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

TL;DR

This paper derives exact solutions for the wavefunction evolution of interacting atoms in a time-dependent trap, revealing universal excitation modes and response characteristics independent of initial state, interaction strength, and atom number.

Contribution

It provides an exact analytical framework for the linear and non-linear response of atoms in a trap with a specific interaction potential, applicable to arbitrary time-dependent trap frequencies.

Findings

Identifies universal excitation frequencies independent of initial conditions and interaction strength.

Derives an explicit formula for the frequency response to trap perturbations.

Shows certain response features are insensitive to interaction parameters and atom number.

Abstract

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 $\omega(t)$. The interaction potential between atoms is taken to be of the form $\xi/r^2$ with $\xi>0$. For a constant trap potential $\omega(t)=\omega_0$, 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 $\xi$ 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 $\omega(t)$ 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 $\omega(t)$. 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.