.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/optics/quantum_optics/plot_photon_antibunching.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_optics_quantum_optics_plot_photon_antibunching.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_optics_quantum_optics_plot_photon_antibunching.py:


Kimble, Dagenais, and Mandel: photon antibunching and the Fano factor
==========================================================================

Kimble, Dagenais, and Mandel measured the first direct evidence that
light can arrive one photon at a time, with a vanishing probability of
detecting two photons simultaneously -- no classical wave, however dim,
can produce sub-Poissonian photon statistics. The Fano factor :math:`F =
\mathrm{Var}(\hat n)/\langle\hat n\rangle` equals 1 for a Poissonian
(coherent, classical-limit) source and 0 for a number (Fock) state -- the
fully antibunched, sub-Poissonian extreme their photon-counting statistics
approached.
:func:`~physicskit.optics.quantum_optics.fock_state` and
:func:`~physicskit.optics.quantum_optics.coherent_state` give the two
limiting photon-number distributions directly, from which the Fano factor
is computed as an ordinary first/second moment.

.. GENERATED FROM PYTHON SOURCE LINES 18-24

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.optics.quantum_optics import coherent_state, compute_wigner_function, fock_state








.. GENERATED FROM PYTHON SOURCE LINES 25-27

Fock :math:`|3\rangle` (fully antibunched) vs. a coherent state of the same mean photon number
------------------------------------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-42

.. code-block:: Python


    cutoff = 20
    n = np.arange(cutoff)


    def fano_factor(psi):
        p = np.abs(psi) ** 2
        mean_n = np.sum(n * p)
        var_n = np.sum(n**2 * p) - mean_n**2
        return var_n / mean_n, p


    F_fock, P_fock = fano_factor(fock_state(3, cutoff))
    F_coh, P_coh = fano_factor(coherent_state(np.sqrt(3.0), cutoff))








.. GENERATED FROM PYTHON SOURCE LINES 43-47

A number state has zero photon-number variance (F=0, fully antibunched:
two photons never arrive together), while a coherent state of the same
mean photon number is Poissonian (F=1) -- the classical boundary that
Kimble, Dagenais, and Mandel's measurement fell decisively below.

.. GENERATED FROM PYTHON SOURCE LINES 47-62

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(8, 3), sharey=True)
    axes[0].bar(n, P_fock)
    axes[0].set_title("Fock |3>: F=0")
    axes[0].set_xlabel("n")
    axes[1].bar(n, P_coh)
    axes[1].set_title(r"Coherent, $\langle n\rangle=3$: F=1")
    axes[1].set_xlabel("n")
    fig.tight_layout()

    print(f"Fock |3>:        Fano factor = {F_fock:.3f}  (fully antibunched, sub-Poissonian)")
    print(f"Coherent <n>=3:  Fano factor = {F_coh:.3f}  (classical Poissonian boundary)")
    print("no classical field, however attenuated, can produce F < 1: only the")
    print("discreteness of the quantized field (the Fock-state extreme) allows it.")




.. image-sg:: /api/gallery/optics/quantum_optics/images/sphx_glr_plot_photon_antibunching_001.png
   :alt: Fock |3>: F=0, Coherent, $\langle n\rangle=3$: F=1
   :srcset: /api/gallery/optics/quantum_optics/images/sphx_glr_plot_photon_antibunching_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Fock |3>:        Fano factor = 0.000  (fully antibunched, sub-Poissonian)
    Coherent <n>=3:  Fano factor = 1.000  (classical Poissonian boundary)
    no classical field, however attenuated, can produce F < 1: only the
    discreteness of the quantized field (the Fock-state extreme) allows it.




.. GENERATED FROM PYTHON SOURCE LINES 63-72

The same two states in phase space: a ring vs. a displaced blob
----------------------------------------------------------------------
A photon-number bar chart hides *where* the two states sit in phase
space. :func:`~physicskit.optics.quantum_optics.compute_wigner_function`
shows the qualitative difference directly: the fully antibunched Fock
state :math:`|3\rangle` has no well-defined phase and is rotationally symmetric --
a ring in :math:`(x,p)` -- while the coherent state of the same mean
photon number is a single Gaussian blob displaced from the origin, the
"most classical" phase-space shape available to a quantum state.

.. GENERATED FROM PYTHON SOURCE LINES 72-87

.. code-block:: Python


    x = np.linspace(-4, 4, 121)
    p = np.linspace(-4, 4, 121)
    W_fock = compute_wigner_function(fock_state(3, cutoff), x, p)
    W_coh = compute_wigner_function(coherent_state(np.sqrt(3.0), cutoff), x, p)

    fig2, axes2 = plt.subplots(1, 2, figsize=(9, 4))
    vmax = max(np.abs(W_fock).max(), np.abs(W_coh).max())
    for ax, W, title in zip(axes2, [W_fock, W_coh], ["Fock |3>: rotationally symmetric ring", r"Coherent, $\langle n\rangle=3$: displaced Gaussian blob"]):
        im = ax.contourf(x, p, W.T, levels=40, cmap="RdBu_r", vmin=-vmax, vmax=vmax)
        ax.set_title(title, fontsize=9)
        ax.set_xlabel("x")
        ax.set_ylabel("p")
        ax.set_aspect("equal")
    fig2.colorbar(im, ax=axes2, label="W(x, p)", shrink=0.8)



.. image-sg:: /api/gallery/optics/quantum_optics/images/sphx_glr_plot_photon_antibunching_002.png
   :alt: Fock |3>: rotationally symmetric ring, Coherent, $\langle n\rangle=3$: displaced Gaussian blob
   :srcset: /api/gallery/optics/quantum_optics/images/sphx_glr_plot_photon_antibunching_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.colorbar.Colorbar object at 0x34d9d0590>




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

   **Total running time of the script:** (0 minutes 0.248 seconds)


.. _sphx_glr_download_api_gallery_optics_quantum_optics_plot_photon_antibunching.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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