.. 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_coherent_state_statistics.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_coherent_state_statistics.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_optics_quantum_optics_plot_coherent_state_statistics.py:


Glauber's coherent states: displaced vacuum, Poissonian statistics
========================================================================

Roy Glauber identified the coherent states :math:`|\alpha\rangle` --
eigenstates of the annihilation operator, :math:`\hat a|\alpha\rangle =
\alpha|\alpha\rangle` -- as the quantum states that most closely reproduce
a classical, stable-amplitude light wave: a positive Gaussian in phase
space, displaced away from the origin, whose photon number follows a
Poissonian distribution -- the quantum-mechanical signature of what an
ideal, shot-noise-limited laser beam actually is.
:func:`~physicskit.optics.quantum_optics.coherent_state` constructs this
state directly in a truncated Fock basis (dimension ``cutoff``),

.. math::

    \lvert\alpha\rangle = e^{-\lvert\alpha\rvert^2/2}
        \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}}\,\lvert n\rangle,

for use with :func:`~physicskit.optics.quantum_optics.compute_wigner_function`
and any of the package's other state-based quantum-optics tools. Below,
:math:`\alpha = 2+i` (mean photon number :math:`\langle n\rangle =
|\alpha|^2 = 5`) is used to compute both the phase-space Wigner function
:math:`W(x,p)` and the photon-number distribution :math:`P(n) =
|\langle n|\alpha\rangle|^2`.

.. GENERATED FROM PYTHON SOURCE LINES 27-33

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.optics.quantum_optics import coherent_state, compute_wigner_function








.. GENERATED FROM PYTHON SOURCE LINES 34-36

A coherent state :math:`|\alpha\rangle`, its phase-space blob and photon statistics
-------------------------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 36-50

.. code-block:: Python


    cutoff = 30
    alpha = 2.0 + 1.0j
    psi = coherent_state(alpha, cutoff)

    n = np.arange(cutoff)
    P_n = np.abs(psi) ** 2
    mean_n = np.sum(n * P_n)
    var_n = np.sum(n**2 * P_n) - mean_n**2

    x = np.linspace(-6, 6, 121)
    p = np.linspace(-6, 6, 121)
    W = compute_wigner_function(psi, x, p)








.. GENERATED FROM PYTHON SOURCE LINES 51-54

A coherent state's Wigner function is a Gaussian bump displaced from the
origin (never negative), and its photon-number distribution is
Poissonian: mean and variance both equal :math:`|\alpha|^2`.

.. GENERATED FROM PYTHON SOURCE LINES 54-71

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(9, 4))
    axes[0].contourf(x, p, W.T, levels=40, cmap="RdBu_r")
    axes[0].set_title(r"$W(x,p)$ for a displaced coherent state")
    axes[0].set_xlabel("x")
    axes[0].set_ylabel("p")
    axes[1].bar(n, P_n)
    axes[1].axvline(mean_n, color="r", ls="--", label=r"$\langle n\rangle=|\alpha|^2$")
    axes[1].set_title("Poissonian photon-number distribution")
    axes[1].set_xlabel("n")
    axes[1].legend()
    fig.tight_layout()

    print(f"alpha = {alpha}, |alpha|^2 = {abs(alpha) ** 2:.4f}")
    print(f"measured <n> = {mean_n:.4f}   measured Var(n) = {var_n:.4f}")
    print("Poissonian statistics: <n> = Var(n) = |alpha|^2, confirmed above.")
    print(f"minimum of W(x,p) over the grid: {W.min():.6f} (non-negative: classical state)")



.. image-sg:: /api/gallery/optics/quantum_optics/images/sphx_glr_plot_coherent_state_statistics_001.png
   :alt: $W(x,p)$ for a displaced coherent state, Poissonian photon-number distribution
   :srcset: /api/gallery/optics/quantum_optics/images/sphx_glr_plot_coherent_state_statistics_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    alpha = (2+1j), |alpha|^2 = 5.0000
    measured <n> = 5.0000   measured Var(n) = 5.0000
    Poissonian statistics: <n> = Var(n) = |alpha|^2, confirmed above.
    minimum of W(x,p) over the grid: -0.000000 (non-negative: classical state)





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

   **Total running time of the script:** (0 minutes 0.327 seconds)


.. _sphx_glr_download_api_gallery_optics_quantum_optics_plot_coherent_state_statistics.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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