Gross-Pitaevskii equation in Bose Einstein condensates

[alexvas]

Hello,

I was hoping someone might be able to give a approachable explanation of the Gross-Pitaevskii equation. All the sources I've been able to find seem to concentrate on the derivation, and I don't have the physics background to follow. From what I understand, however, is that the GP equation models the ground state of a collection of bosons (all in the same, lowest energy state) by using "some mean field theory approach". My main questions are

1) What are some of the assumptions that lead to the GP equation?
2) I've read that the GP is 'a nonlinear Schrödinger equation'. I recognize the form of the Schrödinger equation in the GP, but is the extra magnitude-of-psi-squared term part of the potential or something completely different? Intuitively, where does it come from and what does it mean?
3) What are some interesting and/or physically relevant potentials for the GP equation? I guess this is a vague question, but what kinds of potentials are Bose Einstein condensates frequently subjected to? For instance, in quantum mechanics we study the particle in a box and harmonic oscillator models - are these interesting/relevant in BE condensates as well?

Thanks!

 

[DrDu]

ad 1) Only the interaction of the particles in the low density-low energy region is relevant. This interaction can be described in terms of the scattering length a.
This interaction can be approximated by a delta function like potential.
ad 2) Yes, the nonlinear psi squared term is part of the potential, namely of the particle-particle repulsion. Besides this, there is the external potential e.g. of the trap.
ad 3) The first traps corresponded to more or less harmonical external potentials.
Nowadays, very complex confining potentials can be realized, e.g. potential lattices.

 

[alexvas]

Thanks for the response. I'd appreciate some clarification, if possible:

1) Why is the interaction of particles in low-density energy regions the only important interaction? Aren't (ideally) all the bosons in the same state, so all of the interactions are occurring at the ground state (which has a high density)? Or am I misunderstanding what you mean by "density-low energy region"? Does modeling the interaction as a delta function mean that individual bosons don't interact with anything but themselves?

2) Does particle repulsion scale as psi(x)^2 because it's a function of boson density? The expression for this repulsion seems a little too simple - shouldn't it have some kind of mean-field component?

3) Potential lattices are basically just plane waves, correct? What happens to bosons in such a potential? Can this be a stationary state of the bosons?

Thanks!