Multimode NOON-state generation with ultracold atoms via geodesic counterdiabatic driving

We present a protocol for the generation of NOON states with ultracold atoms, leveraging the Bose-Hubbard model in the self-trapping regime. By means of an optimized adiabatic protocol, we achieve a significant reduction in the time required for the preparation of highly entangled NOON states, involving two or more modes. Our method saturates the quantum speed limit, ensuring both efficiency and high fidelity in state preparation. A detailed analysis of the geodesic counterdiabatic driving protocol and its application to the Bose-Hubbard system highlights its ability to expand the energy gap, facilitating faster adiabatic evolution. Through perturbation theory, we derive effective parameters that emulate the counterdiabatic Hamiltonian, enabling experimentally viable implementations with constant physical parameters. This approach is demonstrated to yield exponential time savings compared with standard geodesic driving, making it a powerful tool for creating complex entangled states for applications in quantum metrology and quantum information. Our findings pave the way for scalable and precise quantum state control in ultracold atomic systems.