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with an angular momentum rotation term and/or long-range DDI, state the two-dimensional (2D) GPE obtained from the 3D GPE via dimension reduction under anisotropic external potential, and review some dynamical laws related to the 2D and 3D GPEs. By introducing a rotating Lagrangian coordinate system, the original GPEs are reformulated to GPEs without the angular momentum rotation, which is replaced by a time-dependent potential in the new coordinate system. We then cast the conserved quantities and dynamical laws in the new rotating Lagrangian coordinates. Based on the new formulation of the GPE for rotating BECs in the rotating Lagrangian coordinates, a time-splitting spectral method is presented for computing the dynamics of rotating BECs. The new numerical method is explicit, simple to implement, unconditionally stable, and very efficient in computation. It is spectral-order accurate in space and second-order accurate in time and conserves the mass on the discrete level. We compare our method with some representative methods in the literature to demonstrate its efficiency and accuracy. In addition, the numerical method is applied to test the dynamical laws of rotating BECs such as the dynamics of condensate width, angular momentum expectation, and center of mass, and to investigate numerically the dynamics and interaction of quantized vortex lattices in rotating BECs without or with the long-range DDI.

Mathematics and Statistics
Keywords and Phrases
  • Bose-Einstein condensates,
  • Dipole dipole interactions,
  • Gross-Pitaevskii equation,
  • Lagrangian coordinate,
  • Time splitting,
  • Angular momentum,
  • Dipole moment,
  • Drug interactions,
  • Dynamics,
  • Electric dipole moments,
  • Lagrange multipliers,
  • Numerical methods,
  • Three dimensional,
  • Bose-Einstein condensation,
  • Angular momentum rotation,
  • Dipole-dipole interaction,
  • Rotating Bose-Einstein condensate,
  • Rotating Lagrangian coordinates,
  • Time-splitting
Document Type
Article - Journal
Document Version
File Type
© 2013 Society for Industrial and Applied Mathematics, All rights reserved.
Publication Date
Citation Information
Weizhu Bao, Daniel Marahrens, Qinglin Tang and Yanzhi Zhang. "A Simple and Efficient Numerical Method for Computing the Dynamics of Rotating Bose-Einstein Condensates Via Rotating Lagrangian Coordinates" SIAM Journal on Scientific Computing Vol. 35 Iss. 6 (2013) p. A2671 - A2695 ISSN: 1064-8275; 1095-7197
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