Undergrad Quantized vortices in superfluid

[korea_mania]

I am doing a final year project on vortex interactions and have searched for several research articles about quantum hydrodynamics. Most said that ''Any rotational motion of a superfluid is sustained only by quantized vortices.'' Is this something provable from the Gross-Pitaevskii Equation or the hydrodynamic equations? It seems that most texts are assuming this without explanations.

 

[SpinFlop]

Begin by writing the density of your superfluid condensate in terms of its normalized macroscopic wave function

$$n_{0}(\mathbf r) = \left|\psi_{0}(\mathbf r)\right|^{2}$$

In general the wave function is complex, so we can write

$$\psi_{0}(\mathbf r) = \sqrt{n_{0}(\mathbf r)}\exp^{i\theta(\mathbf r)}$$

This is all you need to start with in order to get your desired results. With this in hand first compute the current density using the standard prescription from quantum mechanics:

$$\mathbf j_{0}(\mathbf r) = \frac{\hbar}{2mi}\left[\psi_{0}^{*}(\mathbf r)\nabla\psi_{0}(\mathbf r) - \psi_{0}(\mathbf r)\nabla\psi_{0}^{*}(\mathbf r) \right]$$

You should get the result

$$\mathbf v_{s}(\mathbf r) = \frac{\hbar}{m}\nabla\theta(\mathbf r)$$

where $\mathbf v_{s}(\mathbf r)$ is the velocity of the superfluid and is given by $\mathbf j_{0}(\mathbf r) = n_{0}(\mathbf r)\mathbf v_{s}(\mathbf r)$.

An important takeaway point here is that superflow only takes place when the phase $\theta(\mathbf r)$ varies in space. As well, since the curl of a gradient is always zero we immediately see that the flow is also irrotational, ie:

$$\nabla\times\mathbf v_{s}(\mathbf r) = 0$$

Of course, if you had a closed tube then you can still get a finite circulation around it, which we can define as:

$$\kappa = \oint\mathbf v_{s}(\mathbf r)\cdot d\mathbf r = \frac{h}{m}\delta\theta$$

Where the last result follow immediately from the fundamental theorem of vector calculus and $\delta\theta$ is just the change in phase angle going around the tube. However, for the macroscopic wave function to be uniquely defined we must have that $\delta\theta = 2\pi n$ where n is the number of times the phase winds through $2\pi$ around the closed path, the so called topological winding number. So now we see that the transfer of angular momentum into a superfluid is quantized. Indeed, if you were to rotate the normal state fluid in the tube and then cool it into the superfluid phase, then you would see that the circulation of the superfluid would not increase continuously, but jump in steps of h/m known as phase slip events.

Now suppose you have a cup (cylindrical container) instead of a closed tube and you rotated the normal state fluid and cool it down. It turns out that you can still get circulation in the superfluid. To see this, note that in cylindrical coordinates the circular flow is given by $v_{\phi}$ and in order for this to satisfy $\nabla\times\mathbf v_{s}(\mathbf r) = 0$ we must have that

$$\frac{1}{r}\frac{\partial}{\partial r}(rv_{\phi}) = 0$$

Solving this we get that the vortex result

$$\mathbf v_{s}(r) = \frac{\kappa}{2\pi r}\hat\phi$$

where, from before, our flow quantization guarantees $\kappa = n(h/m)$. Since $n = \pm 1$ corresponds to the lowest energy state, these are what we observe in practice.