Analytic results for the linear and nonlinear response of atoms in a trap with a model interaction

We present an exact expression for the evolution of the wave function 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/[Formula Presented] with ξ>0. For a constant trap potential ω(t)=[Formula Presented], 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 one-dimensional 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 “nonlinear” regimes. Certain features of the response spectrum are found to be insensitive to interaction strength and atom number.