Second-order flows for computing the ground states of rotating Bose-Einstein condensates
arXiv:2205.00805 · doi:10.1016/j.jcp.2022.111872
Abstract
Second-order flows in this paper refer to some artificial evolutionary differential equations involving second-order time derivatives distinguished from gradient flows which are considered to be first-order flows. This is a popular topic due to the recent advances of inertial dynamics with damping in convex optimization. Mathematically, the ground state of a rotating Bose-Einstein condensate (BEC) can be modeled as a minimizer of the Gross-Pitaevskii energy functional with angular momentum rotational term under the normalization constraint. We introduce two types of second-order flows as energy minimization strategies for this constrained non-convex optimization problem, in order to approach the ground state. The proposed artificial dynamics are novel second-order nonlinear hyperbolic partial differential equations with dissipation. Several numerical discretization schemes are discussed, including explicit and semi-implicit methods for temporal discretization, combined with a Fourier pseudospectral method for spatial discretization. These provide us a series of efficient and robust algorithms for computing the ground states of rotating BECs. Particularly, the newly developed algorithms turn out to be superior to the state-of-the-art numerical methods based on the gradient flow. In comparison with the gradient flow type approaches: When explicit temporal discretization strategies are adopted, the proposed methods allow for larger stable time step sizes; While for semi-implicit discretization, using the same step size, a much smaller number of iterations are needed for the proposed methods to reach the stopping criterion, and every time step encounters almost the same computational complexity. Rich and detailed numerical examples are documented for verification and comparison.
arXiv comment37 pages, 11 figures
References in corpus (23)
- Theory of Bose-Einstein condensation in trapped gases
- Vortex formation in a stirred Bose-Einstein condensate
- Vortices in a Bose-Einstein Condensate
- Bose-Einstein condensation of photons in an optical microcavity
- Mathematical theory and numerical methods for Bose-Einstein condensation
- Vortices in a trapped dilute Bose-Einstein condensate
- Bosons in a Trap: A Rigorous Derivation of the Gross-Pitaevskii Energy Functional
- Phenomenological damping in trapped atomic Bose-Einstein condensates
- Two-dimensional supersolidity in a dipolar quantum gas
- Observation of Long-lived Vortex Aggregates in Rapidly Rotating Bose-Einstein Condensates
- Nonequilibrium effects of anisotropic compression applied to vortex lattices in Bose-Einstein condensates
- Ground state solution of Bose-Einstein condensate by directly minimizing the energy functional
- Efficient spectral computation of the stationary states of rotating Bose-Einstein condensates by the preconditioned nonlinear conjugate gradient method
- Optimizing Schroedinger functionals using Sobolev gradients: Applications to Quantum Mechanics and Nonlinear Optics
- Giant vortices in combined harmonic and quartic traps
- Gross-Pitaevskii Theory of the Rotating Bose Gas
- Computation of Ground States of the Gross-Pitaevskii Functional via Riemannian Optimization
- A Regularized Newton Method for Computing Ground States of Bose-Einstein condensates
- A finite element method with mesh adaptivity for computing vortex states in fast-rotating Bose-Einstein condensates
- Vortex patterns and the critical rotational frequency in rotating dipolar Bose-Einstein condensates
- A class of second-order geometric quasilinear hyperbolic PDEs and their application in imaging science
- Dynamical representations of constrained multicomponent nonlinear Schrödinger equations in arbitrary dimensions
- A numerical damped oscillator approach to constrained Schrödinger equations