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Numerical Analysis of Vortex Formation in Rotating Two-Dimensional Bose-Einstein Condensates

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.
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.
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.
"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."

Deeper Inquiries

How can the proposed numerical method be extended to study the dynamics of rotating Bose-Einstein condensates, such as the time evolution of vortex patterns

The proposed numerical method for studying vortex nucleation in rotating Bose-Einstein condensates can be extended to investigate the dynamics of these systems by incorporating time evolution into the computational model. This extension would involve solving the time-dependent Gross-Pitaevskii equation, which describes the evolution of the wave function of the condensate over time. By numerically solving this equation with the proposed method, one can track the formation, movement, and interactions of vortices in the condensate as it evolves. This would provide valuable insights into the temporal behavior of vortex patterns, shedding light on phenomena such as vortex shedding, vortex reconnections, and the stability of vortex configurations over time.

What are the potential applications of the observed vortex sheet structures in Bose-Einstein condensates, and how could they be exploited in practical settings

The observed vortex sheet structures in Bose-Einstein condensates have several potential applications across different fields. One application could be in the field of quantum information processing, where the controlled manipulation of vortices in condensates could be utilized for quantum computing and information storage. The vortex sheets could also be exploited in precision sensing applications, where the sensitivity of the condensate to external perturbations could be harnessed for high-precision measurements. Additionally, the unique properties of vortex sheets could find applications in creating novel materials with tailored properties, such as superconductors with enhanced critical currents or metamaterials with exotic electromagnetic responses.

Given the strong connection between Bose-Einstein condensates and superfluidity, how could the insights from this work on vortex formation be leveraged to understand the behavior of superfluid systems more broadly

The insights gained from studying vortex formation in Bose-Einstein condensates can be leveraged to enhance our understanding of superfluid systems more broadly. Superfluidity, characterized by the absence of viscosity and the ability to flow without dissipative loss of energy, is a key phenomenon observed in both Bose-Einstein condensates and superfluid helium. By studying the behavior of vortices in condensates, researchers can gain valuable insights into the dynamics of superfluid flow, vortex interactions, and the transition between different flow regimes. This knowledge can be applied to improve our understanding of superfluid helium, superconductors, and other systems exhibiting superfluid behavior, leading to advancements in areas such as energy transport, quantum technologies, and fundamental physics research.