Oscillation of a Bose Einstein condensate in a harmonical trap

Question

We were asked to try to make a theoretical description of the following phenomenon:

Imagine a 2D Bose Einstein condensate in equilibrium in an harmonical trap with frequency $\omega$. Suddenly the trap is shifted over a distance a along the x-axis. The condensate is no longer in the center of the trap and will start oscillating in the trap.

First I thought about using a 2D trial wavefunction in the Gross-Pitaevski equation or the hydrodynamical equations for condensates, but then we were told that we should actually look at how the energy of the condensate depends on certain parameters (position, width,...) and do something with the fact that, for small deviations of such a parameter, a second order expansion can be made, which will introduce a restoring force.

This makes sense for classical motion, but in this case it confused me, cause I don't know if the energy that is meant here is the original potential energy of the harmonic trap or the Gross-Pitaevskii energy that is calculated with the GP energy functional. This last one, that was calculated in an earlier exercise for a variational Gaussian wave function, turned out to be $E = \hbar \omega \sqrt{1+Na_s}$ (with $a_s$ the scattering length for the interaction energy) and so it doesn't even depend on the position.

Does anyone has got any idea how I should start or approach this theoretical description?

Answer 1

From the way the question is worded, I would assume you can treat this system as a BEC wavepacket in a potential barrier, where the potential barrier is given by your harmonical trap. Since the trap is harmonic, this is in analogy to the well-known vibrational wavepackets. I would also recommend reading this paper for some more insight:

http://arxiv.org/pdf/quant-ph/9708009v1.pdf

Answer 2

As long as you consider the BEC without inter-particle interactions (because they are negligible for instance) you can simply use the Schrodinger equation.

However, if you want to take interactions into account you may want to consider to take the Thomas-Fermi approximation. When interactions are dominant in the dynamics of the system and the kinetic energy is small then this approximation works.