Oscillation of a Bose Einstein condensate in an harmonic trap

[SuperPhysicist]

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

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$_{s}$.

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 = $\hbar ω N \sqrt{1 + 2 N a_{s}}$

with N the total amount of particles and a$_{s}$ 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 $\frac{1}{2}$ m $\omega$² ((x+a)² + y²). The result for the energy is then:

E = $\hbarω N \sqrt{1 + 2 N a_{s}}$ + $\frac{N m}{2}$ $\omega² a²$.
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 $\omega_{e}$ = √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!

 

[SuperPhysicist]

I just realized I made a small mistake in my second question. The linear force expressions suggests that every particle undergoes a harmonic oscillation with frecuency ω, not √2 ω. I still don't know if my overall method is correct, though.

 

[DrClaude]

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.

That is correct.

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)?

I don't see why you make an analogy based on the fact that the energy varies quadraticly with $a$. I would rather argue that the GP energy acts here like a potential energy, and therefore use $F = -\nabla U$. You then indeed get $F = -N m \omega^2 a$.

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

I think that it is at this point that you should make an analogy with the harmonic oscillator, since you get a force that is linear with respect to displacement.

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.

I'm not sure you can say anything about the modification of the "wave function" at this level of approximation. I'll have to think some more about it.

(I put "wave function" in scare quotes because, although many use that vocabulary, it is actually incorrect. The $\psi$ following the Gross-Pitaevskii equation is not the wave function of the condensate, but rather the order parameter.)

 

[SuperPhysicist]

Thank you for your reply! Now I know at least that I'm on the right trail. There is one more thing that I was thinking about today that confuses me, though.

For the harmonic oscillator, the restoring force is lineair in the position x of the oscillator. Because this position is a continuously varying parameter during the oscillation, the restoring force constantly changes dependent on this position, and thus one gets the typical harmonic movement around the equilibrium. In this case however, the parameter a is a constant shift of the potential, which does not change anymore after the potential has been translated. Doesn't this mean that the force I have derived keeps a constant value and sign all the time? If so, I don't immediately see how it can be used to further describe the oscillation of the condensate, since I would say one needs a varying parameter to do this. Or am I completely missing something here?

EDIT: Or maybe I should not see a as a constant translation value, but as a variable distance between the condensate and the center of the potential?

 

[DrClaude]

EDIT: Or maybe I should not see a as a constant translation value, but as a variable distance between the condensate and the center of the potential?

This. $a$ is a general displacement, not only the initial displacement. Once it is shifted, the condensate will start moving towards the center of the trap, due to the restoring force. It is now at another displacement, will feel a slightly smaller force, and so on.

 

[SuperPhysicist]

Thanks again for your reply!
I guess it is pretty straightforward now to make the analogy with the harmonic oscillator. I have a force $F = -N m \omega^2 a$ which is linear in the displacement from the equilibrium. Since $m$ is the mass of one particle in the condensate, and $N$ is the total amount of particles, it should be correct to state $N m = M$, the total mass of the condensate. Since the force works on all of the condensate, we get then $M \ddot{a} = - M \omega^2 a$, which then leads to $a(t) = a(0) cos (\omega t)$, since the initial velocity was zero.

One issue I still have, though, is that in this treatment I neglect the fact that the condensate has a spatial extent and isn't just a point mass with mass $M$. Is there a way to include this into the theoretical description? I was thinking about the following:

The total energy I derived for the condensate was:
$E_{tot}(b)= \frac{\hbar^2 N}{2 m b^2}\left( 1 + 2Na \right) + \frac{Nm\omega^2b^2}{2}$
Here $b$ is the variational parameter I mentioned before, but it is also, correct me if I'm wrong, the width of the Gauss and thus of the condensate. So I was thinking, is it possible to make an expansion of the above expression to the second order in $b$, and is it correct to then study again how the energy varies for small changes of $b$, just as we did with $a$? This might give a description of how the width of the condensate will change when it is deviated from it's equilibrium value.