Accurate Numerical Eigenstates of the Gross-Pitaevskii Equation

Abstract

We consider a bosonic gas of N bosons. Hartree-Fock approximation allows for a product wave function of single particle solutions ${\Psi (\vec {x_i})}$

$$\displaystyle \Psi (x_1, x_2, \cdots , x_N) = \prod _i^N \Psi (\vec {x}_i) $$

Using a pseudo-potential to account for the condensate self-interaction, the Hamiltonian is found to be

$$\displaystyle H=\sum _{{i=1}}^{N}\left (-{\hbar ^{2} \over 2m}{\partial ^{2} \over \partial {\mathbf {x}}_{i}^{2}}+V({\mathbf {x}}_{i})\right )+\sum _{{i< j}}{4\pi \hbar ^{2}a_{s} \over m}\delta ({\mathbf {x}}_{i}-{\mathbf {x}}_{j}), $$

In this setting m is the mass of the particles, a _s is the scattering length of the bosons and $\hbar = \frac {h}{2\pi }$. If all single particle solutions satisfy the governing equation, we arrive at

$$\displaystyle \begin{aligned} \underbrace {\left (-{\frac {\hbar ^{2}}{2m}}{\partial ^{2} \over \partial {\mathbf {x}}^{2}}+V({\mathbf {x}})+\gamma \vert \psi ({\mathbf {x}})\vert ^{2}\right )}_{H_{\text{GPE}}[\psi ]({\mathbf {x}})} \psi ({\mathbf {x}})=\mu \psi ({\mathbf {x}}),{} \end{aligned} $$

(1)

where μ is the chemical potential. Equation (1) is the non-linear Gross-Pitaevskii equation and $\gamma ={4\pi \hbar ^{2}a_{s} \over m}$. We use a spectral element method to discretise (1). To compute the eigenstates {ψ(x)} of the nonlinear Hamiltonian H _GPE, we use two different methods the first is an iterative eigenstate solver and in the second we use a constrained Newton method.