.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "api/gallery/chaos/quantum_chaos/plot_quantum_bakers_map.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_api_gallery_chaos_quantum_chaos_plot_quantum_bakers_map.py: Quantum Baker's Map ======================== The quantum baker's map is the quantization of physicskit.chaos's classical :class:`~physicskit.chaos.systems.maps.BakersMap`, :math:`T(x,y)=(x/\alpha,\ \alpha y)` for :math:`x<\alpha` and :math:`T(x,y)=((x-\alpha)/(1-\alpha),\ \alpha+(1-\alpha)y)` for :math:`x\ge\alpha`. It is built (following Balazs-Voros/Saraceno) as a one-period unitary Floquet operator that applies the discrete Fourier transform separately to the :math:`q < \alpha` and :math:`q \ge \alpha` halves of the (finite, ``dim``-dimensional) position-basis Hilbert space, then transforms the result back to the full position representation -- the quantum-mechanical echo of the classical map's "stretch, cut, and stack" mechanism. It is one of the simplest exactly-solvable models of quantum chaos: the underlying classical map is uniformly hyperbolic everywhere (see :doc:`/api/gallery/chaos/maps/plot_bakers_map`), with none of the mixed regular/chaotic phase space the kicked rotor has. .. GENERATED FROM PYTHON SOURCE LINES 20-28 .. code-block:: Python import matplotlib.pyplot as plt from physicskit.chaos.quantum import QuantumBakersMap from physicskit.chaos.visualizers import animate_husimi_evolution, plot_husimi, plot_quantum_spectrum qbm = QuantumBakersMap(dim=120, alpha=0.5) .. GENERATED FROM PYTHON SOURCE LINES 29-33 The Floquet spectrum ------------------------ As for the kicked rotor, one map iteration is a unitary Floquet operator whose eigenvalues live on the unit circle. .. GENERATED FROM PYTHON SOURCE LINES 33-38 .. code-block:: Python fig, ax = plot_quantum_spectrum(qbm.eigenphases()) ax.set_title(f"Quantum baker's map (dim={qbm.dim}) Floquet eigenphases") plt.show() .. image-sg:: /api/gallery/chaos/quantum_chaos/images/sphx_glr_plot_quantum_bakers_map_001.png :alt: Quantum baker's map (dim=120) Floquet eigenphases :srcset: /api/gallery/chaos/quantum_chaos/images/sphx_glr_plot_quantum_bakers_map_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 39-47 Animation: a wavepacket mixing across the unit square ------------------------------------------------------------ Start a minimum-uncertainty wavepacket in a corner of phase space and iterate the map: the same stretch/cut/stack mechanism that mixes the classical baker's map (see the classical example's stripe-thinning figure) mixes the wavepacket's Husimi distribution, which spreads across the whole unit square within just a few iterations -- there is no room for it to settle onto a regular island, because the classical map has none. .. GENERATED FROM PYTHON SOURCE LINES 47-53 .. code-block:: Python psi0 = qbm.coherent_state(q0=0.15, p0=0.15) states = qbm.evolve(psi0, n_steps=6) anim = animate_husimi_evolution(qbm, states, resolution=90, interval=350, title=f"QuantumBakersMap (dim={qbm.dim})") plt.show() .. container:: sphx-glr-animation .. raw:: html

.. GENERATED FROM PYTHON SOURCE LINES 54-58 To save the animation to a file instead of (or in addition to) displaying it interactively, use e.g.:: anim.save("quantum_bakers_map_animation.gif", writer="pillow", fps=3) .. GENERATED FROM PYTHON SOURCE LINES 60-66 A single iteration, up close --------------------------------- Before mixing takes over, a single iteration already shows the map's signature move: the coherent state's phase-space cell -- initially a small round blob -- gets stretched along ``q`` and squeezed along ``p``, then cut and stacked, in exact correspondence with the classical map's action. .. GENERATED FROM PYTHON SOURCE LINES 66-75 .. code-block:: Python fig2, axes = plt.subplots(1, 2, figsize=(12, 5.5)) q_grid, p_grid, husimi0 = qbm.husimi(states[0], resolution=100) plot_husimi(q_grid, p_grid, husimi0, ax=axes[0], title="Before (iteration 0)") q_grid, p_grid, husimi1 = qbm.husimi(states[1], resolution=100) plot_husimi(q_grid, p_grid, husimi1, ax=axes[1], title="After (iteration 1)") fig2.suptitle("Quantum baker's map: one iteration's stretch-cut-stack, in Husimi phase space") fig2.tight_layout() plt.show() .. image-sg:: /api/gallery/chaos/quantum_chaos/images/sphx_glr_plot_quantum_bakers_map_003.png :alt: Quantum baker's map: one iteration's stretch-cut-stack, in Husimi phase space, Before (iteration 0), After (iteration 1) :srcset: /api/gallery/chaos/quantum_chaos/images/sphx_glr_plot_quantum_bakers_map_003.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 7.473 seconds) .. _sphx_glr_download_api_gallery_chaos_quantum_chaos_plot_quantum_bakers_map.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_quantum_bakers_map.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_quantum_bakers_map.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_quantum_bakers_map.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_