Some exact analytic results for the linear and non-linear response of atoms in a trap with a model interaction

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We present an exact expression for the evolution of the wavefunction of N interacting atoms in an arbitrarily time-dependent, d-dimensional parabolic trap potential ω(t). The interaction potential between atoms is taken to be of the form ξ/r² with ξ>0. For a constant trap potential ω(t)=ω₀, we find an exact, infinite set of relative mode excitations. These excitations are relevant to the linear response of the system; they are universal in that their frequencies are independent of the initial state of the system (e.g. Bose-Einstein condensate), the strength ξ of the atom-atom interaction, the dimensionality d of the trap and the number of atoms N. The time evolution of the system for general ω(t) derives entirely from the solution to the corresponding classical 1D single-particle problem. An analytic expression for the frequency response of the N-atom cluster is given in terms of ω(t). We consider the important example of a sinusoidally-varying trap perturbation. Our treatment, being exact, spans the `linear' and `non-linear' regimes. Certain features of the response spectrum are found to be insensitive to interaction strength and atom number.