Detecting a single quantum of circulation

Lars Onsager (1949) and, independently, Richard Feynman (1955) proposed that frictionless superfluid helium-4 can only rotate by threading itself with discrete vortex lines, each carrying exactly one quantum of circulation,

\[\oint \mathbf{v}\cdot d\boldsymbol{\ell} = \frac{h}{m},\]

because the superfluid velocity \(\mathbf{v}=(\hbar/m)\nabla\theta\) is the gradient of the condensate’s phase \(\theta\), and a single-valued wavefunction only allows \(\theta\) to wind around a core by an integer multiple of \(2\pi\). gpe_imprint_vortex() builds exactly this winding by multiplying the wavefunction by \((x-x_0)+i(y-y_0)\), and count_vortices() detects the same quantization directly, as an exact integer phase winding summed around each grid plaquette, imprinted here on a Gaussian condensate exactly as Onsager and Feynman’s argument requires.

import matplotlib.pyplot as plt
import numpy as np

from physicskit.fields import count_vortices, gpe_imprint_vortex, harmonic_trap_grid, plot_bec_density, plot_bec_phase

A Gaussian condensate with a single vortex imprinted at the center

n, length = 48, 10.0
X, Y, KX, KY, K2 = harmonic_trap_grid(n, length)
psi0 = np.exp(-0.5 * (X**2 + Y**2)).astype(complex)
psi = gpe_imprint_vortex(psi0, X, Y, [(0.0, 0.0)])

Sum the phase winding around every plaquette of the grid

winding = count_vortices(psi)

Exactly one quantum of circulation, \(\oint\mathbf{v}\cdot d\boldsymbol{\ell} = h/m\), is detected – the discrete vortex line Onsager and Feynman predicted, decades before it was observed directly.

fig, ax = plot_bec_phase(X, Y, psi)
ax.set_title(f"total winding detected: {np.sum(np.abs(winding))}")
fig.tight_layout()

print(f"total quantized circulation detected: {np.sum(np.abs(winding))} (expected: 1)")
print(f"density at the vortex core: |psi(0,0)|^2 = {np.abs(psi[n // 2, n // 2]) ** 2:.2e} (expected: ~0)")

total quantized circulation detected: 1 (expected: 1)
density at the vortex core: |psi(0,0)|^2 = 0.00e+00 (expected: ~0)

The two complementary views of the same vortex core

The phase plot above shows the \(2\pi\) winding; the density, plotted alongside it, shows the same vortex core the complementary way – as the zero that a single-valued wavefunction’s phase singularity necessarily forces the amplitude down to (see plot_bec_density()).

fig2, (ax_density, ax_phase) = plt.subplots(1, 2, figsize=(10, 4))
plot_bec_density(X, Y, psi, ax=ax_density)
ax_density.set_title(f"density: core |psi|^2 = {np.abs(psi[n // 2, n // 2]) ** 2:.1e}")
plot_bec_phase(X, Y, psi, ax=ax_phase)
ax_phase.set_title(f"phase: winding = {np.sum(np.abs(winding))}")
fig2.tight_layout()