non linear systems - Gross-Pitaevskii equation in Bose-Einstein condensates

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 Schrodinger equation'. I recognize the form of the Schrodinger 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?

Answer

1) Some of the assumptions of the Gross-Pitaevskii equation (GPE) are:

1. all atoms are in the same condensate wave function,


2. the condensate is at $T=0$,


3. collisions between atoms are sufficiently low energy that the interactions can be well described by the $s$-wave scattering length, so that the interaction can be written $g\delta(\mathbf{x}_i-\mathbf{x}_j)$.

Generalized GPEs can also be solved, allowing for thermal and quantum depletion (some atoms not in the condensate) and allowing for other forms of interaction, such as dipolar.

2) The interaction term, $g|\Psi(\mathbf{x})|^2$, is in addition to the external potential $V_\mathrm{ext}(\mathbf{x})$, the effective potential is the sum of both: $V_\mathrm{ext}(\mathbf{x})+g|\Psi(\mathbf{x})|^2$. The condensate density is $n_0(\mathbf{x})=|\Psi(\mathbf{x})|^2$, so the interaction term is $gn_0(\mathbf{x})$ which is the potential due to interaction with the condensate itself.

More detail in response to the OP's comment:

The interaction potential between two atoms can usually be written as $V(\mathbf{r}_{ij})$ where $\mathbf{r}_{ij} =\mathbf{x}_i-\mathbf{x}_j$. For neutral atoms without a significant magnetic dipole moment, the dominant interaction is van der Waals so $V(\mathbf{r}_{ij})\propto r_{ij}^{-6}$.

When considering the scattering between two atoms, we can do a partial wave expansion (matching incoming and outgoing wave functions and expanding in terms of Legendre polynomials, e.g. "Quantum Mechanics", Ch. 17, Landau and Lifshitz). For slow particles with van der Waals interaction, the $s$-wave term is dominant and the interaction can be simplified to $V(\mathbf{r}_{ij}) = g \delta(\mathbf{r}_{ij})$ where $g=4\pi\hbar^2 a_s/m$ and $a_s$ is the $s$-wave scattering length. To get a feel for the scattering length, in the the $s$-wave approximation, the cross section is $\sigma=4\pi a_s^2$, so $a_s$ is a length scale for the interaction.

The interaction potential in the GPE can be written $$\int d\mathbf{x'} V(\mathbf{x}'-\mathbf{x})|\Psi(\mathbf{x'})|^2$$ When $V(\mathbf{x}'-\mathbf{x})=g\delta(\mathbf{x}'-\mathbf{x})$, this simplifies to $$\int d\mathbf{x'} g\delta(\mathbf{x}'-\mathbf{x})|\Psi(\mathbf{x'})|^2 = g|\Psi(\mathbf{x})|^2$$

3) The external potential $V_\mathrm{ext}(\mathbf{x})$ is generally due to applied optical or magnetic fields, and is often approximately a harmonic oscillator. The oscillator strength may be very strong in some directions creating quasi one or two dimensional confinement. A particle in a box is not possible yet (the atoms would interact with the "walls"), but the external potential may be locally approximately uniform near the center of the trap. Lattice potentials are also common, where (in addition to harmonic confinement) the atoms are trapped in a standing wave created by counterpropogating lasers resulting in a periodic potential. Many other shapes are possible, such as toroids.

A good reference is the book "Bose-Einstein Condensation in Dilute Gases" by Pethick and Smith. This slightly dated review is also good (free arXiv version here): section III is relevant to your question 2.