Core Concepts
This article presents a numerical method for minimizing the Gross-Pitaevskii energy functional modeling rotating two-component Bose-Einstein condensates in a strong confinement regime, with the ability to handle both segregation and coexistence regimes between the components. The method includes discretization of the continuous energy in 2D and a gradient algorithm with adaptive time step and projection for the minimization. The goal is to study the structures of the minimizers, which can display different patterns including vortices, vortex sheets, and giant holes.
Abstract
The article introduces a numerical method for studying the behavior of rotating two-component Bose-Einstein condensates (BECs) in a strong confinement regime. The key highlights are:
- Discretization of the Gross-Pitaevskii (GP) energy functional in 2D using the Fast Fourier Transform (FFT) scheme, which allows efficient computation of the gradient.
- Implementation of an explicit L2 gradient method with adaptive step size and projection to handle the constraints on the wave functions.
- Derivation of a stopping criterion based on the residue of the Euler-Lagrange equation corresponding to the constraints.
- Development of post-processing algorithms to compute the indices of vortices and vortex sheets in the minimizers.
- Numerical results that validate recent theoretical findings and support conjectures on the structures of minimizers, such as the existence of vortex sheets in segregation regimes.
- Comparison of the efficiency of the proposed explicit projected gradient (EPG) method with an alternative implicit method from the literature.
The article covers different regimes for one-component and two-component BECs, depending on the rotation speed and the interaction between the components (segregation vs. coexistence). The numerical results illustrate the rich variety of structures that can emerge in the minimizers, including singly quantized vortices, vortex sheets, and giant holes.
Stats
The article does not provide specific numerical data or statistics to support the key findings. The focus is on the development and validation of the numerical method.
Quotes
"The goal of this paper is to study numerically the structures of the minimizers."
"One of the interests of this method is that it allows for the derivation of a stopping criterion which we develop in the same section."
"The numerical results of the test cases in this paper will illustrate how an explicit projected gradient method together with an energy discretization allowing for the use of FFT in the computation of its gradient makes it possible to outperform (linearly) implicit methods such as that of GPELab."