Trapped cold atoms with resonant interactions: unitary gas and three-body problem

Near a Feshbach resonance, cold atoms are strongly interacting because the scattering length diverges. Moreover the interactions are short ranged, which generally allows to model them by a zero-range pseudopotential. The limit of infinite scattering length and zero range is called unitary limit. We solve the 3-body problem at the unitary limit in an isotropic harmonic trap. For bosonic particles, we find two types of eigenstates: universal states which only depend on the oscillation frequency of a particle in the trap, the particles' mass and Planck's constant; and efimovian states which also depend on a 3-body parameter, similarly to the 3-body bound states in free space discovered by Efimov. In an experiment, we predict that the universal states are long-lived, which is unusual for bosonic atoms. This lifetime is limited by the coupling between universal and efimovian states induced by the non-zero range of interactions. In the N-body case, we find that the hyper-radius, a collective degree of freedom describing the global size of the gas, is separable. We determine the dependence of the many-body wave functions on the hyper-radius. We deduce a relation on the thermal fluctuations of the gas size. Our results generalize to some N-body resonances. In a trap which rotates sufficiently fast, we find within superfluid hydrodynamics that the unitary gas becomes dynamically unstable. For an arbitrary trap and space dimension, we obtain new virial theorems for several relevant interactions. For the BEC-BCS crossover in a harmonic trap, we deduce that the trapping potential energy has an inflexion point at the unitary limit if the scattering length is varied adiabatically. (author)