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## Homework Statement

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 ω. 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.*

The hint that was given was that it would be easiest to study how the total energy of the condensate depends on certain parameters, 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.

One of the earlier questions on the task also told us to make use of a variational wavefunction of the following form to describe the ground state:

ψ = [itex]\frac{\sqrt{N/π}}{b}[/itex][itex] Exp[-\frac{x^2+y^2}{2b^2}]

[/itex]

with b the variational parameter.

## Homework Equations

The most relevant equation here is probably the Gross-Pitaevskii energy functional, in this case for a harmonic potential with frequency ω and a contact potential with scattering length a[itex]_{s}[/itex].

## The Attempt at a Solution

First of all, I calculated the total GP energy with the varational wavefunction that was given, and minimized the energy to find an optimal value for b. The final result for the energy is:

E = [itex] \hbar ω N \sqrt{1 + 2 N a_{s}}[/itex]

with N the total amount of particles and a[itex]_{s}[/itex] the scattering length. I know this much is correct, because other people have the same result.

Next I thought I would have to use the GP equation or the hydrodynamical equation to describe the problem, but then the hint was given to study how the energy depends on certain parameters, and do something with the fact that, for small deviations of such a parameter, a second order expansion can be made. This will introduce a linear restoring force, so that we can make an analogy with the harmonic oscillator.

The first thing I did was recalculate the energy for the case of the shifted harmonic potential [itex]\frac{1}{2}[/itex] m [itex]\omega[/itex]² ((x+a)² + y²). The result for the energy is then:

E = [itex]\hbarω N \sqrt{1 + 2 N a_{s}}[/itex] + [itex]\frac{N m}{2}[/itex] [itex]\omega² a²[/itex].

This expression goes to second order in a (namely a²).

Now this is were I am at loss a bit. I know about vibrations and the harmonic oscillator in the classical case, but I'm not that sure at all for the oscillation of the BEC in the trap. A few questions I have:

1) Can I just state that the equilibrium of the system in function of the shifting distance

**a**is found by setting the derivative of the energy to the parameter

**a**equal to zero? This would give the result that the system has an equilibrium for a = 0, which seems logical enough, since then we have the normal ground state energy again.

2) The expression depends quadraticly on

**a**, just like the energy of the harmonic oscillator depends quadraticly on the position, so is it correct to state that the total restoring force on the system can be found by taking the derivative of the energy to

**a**(like you would take the derivative to the position of the potential energy in the classical case)? This would give a linear force F = - N m ω² a , like every particle undergoes a harmonic oscillation with a frequency [itex]\omega_{e}[/itex] = √2 ω.

3) Even if my above statemens are correct, which I doubt, I'm at loss at how I should continue the analogy with the harmonic oscillator, because, right now, I don't have that much of a theoretical description yet. Furthermore, images in the course notes show that not only the position, but also the width of the condensate can oscillate (b in this case?), but I don't know if that is also the case here.

It would be great if someone can help me with some of these questions! Thanks in advance!