Superfluid Hydrodynamics

We may transform the GPE into a hydronamic equation for a superfluid,

${\frac {\partial |\psi |^{2}}{\partial t}}+\nabla {\frac {\hbar }{2mi}}\left({\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*}}\right)\,,$

by introducing flow, from current $j$,

$v={\frac {j}{n}}={\frac {\psi ^{*}\nabla \psi -\psi \nabla \psi ^{*}}{2mi|\psi |^{2}}}\,.$

This gives the continuity equation

${\frac {\partial n}{\partial t}}+\nabla (nv)=0\,.$

Writing $\psi =fe^{i\phi }$, and noting that the gradient of the phase gives us the velocity field, we get equations of motion for $f$ and $\phi$,

$-\hbar {\frac {\partial \phi }{\partial t}}=-frac{\hbar ^{2}}{2mf}\nabla ^{2}f+{\frac {1}{2}}mv^{2}+V(r)+U_{0}f^{2}\,.$

This reduces to

${\begin{array}{rcl}m{\frac {\partial v}{\partial t}}&=&-\nabla (\delta \mu +{\frac {1}{2}}mv^{2}\\\delta \mu &=&v+U_{0}n{\frac {\hbar ^{2}}{2m{\sqrt {n}}}}\nabla ^{2}{\sqrt {n}}-\mu _{0}\,.\end{array}}$

The Thomas-Fermi approximation is now applied, neglecting $\nabla f$, but keeping $\nabla \phi$, giving

$m{\frac {\partial ^{2}\delta n}{\partial t^{2}}}=U_{0}\nabla (n_{0}\nabla (\delta n))\,,$

a wave equation for the density. For $n_{0}$ constant, $U_{0}$ is the speed of sound squared, $c={\sqrt {U_{0}/m}}$. The Thomas-Fermi solution for $n_{0}$ gives collective modes of the condensate. A droplet of condensate can have shape resonances, waves, and many other physical behaviors, captured by these solutions.