Generalized Riemannian Gradient Methods for Spinor Bose-Einstein Condensates

• Applicant: Professor Dr. Patrick Henning
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 551527112

Bose-Einstein condensates (BECs) can be seen as super particles that provide the intriguing possibility to explore the hidden world of quantum mechanics on observable scales. Of particular interest are BECs with a spin degree of freedom (pseudo-spinor and spinor BECs) which allow to study phenomena such as quantum magnetism. Due to the high complexity of the experimental physical setups, numerical simulations of BECs are crucial for investigating their fascinating properties. In this project we consider the computation of spinor ground states. This involves the minimization of an energy functional on a Riemannian manifold. The manifold incorporates constraints for the mass and the magnetization of the BEC. To tackle this problem numerically, we will develop and apply suitable Riemannian gradient methods which are based on an adaptive choice of the metric on the tangent space. By incorporating information about the energy into the metric, global energy dissipation shall be ensured. One of the major objectives of this project is a rigorous error analysis of the resulting methods. In particular, we want to prove global convergence and we want to develop new technqiues that allow us to quantify and understand their local convergence behavior. In that vein, we also want to uncover mechanisms that can hinder or amplify the convergence of these generalized Riemannian gradient methods. As a corner stone of our convergence analysis we will develop and exploit connections to nonlinear eigenvalue problems and their solution by generalized inverse iterations with Rayleigh shifts. With this, we bridge different fields of numerics both for the design of new methods and their error analysis. Finally, in order to accelerate the convergence in a neighborhood of the ground state, we will also investigate and analyze extensions of our approach to energy-adaptive conjugate gradients, as well as modified Riemannian Newton methods. In essence, the ultimate goal of this project is the design of novel iterative methods for an efficient computation of ground states of spinor BECs and the development of a corresponding error analysis. With this, we do not only want to improve the performance compared to existing methods for this problem class, but we especially want to provide the first rigorous analytical convergence results in this setting.

• DFG Programme: Research Grants