Effective-Scattering-Length Model of Ultracold Atomic Collisions and Feshbach Resonances in Tight Harmonic Traps

We consider the problem of cold atomic collisions in tight traps, where the absolute scattering length may be larger than the trap size. As long as the size of the trap ground state is larger than a characteristic length of the van der Waals potential, the energy eigenvalues can be computed self-consistently from the scattering phase shift for untrapped atoms. By comparing with the exact numerical eigenvalues of the trapping plus interatomic potentials, we verify that our model gives accurate eigenvalues up to milliKelvin energies for single-channel s-wave scattering of 2 3 Na atoms in an isotropic harmonic trap, even when outside the Wigner threshold regime. Our model works also for multichannel scattering, where the scattering length can be made large due to a magnetically tunable Feshbach resonance.