Lellouch-Lüscher relation for ultracold few-atom systems under confinement

We derive an analog of the Lellouch-Lüscher (LL) relation for few-body bosonic systems, linking few-body scattering loss rates to the energies and widths of the corresponding harmonically trapped few-body states. Three-body numerical simulations show that the LL relation applies across a broad range of interaction strengths and energies and allows the determination of scattering rates within a single partial wave. Our work establishes a robust theoretical framework for understanding the role of the finite volume effect in few-body observables in optical lattice and tweezer experiments, enabling precise determination of multi-body scattering rates.