.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "api/gallery/quantum/entanglement/plot_bloch_sphere_spin_dynamics.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_quantum_entanglement_plot_bloch_sphere_spin_dynamics.py: Bloch-sphere spin dynamics ============================= A spin-1/2 (qubit) state evolves under :math:`H=-(\omega/2)\sigma_z` (a static field along z) or :math:`H=-(\omega_R/2)\sigma_x` (a resonant drive along x). Since :math:`\sigma_z^2=\sigma_x^2=I`, the propagator has the closed form :math:`e^{-i\theta\sigma} = \cos\theta\, I - i\sin\theta\, \sigma`, used directly here instead of a general matrix exponential. A final section drives the same kind of two-level system with Rabi's own closed-form resonance formula (:class:`~physicskit.quantum.chapters.spin.RabiProblem`), animating the Bloch vector itself as it nutates under the drive. .. GENERATED FROM PYTHON SOURCE LINES 15-51 .. code-block:: Python import matplotlib.pyplot as plt import numpy as np from physicskit.quantum.chapters.spin import RabiProblem from physicskit.quantum.core.operators import sigma_x, sigma_z from physicskit.quantum.visualizers.bloch_sphere import ( animate_bloch_sphere, plot_bloch_sphere, state_to_bloch_trajectory, ) def evolve(psi0, sigma, omega, t): """psi(t) = exp(-i*(omega/2)*sigma*t) psi0, for sigma a Pauli matrix.""" theta = omega * t / 2 I = np.eye(2) states = np.array([(np.cos(th) * I - 1j * np.sin(th) * sigma) @ psi0 for th in theta]) return states t = np.linspace(0, 4 * np.pi, 300) # Larmor precession: static field along z, initial state on the equator omega0 = 1.0 psi0_larmor = np.array([1, 1]) / np.sqrt(2) # |+x>, on the equator states_larmor = evolve(psi0_larmor, sigma_z, omega0, t) traj_larmor = state_to_bloch_trajectory(states_larmor) # Rabi oscillation: resonant drive along x, initial state |0> omega_R = 1.0 psi0_rabi = np.array([1, 0]) # |0>, north pole states_rabi = evolve(psi0_rabi, sigma_x, omega_R, t) traj_rabi = state_to_bloch_trajectory(states_rabi) P1 = np.abs(states_rabi[:, 1]) ** 2 # population in |1> .. GENERATED FROM PYTHON SOURCE LINES 52-54 Interactive 3D Bloch-sphere trajectories -------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 54-61 .. code-block:: Python fig_larmor = plot_bloch_sphere(trajectory=traj_larmor) fig_larmor.update_layout(title="Larmor precession: |+x> under H=-(omega/2)sigma_z") fig_rabi = plot_bloch_sphere(trajectory=traj_rabi) fig_rabi.update_layout(title="Rabi oscillation: |0> under H=-(omega_R/2)sigma_x") .. raw:: html

.. GENERATED FROM PYTHON SOURCE LINES 62-64 Bloch-vector components and the Rabi population curve ----------------------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 64-85 .. code-block:: Python fig, axes = plt.subplots(1, 2, figsize=(11, 4.5)) axes[0].plot(t, traj_larmor[:, 0], label="x") axes[0].plot(t, traj_larmor[:, 1], label="y") axes[0].plot(t, traj_larmor[:, 2], label="z") axes[0].set_title("Larmor precession: Bloch vector components\n(x,y rotate at omega, z fixed)") axes[0].set_xlabel("t") axes[0].legend(fontsize=8) axes[1].plot(t, P1, label=r"$P(|1\rangle)$ numeric") axes[1].plot(t, np.sin(omega_R * t / 2) ** 2, "--", label=r"$\sin^2(\omega_R t/2)$ analytic") axes[1].set_title("Rabi oscillation: population flopping") axes[1].set_xlabel("t") axes[1].legend(fontsize=8) fig.tight_layout() print("Larmor |Bloch vector| (should stay 1):", np.round(np.linalg.norm(traj_larmor, axis=1)[:5], 6)) print("Rabi max population transfer:", P1.max()) .. image-sg:: /api/gallery/quantum/entanglement/images/sphx_glr_plot_bloch_sphere_spin_dynamics_001.png :alt: Larmor precession: Bloch vector components (x,y rotate at omega, z fixed), Rabi oscillation: population flopping :srcset: /api/gallery/quantum/entanglement/images/sphx_glr_plot_bloch_sphere_spin_dynamics_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none Larmor |Bloch vector| (should stay 1): [1. 1. 1. 1. 1.] Rabi max population transfer: 0.9999724009977419 .. GENERATED FROM PYTHON SOURCE LINES 86-98 Rabi's resonance method: a driven two-level system, animated ------------------------------------------------------------------ :class:`~physicskit.quantum.chapters.spin.RabiProblem` implements Rabi's closed-form rotating-wave-approximation propagator directly (rather than the ``sigma_z``/``sigma_x``-only shortcut used above), including detuning. On resonance a "pi pulse" fully inverts the population; here :func:`~physicskit.quantum.visualizers.bloch_sphere.animate_bloch_sphere` sweeps the Bloch vector itself, frame by frame with a matplotlib ``FuncAnimation``, tracing the nutation from the north pole toward the south pole as the resonant drive acts -- the same geometric picture Bloch introduced alongside his 1946 nuclear induction experiment. .. GENERATED FROM PYTHON SOURCE LINES 98-110 .. code-block:: Python rabi = RabiProblem(omega0=1.0, omega_d=1.0, Omega=0.5) # on resonance (Delta=0) t_pi = np.pi / rabi.Omega # a resonant pi-pulse fully inverts the population t_rabi_anim = np.linspace(0, t_pi, 60) states_rabi_anim = rabi.state_trajectory(t_rabi_anim) traj_rabi_anim = state_to_bloch_trajectory(states_rabi_anim) anim = animate_bloch_sphere(traj_rabi_anim, times=t_rabi_anim) # anim.save("rabi_bloch_sphere.gif", writer="pillow", fps=15) print(f"population in |1> at the pi-pulse time: {rabi.excited_state_population(np.array([t_pi]))[0]:.6f}") .. container:: sphx-glr-animation .. raw:: html

.. rst-class:: sphx-glr-script-out .. code-block:: none population in |1> at the pi-pulse time: 1.000000 .. GENERATED FROM PYTHON SOURCE LINES 111-121 The Rabi chevron: population vs. detuning and time ------------------------------------------------------- Off resonance, :meth:`~physicskit.quantum.chapters.spin.RabiProblem.excited_state_population` caps the population-transfer amplitude at :math:`\Omega^2/\Omega_R^2` and speeds the oscillation up to the generalized Rabi frequency :math:`\Omega_R=\sqrt{\Delta^2+\Omega^2}` as the drive's detuning :math:`\Delta=\omega_d-\omega_0` grows -- sweeping both time and detuning traces out the "Rabi chevron", the same 2D diagnostic pattern used to calibrate real qubit drives. .. GENERATED FROM PYTHON SOURCE LINES 121-135 .. code-block:: Python detunings = np.linspace(-2.0, 2.0, 121) t_chevron = np.linspace(0, 4 * np.pi, 200) chevron = np.array([RabiProblem(omega0=rabi.omega0, omega_d=rabi.omega0 + delta, Omega=rabi.Omega).excited_state_population(t_chevron) for delta in detunings]) fig_chevron, ax_chevron = plt.subplots(figsize=(7, 4.5)) im = ax_chevron.pcolormesh(t_chevron, detunings, chevron, shading="auto", cmap="inferno") ax_chevron.set_xlabel("t") ax_chevron.set_ylabel(r"detuning $\Delta=\omega_d-\omega_0$") ax_chevron.set_title("Rabi chevron: $P(|1\\rangle)$ vs. detuning and time") fig_chevron.colorbar(im, ax=ax_chevron, label=r"$P(|1\rangle)$") fig_chevron.tight_layout() print(f"on-resonance (Delta=0) max population: {chevron[np.argmin(np.abs(detunings))].max():.6f}") .. image-sg:: /api/gallery/quantum/entanglement/images/sphx_glr_plot_bloch_sphere_spin_dynamics_003.png :alt: Rabi chevron: $P(|1\rangle)$ vs. detuning and time :srcset: /api/gallery/quantum/entanglement/images/sphx_glr_plot_bloch_sphere_spin_dynamics_003.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none on-resonance (Delta=0) max population: 0.999938 .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 2.645 seconds) .. _sphx_glr_download_api_gallery_quantum_entanglement_plot_bloch_sphere_spin_dynamics.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_bloch_sphere_spin_dynamics.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_bloch_sphere_spin_dynamics.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_bloch_sphere_spin_dynamics.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_