.. _sphx_glr_api_gallery_fields_quantum_fields:

Quantum fields
-----------------

Gross-Pitaevskii mean-field theory for a trapped, rotating Bose-Einstein
condensate: quantized vortex detection, a bound vortex-antivortex pair with
canceling net winding, ground-state relaxation, the critical rotation
frequency for vortex nucleation, a vortex genuinely precessing in real time,
self-focusing wave collapse, and the Casimir vacuum force.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

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    <div class="sphx-glr-thumbcontainer" tooltip="Following the first dilute-gas Bose-Einstein condensates in 1995 (Cornell, Wieman, Ketterle), experimentalists set condensates rotating and watched them nucleate quantized vortices above a critical rotation frequency, arranging themselves into a triangular Abrikosov-like lattice -- exactly as Gross-Pitaevskii theory, by then over three decades old, had predicted. Both the vortex-free and single-vortex states compared here are relaxed (at zero rotation) via imaginary-time propagation of the Gross-Pitaevskii equation for a harmonically trapped condensate,">

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  .. image:: /api/gallery/fields/quantum_fields/images/thumb/sphx_glr_plot_bec_vortex_lattice_nucleation_thumb.png
    :alt:

  :doc:`/api/gallery/fields/quantum_fields/plot_bec_vortex_lattice_nucleation`

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      <div class="sphx-glr-thumbnail-title">The critical rotation frequency for BEC vortex nucleation</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Hendrik Casimir showed in 1948 that two uncharged, perfectly conducting parallel plates in vacuum should attract each other: only electromagnetic vacuum modes that fit an integer number of wavelengths between the plates are allowed there, so the confined zero-point energy is slightly lower than in the unbounded vacuum outside, producing a net inward force. This is a simplified, standard textbook treatment -- a 1D scalar field confined between two &quot;plates&quot; a distance d apart, not a full 3D electromagnetic calculation. casimir_mode_frequencies builds the resulting discrete standing-wave spectrum,">

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  .. image:: /api/gallery/fields/quantum_fields/images/thumb/sphx_glr_plot_casimir_effect_thumb.gif
    :alt:

  :doc:`/api/gallery/fields/quantum_fields/plot_casimir_effect`

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      <div class="sphx-glr-thumbnail-title">The Casimir energy grows weaker as plates separate</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Eugene Gross and Lev Pitaevskii independently derived, in 1961, a mean-field equation for the macroscopic wavefunction of a dilute Bose-Einstein condensate: a nonlinear Schrodinger equation with a cubic self-interaction term, i\hbar\partial_t\psi = [-\tfrac{\hbar^2}{2m}\nabla^2 + V + g|\psi|^2]\psi. Decades before a real BEC existed in a laboratory, this gave theorists a concrete tool for its structure. In the natural units used here (\hbar=m=1) for a condensate held in a 2D harmonic trap V(\mathbf{r})=\tfrac12(x^2+y^2),">

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  .. image:: /api/gallery/fields/quantum_fields/images/thumb/sphx_glr_plot_gpe_ground_state_relaxation_thumb.png
    :alt:

  :doc:`/api/gallery/fields/quantum_fields/plot_gpe_ground_state_relaxation`

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      <div class="sphx-glr-thumbnail-title">Relaxing to the Gross-Pitaevskii interacting ground state</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="gpe_relax only ever finds a stationary vortex state, by imaginary-time relaxation -- it cannot show what a vortex actually does as time passes. gpe_evolve solves the same Gross-Pitaevskii equation in genuine real (lab-frame) time instead,">

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  .. image:: /api/gallery/fields/quantum_fields/images/thumb/sphx_glr_plot_gpe_real_time_vortex_precession_thumb.gif
    :alt:

  :doc:`/api/gallery/fields/quantum_fields/plot_gpe_real_time_vortex_precession`

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      <div class="sphx-glr-thumbnail-title">A vortex genuinely precessing in real time</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Vladimir Zakharov showed in &quot;Collapse of Langmuir Waves&quot; (1972) that the focusing nonlinear Schrodinger equation has a regime with no shape-preserving soliton at all: a sufficiently intense, narrow wave packet can self-focus so strongly that it contracts toward a genuine singularity in finite time, rather than settling into a stable envelope. gpe_evolve solves the 2D Gross-Pitaevskii / cubic NLS equation">

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  .. image:: /api/gallery/fields/quantum_fields/images/thumb/sphx_glr_plot_nls_self_focusing_collapse_thumb.gif
    :alt:

  :doc:`/api/gallery/fields/quantum_fields/plot_nls_self_focusing_collapse`

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      <div class="sphx-glr-thumbnail-title">Self-focusing collapse of an attractive condensate</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="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,">

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  .. image:: /api/gallery/fields/quantum_fields/images/thumb/sphx_glr_plot_superfluid_vortex_quantization_thumb.png
    :alt:

  :doc:`/api/gallery/fields/quantum_fields/plot_superfluid_vortex_quantization`

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      <div class="sphx-glr-thumbnail-title">Detecting a single quantum of circulation</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="John Kosterlitz and David Thouless explained how a two-dimensional superfluid can support order despite thermal fluctuations: below a critical temperature, vortices bind tightly into vortex-antivortex pairs of opposite circulation, whose combined velocity fields cancel at long range, leaving superfluid order intact; above it, the pairs unbind into a free plasma of independent vortices and antivortices. This package does not simulate the thermal binding-unbinding transition itself, but the two ingredients the picture is built from -- an isolated vortex and its oppositely circulating antivortex -- are both exactly representable as phase windings of a condensate wavefunction.">

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  .. image:: /api/gallery/fields/quantum_fields/images/thumb/sphx_glr_plot_vortex_antivortex_pair_thumb.png
    :alt:

  :doc:`/api/gallery/fields/quantum_fields/plot_vortex_antivortex_pair`

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      <div class="sphx-glr-thumbnail-title">A bound vortex-antivortex pair: opposite windings that cancel at long range</div>
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    </div>


.. toctree::
   :hidden:

   /api/gallery/fields/quantum_fields/plot_bec_vortex_lattice_nucleation
   /api/gallery/fields/quantum_fields/plot_casimir_effect
   /api/gallery/fields/quantum_fields/plot_gpe_ground_state_relaxation
   /api/gallery/fields/quantum_fields/plot_gpe_real_time_vortex_precession
   /api/gallery/fields/quantum_fields/plot_nls_self_focusing_collapse
   /api/gallery/fields/quantum_fields/plot_superfluid_vortex_quantization
   /api/gallery/fields/quantum_fields/plot_vortex_antivortex_pair