N ov 2 00 3 Dynamics of a Bose - Einstein Condensate in an Anharmonic Trap

We present a theoretical model to describe the dynamics of Bose-Einstein condensates in anharmonic trapping potentials. To first approximation the center-of-mass motion is separated from the internal condensate dynamics and the problem is reduced to the well known scaling solutions for the Thomas-Fermi radii. We discuss the validity of this approach and analyze the model for an anharmonic waveguide geometry which was recently realized in an experiment [1]. It is well known, that for a cloud of particles that is moving in a parabolic potential the center-of-mass motion is completely decoupled from the internal excitations [2, 3]. This holds in the presence of interaction between the particles and therefore also applies to Bose-Einstein condensates. Most of the experimental work with Bose-Einstein condensates has been done in parabolic potentials. The static properties of a condensate, see e. g. Ref. [4, 5], as well as the dynamical properties [6, 7] have been investigated and were found to be in good agreement with the theoretical predictions [8, 9, 10, 11]. Higher order contributions to the trapping potential were either avoided or turned out to be negligible due to the large scale magnetic field generating elements. Anharmonic configurations were theoretically considered for a Gaussian [12] and a box-like [13] potential and were experimentally realized in the case of a magnetic mirror [14]. Current progress in generating Bose-Einstein condensates in magnetic microtraps [15, 16] in which the trapping potential in general differs from the trivial parabolic shape raises the questions how the dynamics of the trapped condensate are affected by the anharmonicity of the potential. In a recent experiment, the dynamics of a Bose-Einstein condensate in an anharmonic waveguide were studied in detail [1] and the theory in this paper has been developed to describe the experimental data. In anharmonic traps the center-of-mass motion of a condensate is coupled to the internal dynamics. In the rest frame of the condensate, the local potential curvature changes during the center-of-mass oscillation and collective modes of the condensate are excited. We present a theoretical model which allows us to describe the experimental data to high accuracy. The model is based on existing exact solutions of the Gross-Pitaevskii equation. Kagan et al. [17] and Castin and Dum [11] developed an analytical solution of the time-dependent Gross-Pitaevskii equation for anisotropic time-dependent harmonic potentials in the Thomas-Fermi approximation. The time evolution of the Thomas-Fermi radii is thereby described by three ordinary …