Gross-Pitaevskii Model of the Condensate

Abstract

The Gross-Pitaevskii equation (GPE) is a successful and well-established model for describing an atomic Bose-Einstein condensate. Here we introduce this model, along with its assumptions. Throughout the rest of this chapter we explore its properties and key time-independent solutions.

Problems

3.1

(a) Using the normalization condition, determine the dimensions of the wavefunction $\varPsi$ in S.I. units (metres, kilograms, seconds).

(b) Verify that all terms of the GPE have the same dimension.

(c) Show that $g \vert \varPsi \vert ^2$ has dimension of energy.

3.2

Consider a BEC in the Thomas-Fermi limit, confined within a three-dimensional spherical harmonic trap.

(a) Normalize the wavefunction, and hence determine an expression for the Thomas-Fermi radius $R_r$ in terms of N, $a_\mathrm{s}$ and $\ell _r$.

(b) Determine an expression for the peak density in terms of N and $R_r$.

(c) Find an expression for the ratio $R_r/\ell _r$, and comment on its behaviour for large N.

(d) What is the energy of the condensate?

3.3

Derive the expression for the variational energy of a three-dimensional trapped condensate, Eq. (3.35). Repeat in two dimensions (for a potential $V(x,y)=m\omega _r^2 (x^2+y^2)/2$) and in one dimension (for a potential $V(x)=m\omega _r^2 x^2/2$). For each case plot $E/N\hbar \omega _r$ versus the variational width $\sigma$, for some different values of the interaction parameter $N a_\mathrm{s}/\ell _r$. What effect does dimensionality have on the shape of the curves? How do this change the qualitative behaviour described in Sect. 3.5.3?

3.4

Consider a BEC in the non-interacting limit with wavefunction

$$\begin{aligned} \psi (x,y,z)= \sqrt{n_0} ~e^{-x^2/2 \ell _x^2}e^{-y^2/2 \ell _y^2}e^{-z^2/2 \ell _z^2}, \end{aligned}$$

(3.57)

where $n_0$ is the peak density and $\ell _x$, $\ell _y$ and $\ell _z$ are the harmonic oscillator lengths in three Cartesian directions. The BEC is imaged along the z-direction. Determine the form of the column-integrated density $n_\mathrm{CI}(x,y)$. Hint: $\int ^{\infty }_0 e^{ax^2}=\frac{1}{2}\sqrt{\pi /a}$.

3.5

Consider a 1D uniform static condensate with $V(x)=0$. Obtain an expression for the energy E in a length L of the condensate, in terms of $n_0$, g and L.

Now consider the condensate to be flowing with uniform speed $v_0$, by constructing a solution according to Eq. (3.43). Show that the solution satisfies the 1D GPE, and confirm that the velocity field of this solution is indeed $v(x)=v_0$. What is the corresponding energy for the flowing condensate, and how does it differ from the static result? Finally, what is its momentum?

3.6

Consider a homogeneous condensate. Identify dimensionless variables so that the dimensionless GPE is,

$$\begin{aligned} i \frac{\partial \psi '}{\partial t'}= -\nabla '^2 \psi ' + \vert \psi ' \vert ^2 \psi ' -\psi ', \end{aligned}$$

(3.58)

i.e., without the 1 / 2 factor as in Eq. (3.52).