Tutorial covering the concept of geometric phase for a qubit, tutorial available under https://mareknarozniak.com/2021/01/09/qubit-berry-phase/

plotting.py

import matplotlib.pyplot as plt


def plotQubit(angles, refrel, refimag, resrel, resimag, title, x_axis, filename, url=''):
    fig, ax = plt.subplots(1, 1, constrained_layout=True)

    ax.set_title(title)
    x_axis += '\n' + url
    ax.set_xlabel(x_axis)
    ax.set_ylabel('phase')

    ax.plot(angles, refrel, color='tab:blue', label='ref real')
    ax.plot(angles, refimag, color='tab:orange', label='ref imaginary')

    ax.plot(angles, resrel, color='tab:blue', linestyle='--', label='res real')
    ax.plot(angles, resimag, color='tab:orange', linestyle='--', label='res imaginary')

    ax.grid()
    ax.legend()

    # fig.show()
    fig.savefig(filename)

qm.py

import numpy as np

from qutip import Options, mesolve, sigmax, sigmay, sigmaz


def sweepPhaseDown(psi0, theta_min, theta_max, phi, tau):
    def angle(t):
        return (1-(t/tau))*theta_max + (t/tau)*theta_min
    res = 300
    times = np.linspace(0., tau, res)
    opts = Options(store_final_state=True)
    H = [
        [sigmax(), (lambda t, args: np.cos(-phi)*np.sin(angle(t)))],
        [sigmay(), (lambda t, args: np.sin(-phi)*np.sin(angle(t)))],
        [sigmaz(), (lambda t, args: np.cos(angle(t)))]]
    return mesolve(H, psi0, times, options=opts)


def sweepPhaseUp(psi0, theta_min, theta_max, phi, tau):
    def angle(t):
        return (1-(t/tau))*theta_min + (t/tau)*theta_max
    res = 300
    times = np.linspace(0., tau, res)
    opts = Options(store_final_state=True)
    H = [
        [sigmax(), (lambda t, args: np.cos(-phi)*np.sin(angle(t)))],
        [sigmay(), (lambda t, args: np.sin(-phi)*np.sin(angle(t)))],
        [sigmaz(), (lambda t, args: np.cos(angle(t)))]]
    return mesolve(H, psi0, times, options=opts)


def sweepPhaseRound(psi0, theta, phi_min, phi_max, tau):
    def angle(t):
        return (1-(t/tau))*phi_min + (t/tau)*phi_max
    res = 300
    times = np.linspace(0., tau, res)
    opts = Options(store_final_state=True)
    H = [
        [sigmax(), (lambda t, args: np.cos(-angle(t))*np.sin(theta))],
        [sigmay(), (lambda t, args: np.sin(-angle(t))*np.sin(theta))],
        [sigmaz(), (lambda t, args: np.cos(theta))]]
    return mesolve(H, psi0, times, options=opts)

res_qubit.png

res_qubit_2_1.png

res_qubit_2_2.png

run.py

import numpy as np

from qutip import basis

from qm import sweepPhaseDown, sweepPhaseUp, sweepPhaseRound
from plotting import plotQubit


url = 'https://mareknarozniak.com/2021/01/09/qubit-berry-phase/'

tau = 16.*np.pi
res = 60
angles = np.linspace(0., 2., res)

refrels = np.zeros(res)
refimag = np.zeros(res)

resrels = np.zeros(res)
resimag = np.zeros(res)

for i, angle in enumerate(angles):
    theta = np.pi*angle

    psi0 = np.cos(theta/2.)*basis(2, 0)+np.sin(theta/2.)*basis(2, 1)
    result = sweepPhaseRound(psi0, theta, 0., 2.*np.pi, tau)
    psif = result.final_state

    gamma = -2.*np.pi*np.sin(theta/2.)**2.
    dyn = np.pi

    grefe = psif.overlap(psi0)

    resrels[i] = grefe.real
    resimag[i] = grefe.imag

    ref = np.exp(1j*gamma)

    refrels[i] = ref.real
    refimag[i] = ref.imag

title = '$\\phi_f \\in [0, 2\\pi]$ and $\\theta$ fixed'
x_axis = '$\\theta$ [$\\pi$]'
filename = 'res_qubit.png'
plotQubit(angles, refrels, refimag, resrels, resimag, title, x_axis, filename, url=url)

refrels = np.zeros(res)
refimag = np.zeros(res)

resrels = np.zeros(res)
resimag = np.zeros(res)

# fix theta_min, vary phi
for i, angle in enumerate(angles):
    theta_min = 3.*np.pi/4.
    theta_max = 0.

    phi_min = 0.
    phi_max = np.pi*angle

    psi0 = np.cos(theta_max/2.)*basis(2, 0)+np.sin(theta_max/2.)*basis(2, 1)

    result = sweepPhaseDown(psi0, theta_min, theta_max, phi_min, tau)
    psif = result.final_state

    result = sweepPhaseRound(psif, theta_min, phi_min, phi_max, tau)
    psif = result.final_state

    result = sweepPhaseUp(psif, theta_min, theta_max, phi_max, tau)
    psif = result.final_state

    gamma = -(phi_max-phi_min)*np.sin(theta_min/2.)**2.
    grefe = psif.overlap(psi0)

    ref = np.exp(1j*gamma)

    resrels[i] = grefe.real
    resimag[i] = grefe.imag

    refrels[i] = ref.real
    refimag[i] = ref.imag

title = '$\\phi_f \\in [0, 2\\pi]$ and $\\theta = \\frac{3\\pi}{4}$ fixed'
x_axis = '$\\phi_f$ [$\\pi$]'
filename = 'res_qubit_2_1.png'
plotQubit(angles, refrels, refimag, resrels, resimag, title, x_axis, filename, url=url)

refrels = np.zeros(res)
refimag = np.zeros(res)

resrels = np.zeros(res)
resimag = np.zeros(res)

# fix phi_max, vary theta_min
for i, angle in enumerate(angles):
    theta_min = np.pi*angle
    theta_max = 0.

    phi_min = 0.
    phi_max = 3.*np.pi/4.

    psi0 = np.cos(theta_max/2.)*basis(2, 0)+np.sin(theta_max/2.)*basis(2, 1)

    result = sweepPhaseDown(psi0, theta_min, theta_max, phi_min, tau)
    psif = result.final_state

    result = sweepPhaseRound(psif, theta_min, phi_min, phi_max, tau)
    psif = result.final_state

    result = sweepPhaseUp(psif, theta_min, theta_max, phi_max, tau)
    psif = result.final_state

    gamma = -(phi_max-phi_min)*np.sin(theta_min/2.)**2.
    grefe = psif.overlap(psi0)

    ref = np.exp(1j*gamma)

    resrels[i] = grefe.real
    resimag[i] = grefe.imag

    refrels[i] = ref.real
    refimag[i] = ref.imag

title = '$\\theta \\in [0, 2\\pi]$ and $\\phi_f = \\frac{3\\pi}{4}$ fixed'
x_axis = '$\\theta$ [$\\pi$]'
filename = 'res_qubit_2_2.png'
plotQubit(angles, refrels, refimag, resrels, resimag, title, x_axis, filename, url=url)