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International Journal of Computer Mathematics

Efficient spectral collocation and continuation methods for symmetry-breaking solutions of rotating Bose–Einstein condensates

S.-Y. Chen, Zhenyang Li, Y.-S. Wang, C.‐S. Chien

Published Oct 9, 2012• 1 citation
Publisher

Abstract

We describe an efficient spectral collocation method (SCM) for symmetry-breaking solutions of rotating Bose–Einstein condensates (BECs) which is governed by the Gross–Pitaevskii equation (GPE). The Lagrange interpolants using the Legendre–Gauss–Lobatto points are used as the basis functions for the trial function space. Some formulas for the derivatives of the basis functions are given so that the GPE can be efficiently computed. The SCMs are incorporated in the context of a predictor–corrector continuation algorithm for tracing primary and secondary solution branches of the GPE. Symmetry-breaking solutions are numerically presented for both rotating BECs, BECs in optical lattices, and two-component BECs in optical lattices. Our numerical results show that the numerical algorithm we propose in this paper outperforms the classical orthogonal Legendre polynomials.

DOI
10.1080/00207160.2012.736615

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We describe an efficient spectral collocation method (SCM) for symmetry-breaking solutions of rotating Bose–Einstein condensates (BECs) which is governed by the Gross–Pitaevskii equation (GPE). The Lagrange interpolants using the Legendre–Gauss–Lobatto points are used as the basis functions for the trial function space. Some formulas for the derivatives of the basis functions are given so that the GPE can be efficiently computed. The SCMs are incorporated in the context of a predictor–corrector continuation algorithm for tracing primary and secondary solution branches of the GPE. Symmetry-breaking solutions are numerically presented for both rotating BECs, BECs in optical lattices, and two-component BECs in optical lattices. Our numerical results show that the numerical algorithm we propose in this paper outperforms the classical orthogonal Legendre polynomials.
We describe multi-parameter continuation methods combined with spectral collocation methods for computing numerical solutions of rotating two-component Bose–Einstein condensates (BECs), which are governed by the Gross–Pitaevskii equations (GPEs). Various types of orthogonal polynomials are used as the basis functions for the trial function space. A novel multi-parameter/multiscale continuation algorithm is proposed for computing the solutions of the governing GPEs, where the chemical potential of each component and angular velocity are treated as the continuation parameters simultaneously. The proposed algorithm can effectively compute numerical solutions with abundant physical phenomena. Numerical results on rotating two-component BECs are reported.
We study efficient spectral-collocation and continuation methods (SCCM) for rotating two-component Bose-Einstein condensates (BECs) and rotating two-component BECs in optical lattices, where the second kind Chebyshev polynomials are used as the basis functions for the trial function space. A novel two-parameter continuation algorithm is proposed for computing the ground state and first excited state solutions of the governing Gross-Pitaevskii equations (GPEs), where the classical tangent vector is split into two constraint conditions for the bordered linear systems. Numerical results on rotating two-component BECs and rotating two-component BECs in optical lattices are reported. The results on the former are consistent with the published numerical results.
We study efficient continuation methods for computing the ground state solution of quasi-2D rotating dipolar Bose-Einstein condensates (BECs). First, the highly accurate spectral collocation method is used to discretize the governing Gross-Pitaevskii equation (GPE). Then, we modify the two-level continuation scheme for 3D dipolar BECs described in Jeng et al. (2014) to develop a single-parameter continuation method for quasi-2D rotating dipolar BECs, where the chemical potential is treated as the continuation parameter. Further, by adding the ratio of dipolar interaction strength to contact interaction strength as the second continuation parameter, we propose an efficient two-parameter continuation method which can effectively show the change of the ground-state vortex structures as the dipolar interaction strength gradually increases. Moreover, we also study linear stability analysis for the GPE. Sample numerical results on quasi-2D rotating dipolar BECs are reported.
We propose a simple, efficient and accurate numerical method for simulating the dynamics of rotating Bose-Einstein condensates (BECs) in a rotational frame with/without a long-range dipole-dipole interaction. We begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with an angular momentum rotation term and/or long-range dipole-dipole interaction, 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 GPE. By introducing a rotating Lagrangian coordinate system, the original GPEs are re-formulated 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 in the discrete level. Extensive numerical results are reported to demonstrate the efficiency and accuracy of the new numerical method. Finally, 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/with the long-range dipole-dipole interaction.
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 long-range 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.