Simulation and Numerical Analysis of Multicomponent Bose-Einstein Condensates

• Applicants: Professor Dr. Daniel Peterseim; Professorin Dr. Tatjana Stykel
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 564828373

Multicomponent Bose-Einstein condensates (BECs) represent a frontier research area in atomic and condensed matter physics. Given the challenges of observing these systems experimentally, numerical simulation plays a crucial role in studying their behaviour. However, the multiple interacting species and spin degrees of freedom in multicomponent BECs pose fundamental theoretical and computational challenges. Existing numerical methods, designed primarily for single-component systems, cannot fully capture the complexity of multicomponent condensates, in particular the intricate interplay of spin interactions, rotational dynamics and strong nonlinearities. Furthermore, these methods lack the efficiency required for large-scale simulations and the theoretical guarantees needed to ensure convergence and stability in extreme parameter regimes. This proposal aims to fill this gap by developing and rigorously analysing advanced numerical methods for the coupled Gross-Pitaevskii equations governing multicomponent BECs. First, we will introduce novel Riemannian optimisation methods for ground state computations, formulating the problem on infinite-dimensional manifolds equipped with energy-adaptive metrics to ensure reliable convergence and to prevent significant dependence on the type and resolution of the spatial discretisation. Second, we will design structure-preserving spatial discretisation schemes that preserve key physical properties such as positivity, uniqueness and stability, while eliminating non-physical artefacts. Finally, we will develop conservative time integration algorithms that preserve crucial invariants such as energy, mass and magnetisation, allowing accurate simulation of nonlinear dynamical phenomena. Altogether, by providing a robust computational framework tailored to the complex physics of multicomponent BECs, this project will enable the numerical exploration of ground states of multicomponent BECs and their dynamics, focussing on challenging effects of spinor interactions and rotating frames. Beyond the specific application, the mathematical and algorithmic innovations in Riemannian optimisation and structure-preserving numerical methods are expected to impact broader classes of nonlinear Schrödinger equations and associated nonlinear eigenvalue problems.

• DFG Programme: Research Grants