The Bose-Einstein condensate is a phase of matter that arises when cooling gases of bosons to extremely low temperatures. When studying these condensates one may use the Gross-Pitaevskii equation, which is a non-linear variant of the Schrödinger equation. An interesting phenomenon that arises when rotating a Bose-Einstein condensate is the appearance of vortices. We implement a semi-implicit Euler scheme using spectral methods proposed in [1] to numerically calculate the ground state of a rotating Bose-Einstein condensate. We start with implementing a simpler iterative fixed-point method to solve the Euler scheme but show that this method fails to converge for large rotations. Because of this we implement multiple Krylov subspace solvers that in fact do converge for large rotations and show that the Preconditioned Conjugate Gradient method has better performance than the BiConjugate Gradient Stabilized method. After the implementation we briefly look at the performance of the method and improve it with simple tricks that do not compromise the accuracy or robustness and which reduce the computation time slightly. Lastly we look at the formation of vortices in 2-dimensional and 3-dimensional Bose-Einstein condensates. We show that the number of vortices increases exponentially for increasing angular velocity in 2D until the condensate breaks apart, but in 3D we ultimately find that the required computation time and RAM storage is too large to be able to analyze the vortices in a similar way on our personal computers.
The Bose-Einstein condensate is a phase of matter that arises when cooling gases of bosons to extremely low temperatures. When studying these condensates one may use the Gross-Pitaevskii equation, which is a non-linear variant of the Schrödinger equation. An interesting phenomenon that arises when rotating a Bose-Einstein condensate is the appearance of vortices. We implement a semi-implicit Euler scheme using spectral methods proposed in [1] to numerically calculate the ground state of a rotating Bose-Einstein condensate. We start with implementing a simpler iterative fixed-point method to solve the Euler scheme but show that this method fails to converge for large rotations. Because of this we implement multiple Krylov subspace solvers that in fact do converge for large rotations and show that the Preconditioned Conjugate Gradient method has better performance than the BiConjugate Gradient Stabilized method. After the implementation we briefly look at the performance of the method and improve it with simple tricks that do not compromise the accuracy or robustness and which reduce the computation time slightly. Lastly we look at the formation of vortices in 2-dimensional and 3-dimensional Bose-Einstein condensates. We show that the number of vortices increases exponentially for increasing angular velocity in 2D until the condensate breaks apart, but in 3D we ultimately find that the required computation time and RAM storage is too large to be able to analyze the vortices in a similar way on our personal computers.
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.
In this paper, we propose an efficient and accurate numerical method for computing the dynamics of rotating two-component Bose--Einstein condensates (BECs) which is described by coupled Gross--Pitaevskii equations (CGPEs) with an angular momentum rotation term and an external driving field. By introducing rotating Lagrangian coordinates, we eliminate the angular momentum rotation term from the CGPEs, which allows us to develop an efficient numerical method. Our method has spectral accuracy in all spatial dimensions and moreover it can be easily implemented in practice. To examine its performance, we compare our method with those reported in literature. Numerical results show that to achieve the same accuracy, our method needs much shorter computing time. We also applied our method to study the dynamic properties of rotating two-component BECs. Furthermore, we generalize our method to solve the vector Gross--Pitaevskii equations (VGPEs) which is used to study rotating multi-component BECs.
In this article, we propose an efficient and spectrally accurate numerical method to compute the ground states of three-dimensional (3D) rotating dipolar Bose-Einstein condensates (BEC) under strongly anisotropic trapping potentials.The kernel singularity, convolution non-locality and density anisotropy together complicate the dipolar potential evaluation. The fast rotation mechanism not only induces a complicated energy landscape with many local minima, but also creates a large number of vortices in the condensates. Such factors collectively make the ground state computation challenging in terms of convergence, accuracy and efficiency, especially for 3D anisotropic systems. Coupled with Fourier spectral discretization, we proposed a preconditioned conjugate gradient method (PCG) by integrating the anisotropic truncated kernel method (ATKM) for the dipolar potential evaluation. An adaptive step size control strategy is designed and ATKM allows for a spectral accuracy without introducing any extra anisotropy-dependent memory requirement or computational time. Our algorithm is spectrally accurate, highly efficient and memory-economic. Extensive numerical results are presented to confirm the accuracy and efficiency, together with applications to study impacts of the model parameters on critical rotational frequency, energies and chemical potential. Furthermore, these simulations reveal additional novel ground state patterns, such as bent vortices.
The aim of this paper is to present the extension of the HPC pseudo-spectral solver BEC2HPC to compute the dynamics of 2D/3D rotating Bose-Einstein condensates modeled by the Gross-Pitaevskii equations with a rotation term. Numerical examples are provided to show the efficiency of the solver for large-scale simulations. NEW VERSION PROGRAM SUMMARY Program Title: BEC2HPC CPC Library link to program files: https://doi.org/10.17632/mdzpw4dr4t.2 Licensing provisions: GPLv2 Programming language: C++, Python Journal reference of previous version: Comput. Phys. Commun. 265 (2021) 108007, https://doi.org/10.1016/j.cpc.2021.108007 Does the new version supersede the previous version?: Yes Reasons for the new version: The first version of BEC2HPC provided a stationary-state solver, while this new version also enables the computation of the dynamics of rotating Gross–Pitaevskii equations. Summary of revisions: We have implemented ADI time-splitting and relaxation pseudo-spectral schemes in BEC2HPC, following the same parallel design and domain decomposition strategy as for the stationary-state solver. Nature of problem: This extension of the software BEC2HPC computes the dynamics of rotating Bose–Einstein condensates (BEC) modeled by the Gross–Pitaevskii equation (GPE). The available numerical methods are accurate and efficient for BEC with large rotation speeds and high nonlinearities. The parallel implementation allows performing large-scale simulations of 2D or 3D problems on parallel computing platforms. Solution method: Four time discretization methods are implemented for the dynamics of the GPE: a first-, second- and a fourth-order splitting schemes, and a second-order relaxation scheme. The spatial discretization is based on a Fourier pseudo-spectral method. This solver is implemented in distributed memory using MPI and a decomposition of the computational domain. Additional comments including restrictions and unusual features: The parallel algorithms are implemented in C++ and MPI, with a Python interface available for defining the physics of the problem. A pure Python version of sequential algorithms is also provided for rapid prototyping using the same interface. Results can be exported and visualized with external tools (HDF5 and VTK formats for ParaView or PyVista).
In this paper, we study mathematically and numerically the ground states of rotating spin-orbit coupled two-component Bose-Einstein condensates modeled by the coupled Gross-Pitaevskii equations. Firstly, we rigorously prove the existence and nonexistence results for the ground states and derive some analytical properties, including the virial identity and negativity of SOC energy. Secondly, we propose an efficient and accurate preconditioned nonlinear conjugate gradient (PCG) algorithm to compute the ground states. We truncate the whole space into a bounded rectangular domain and readily apply the Fourier spectral method to approximate the wave function. The PCG method is successfully adapted with appropriate modifications to the adaptive step size control strategy for the one-parameter energy minimization problem and to the choice of preconditioners, achieving great performance in terms of accuracy and efficiency. Lastly, we carry out extensive numerical experiments to confirm the spatial spectral accuracy by traversing the most commonly-used initial guesses for each component. Moreover, we investigate the effects of local interaction, angular velocity, spin-orbit coupling and external trapping potential on the ground states, and predict some interesting physical phenomena, such as vortex lattices.