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.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/semiclassical/propagators/plot_maslov_index_caustics.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_semiclassical_propagators_plot_maslov_index_caustics.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_semiclassical_propagators_plot_maslov_index_caustics.py:


The Maslov index: caustics and Keller's quantization condition
====================================================================

The Van Vleck-Morette semiclassical propagator
(:doc:`/api/gallery/semiclassical/propagators/plot_semiclassical_propagators`)
needs a topological correction beyond its classical action and stability
prefactor: every time a classical trajectory crosses a caustic (a focal
point where :math:`\partial q_t/\partial p_0=0`), the propagator picks up
an extra phase of :math:`-\pi/2`. Joseph Keller's corrected
Bohr-Sommerfeld (EBK) quantization condition built exactly this
caustic-counting correction into the old quantum theory, and Viktor
Maslov later showed that the resulting phase count -- the Maslov index
:math:`\mu` -- is a topological invariant of the trajectory itself,
rather than an artifact of the coordinates used to compute it.

For a unit-mass, unit-frequency harmonic oscillator,
:math:`V(q)=\tfrac12m\omega^2q^2` with :math:`m=\omega=1`, a classical
trajectory crosses exactly one focal point every half-period, so
:math:`\mu(t)` should step upward by one at
:math:`t=T/2,\,T,\,3T/2,\dots` and keep doing so however many periods it
is followed for --
:func:`~physicskit.semiclassical.core.propagators.count_caustics` counts
exactly these sign changes of :math:`\partial q_t/\partial p_0` along the
trajectory
(:func:`~physicskit.semiclassical.core.propagators.propagate_trajectory_monodromy_action`).

.. GENERATED FROM PYTHON SOURCE LINES 28-74

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np
    from numba import njit

    from physicskit.semiclassical.core.propagators import (
        count_caustics,
        propagate_trajectory_monodromy_action,
    )

    m, omega = 1.0, 1.0
    params = np.array([m * omega**2])


    @njit(cache=True)
    def dVdx(q, params):
        return params[0] * q


    @njit(cache=True)
    def d2Vdx2(q, params):
        return params[0]


    @njit(cache=True)
    def V(q, params):
        return 0.5 * params[0] * q**2


    q0, p0 = 1.0, 0.3
    T_period = 2 * np.pi / omega


    def maslov_index(t):
        steps = 4000
        dt = t / steps
        _, _, _, _, Mqp_hist = propagate_trajectory_monodromy_action(q0, p0, dVdx, d2Vdx2, V, m, dt, steps, params)
        return count_caustics(Mqp_hist)


    # Over many periods, crossing several caustics: the Maslov index itself
    # needs no branch-sensitive comparison to an exact quantum formula, just a
    # count of sign changes.
    times_full = np.linspace(0.03, 1.9, 150) * T_period
    mu_history = np.array([maslov_index(t) for t in times_full])








.. GENERATED FROM PYTHON SOURCE LINES 75-78

The Maslov index counting caustic crossings once per half-period, over
many periods
--------------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 78-87

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(6.5, 4.5))
    ax.plot(times_full / T_period, mu_history, drawstyle="steps-post")
    for k in range(1, 4):
        ax.axvline(0.5 * k, color="gray", ls=":", lw=0.8)
    ax.set_xlabel("t / T")
    ax.set_ylabel(r"Maslov index $\mu$")
    ax.set_title("Caustics: one crossing per half-period")
    fig.tight_layout()



.. image-sg:: /api/gallery/semiclassical/propagators/images/sphx_glr_plot_maslov_index_caustics_001.png
   :alt: Caustics: one crossing per half-period
   :srcset: /api/gallery/semiclassical/propagators/images/sphx_glr_plot_maslov_index_caustics_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.432 seconds)


.. _sphx_glr_download_api_gallery_semiclassical_propagators_plot_maslov_index_caustics.py:

.. only:: html

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

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

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