.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fields/quantum_fields/plot_superfluid_vortex_quantization.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_fields_quantum_fields_plot_superfluid_vortex_quantization.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_fields_quantum_fields_plot_superfluid_vortex_quantization.py:


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,

.. math::

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

because the superfluid velocity :math:`\mathbf{v}=(\hbar/m)\nabla\theta`
is the gradient of the condensate's phase :math:`\theta`, and a
single-valued wavefunction only allows :math:`\theta` to wind around a
core by an integer multiple of :math:`2\pi`.
:func:`~physicskit.fields.quantum_fields.gpe_imprint_vortex` builds
exactly this winding by multiplying the wavefunction by
:math:`(x-x_0)+i(y-y_0)`, and
:func:`~physicskit.fields.quantum_fields.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.

.. GENERATED FROM PYTHON SOURCE LINES 26-32

.. code-block:: Python


    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








.. GENERATED FROM PYTHON SOURCE LINES 33-35

A Gaussian condensate with a single vortex imprinted at the center
--------------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 35-41

.. code-block:: Python


    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)])








.. GENERATED FROM PYTHON SOURCE LINES 42-44

Sum the phase winding around every plaquette of the grid
-----------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 44-47

.. code-block:: Python


    winding = count_vortices(psi)








.. GENERATED FROM PYTHON SOURCE LINES 48-51

Exactly one quantum of circulation, :math:`\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.

.. GENERATED FROM PYTHON SOURCE LINES 51-59

.. code-block:: Python


    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)")




.. image-sg:: /api/gallery/fields/quantum_fields/images/sphx_glr_plot_superfluid_vortex_quantization_001.png
   :alt: total winding detected: 1
   :srcset: /api/gallery/fields/quantum_fields/images/sphx_glr_plot_superfluid_vortex_quantization_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

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




.. GENERATED FROM PYTHON SOURCE LINES 60-67

The two complementary views of the same vortex core
------------------------------------------------------
The phase plot above shows the :math:`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
:func:`~physicskit.fields.visualizers.plot_bec_density`).

.. GENERATED FROM PYTHON SOURCE LINES 67-74

.. code-block:: Python


    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()



.. image-sg:: /api/gallery/fields/quantum_fields/images/sphx_glr_plot_superfluid_vortex_quantization_002.png
   :alt: density: core |psi|^2 = 0.0e+00, phase: winding = 1
   :srcset: /api/gallery/fields/quantum_fields/images/sphx_glr_plot_superfluid_vortex_quantization_002.png
   :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.101 seconds)


.. _sphx_glr_download_api_gallery_fields_quantum_fields_plot_superfluid_vortex_quantization.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_superfluid_vortex_quantization.ipynb <plot_superfluid_vortex_quantization.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_superfluid_vortex_quantization.py <plot_superfluid_vortex_quantization.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_superfluid_vortex_quantization.zip <plot_superfluid_vortex_quantization.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_