Abstract
We consider controlled collisions between two ultracold atoms guided by external harmonic potentials. We derive analytical solutions of the Schr\odinger equation for this system and investigate the properties of eigenergies and eigenstates for different trap geometries as a function of a trap separation and of the scattering length. When varying the trap separation the energy spectrum exhibits avoided crossings, corresponding to trap-induced shape resonances. Introducing an energy-dependent scattering length we investigate the behavior of the system in the vicinity of a magnetic Feshbach resonance. Finally, we illustrate our analytical results with two examples: the quantum phase gate controlled by the external magnetic field and a scheme for a coherent transport of atoms in optical lattices into higher Bloch bands.
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