QuTiP example: Landau-Zener-Stuckelberg inteferometry¶
J.R. Johansson and P.D. Nation
For more information about QuTiP see http://qutip.org
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%matplotlib inline
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import matplotlib.pyplot as plt
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import numpy as np
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from qutip import *
from qutip.ui.progressbar import TextProgressBar as ProgressBar
Landau-Zener-Stuckelberg interferometry: Steady state of a strongly driven two-level system, using the one-period propagator.
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# set up the parameters and start calculation
delta = 1.0 * 2 * np.pi # qubit sigma_x coefficient
w = 2.0 * 2 * np.pi # driving frequency
T = 2 * np.pi / w # driving period
gamma1 = 0.00001 # relaxation rate
gamma2 = 0.005 # dephasing rate
eps_list = np.linspace(-20.0, 20.0, 101) * 2 * np.pi
A_list = np.linspace( 0.0, 20.0, 101) * 2 * np.pi
# pre-calculate the necessary operators
sx = sigmax(); sz = sigmaz(); sm = destroy(2); sn = num(2)
# collapse operators
c_op_list = [np.sqrt(gamma1) * sm, np.sqrt(gamma2) * sz] # relaxation and dephasing
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# ODE settings (for list-str format)
options = Options()
options.rhs_reuse = True
options.atol = 1e-6 # reduce accuracy to speed
options.rtol = 1e-5 # up the calculation a bit
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# for function-callback style time-dependence
def hamiltonian_t(t, args):
""" evaluate the hamiltonian at time t. """
H0 = args[0]
H1 = args[1]
w = args[2]
return H0 + H1 * np.sin(w * t)
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# perform the calculation for each combination of eps and A, store the result
# in a matrix
def calculate():
p_mat = np.zeros((len(eps_list), len(A_list)))
pbar = ProgressBar(len(eps_list))
for m, eps in enumerate(eps_list):
H0 = - delta/2.0 * sx - eps/2.0 * sz
pbar.update(m)
for n, A in enumerate(A_list):
H1 = (A/2) * sz
# function callback format
#args = (H0, H1, w); H_td = hamiltonian_t
# list-str format
#args = {'w': w}; H_td = [H0, [H1, 'sin(w * t)']]
# list-function format
args = w; H_td = [H0, [H1, lambda t, w: np.sin(w * t)]]
U = propagator(H_td, T, c_op_list, args, options)
rho_ss = propagator_steadystate(U)
p_mat[m,n] = np.real(expect(sn, rho_ss))
return p_mat
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p_mat = calculate()
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fig, ax = plt.subplots(figsize=(8, 8))
A_mat, eps_mat = np.meshgrid(A_list/(2*np.pi), eps_list/(2*np.pi))
ax.pcolor(eps_mat, A_mat, p_mat)
ax.set_xlabel(r'Bias point $\epsilon$')
ax.set_ylabel(r'Amplitude $A$')
ax.set_title("Steadystate excitation probability\n" +
r'$H = -\frac{1}{2}\Delta\sigma_x -\frac{1}{2}\epsilon\sigma_z - \frac{1}{2}A\sin(\omega t)$' + "\n");
Versions¶
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from qutip.ipynbtools import version_table
version_table()
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