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

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

.. _sphx_glr_api_gallery_semiclassical_propagators_plot_herman_kluk_wavepacket.py:


Herman-Kluk frozen-Gaussian wavepacket propagation
========================================================

Michael Herman and Edward Kluk's 1984 method propagates a quantum
wavepacket by summing many classical trajectories rather than following
just one: launch one fixed-width ("frozen") Gaussian wavepacket from
every point of a phase-space grid under the initial state, propagate
each one along its own classical trajectory, and sum the results with a
monodromy-built prefactor and classical action phase,

.. math::

   \psi(x,t) \approx \int\!\frac{dq_0\,dp_0}{2\pi\hbar}\,
   C_t(q_0,p_0)\,e^{iS(t)/\hbar}\,
   \langle g_{q_0,p_0}|\psi_0\rangle\, g_{q_t,p_t}(x),

(:func:`~physicskit.semiclassical.core.propagators.herman_kluk_propagate_wavepacket`,
built on :func:`~physicskit.semiclassical.core.propagators.frozen_gaussian_1d`
and the monodromy-built prefactor :math:`C_t`,
:func:`~physicskit.semiclassical.core.propagators.herman_kluk_prefactor`).
Unlike the single-trajectory Van Vleck-Morette propagator
(:doc:`/api/gallery/semiclassical/propagators/plot_semiclassical_propagators`),
this multi-trajectory ("initial value representation") sum needs no
troublesome root-finding to connect fixed endpoints, and -- for a
potential at most quadratic in position, as here -- is exact.

Propagates an initial frozen-Gaussian (coherent) wavepacket centered at
:math:`(q_c,p_c)=(2.0,0.0)` for a unit-mass, unit-frequency harmonic
oscillator, :math:`V(q)=\tfrac12m\omega^2q^2` with :math:`m=\omega=1`,
and compares the propagated wavepacket at :math:`t=T/2` (half an
oscillator period) to the exact analytic coherent-state evolution.

.. GENERATED FROM PYTHON SOURCE LINES 34-78

.. code-block:: Python


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

    from physicskit.quantum.chapters.harmonic_spin import HarmonicOscillator
    from physicskit.semiclassical.core.propagators import herman_kluk_propagate_wavepacket

    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


    T_period = 2 * np.pi / omega

    ho = HarmonicOscillator()
    gamma = 1.0 / (2 * ho.x0**2)

    qc0, pc0 = 2.0, 0.0
    alpha = qc0 / (ho.x0 * np.sqrt(2)) + 1j * pc0 / np.sqrt(2 * ho.hbar * ho.m * ho.omega)

    t_hk = 0.5 * T_period
    steps_hk = 2000
    x_eval = np.linspace(-6, 6, 400)

    psi_hk = herman_kluk_propagate_wavepacket(
        qc0, pc0, gamma, dVdx, d2Vdx2, V, ho.m, dt=t_hk / steps_hk, steps=steps_hk, x_eval=x_eval, hbar=ho.hbar, n_grid=51, n_sigma=6.0, params=params
    )
    psi_exact = ho.coherent_wavefunction(alpha, x_eval, t=t_hk)








.. GENERATED FROM PYTHON SOURCE LINES 79-82

The Herman-Kluk multi-trajectory propagator vs. the exact coherent-state
evolution
------------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 82-93

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(6.5, 4.5))
    ax.plot(x_eval, np.abs(psi_exact) ** 2, label="exact coherent state |psi(t)|^2")
    ax.plot(x_eval, np.abs(psi_hk) ** 2, "--", label="Herman-Kluk |psi(t)|^2")
    ax.set_xlabel("x")
    ax.set_title("Herman-Kluk vs. exact coherent-state evolution at t=T/2")
    ax.legend(fontsize=8)
    fig.tight_layout()

    overlap = np.abs(np.trapezoid(np.conj(psi_exact) * psi_hk, x_eval)) ** 2
    print(f"Herman-Kluk fidelity to the exact coherent state at t=T/2: {overlap:.4f}")



.. image-sg:: /api/gallery/semiclassical/propagators/images/sphx_glr_plot_herman_kluk_wavepacket_001.png
   :alt: Herman-Kluk vs. exact coherent-state evolution at t=T/2
   :srcset: /api/gallery/semiclassical/propagators/images/sphx_glr_plot_herman_kluk_wavepacket_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Herman-Kluk fidelity to the exact coherent state at t=T/2: 0.9999





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

   **Total running time of the script:** (0 minutes 1.654 seconds)


.. _sphx_glr_download_api_gallery_semiclassical_propagators_plot_herman_kluk_wavepacket.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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