Describing vortices using Gross-Pitaevskii equation

Question

A lot of the description of vortices starts by saying a vortex in a Bose-Einstein condensate can be generated by imparting an angular momentum to the container. So as I understand it is described by a Hamiltonian of the form

$$ H= (p^2/2m + V_{ext} + g\lvert\psi\rvert^2)\psi - \Omega L_z\psi $$

But then when proceeding to analyse the form of the vortex, $\psi=fe^{i\phi}$ form is used and plugged into the GP equation, without any angular momentum term (for a free vortex solution away from effects of external potentials). So I don't completely understand how GP equation is a good model for describing these vortices. A vortex solution clearly has more energy than the solution without any vortex, and if we remove the angular momentum term from the Hamiltonian, I dont see how solving GP equation with some boundary conditions can lead to vortex solutions. If such a state of the system existed, won’t the system try to minimise its energy and go to the lower energy state which does not have any vortices?

Answer 1

The Hamiltonian with the $-\Omega L_z$ is the one you would use for the thermodynamics of a system that it is in equilibrium in a rotating frame of reference. Clearly it encourages motion with a postive $L_z$ as such motion has lower energy. If $\Omega$ is large enough the ground state in the rotating frame will contain vortices.

The GP equation is perfectly good for describing vortices, however. It is just that the vortices are excited states when seen from the viewpoint of the non-rotating laboratory frame --- they cost energy to make. If you put in a vortex solution by hand (by setting $\psi= f(r)e^{i\theta}$ then you have supplied the necessary energy. The $-\Omega L_z$ with $$ L_z= \int d^3x \frac 1{2m} \psi^* (x\partial_y-y\partial_x) \psi $$
does not affect the vortex solution significantly. In particular the vortices can't just dissappear as they are protected by the topological phase-winding number.