J.R. Johansson and P.D. Nation
For more information about QuTiP see http://qutip.org
Find the steady state of a driven qubit, by finding the eigenstates of the propagator for one driving period
In [1]:
%matplotlib inline
In [2]:
import matplotlib.pyplot as plt
In [3]:
import numpy as np
In [4]:
from qutip import *
In [5]:
def hamiltonian_t(t, args):
#
# evaluate the hamiltonian at time t.
#
H0 = args['H0']
H1 = args['H1']
w = args['w']
return H0 + H1 * np.sin(w * t)
In [6]:
def sd_qubit_integrate(delta, eps0, A, w, gamma1, gamma2, psi0, tlist):
# Hamiltonian
sx = sigmax()
sz = sigmaz()
sm = destroy(2)
H0 = - delta/2.0 * sx - eps0/2.0 * sz
H1 = - A * sx
H_args = {'H0': H0, 'H1': H1, 'w': w}
# collapse operators
c_op_list = []
n_th = 0.5 # zero temperature
# relaxation
rate = gamma1 * (1 + n_th)
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * sm)
# excitation
rate = gamma1 * n_th
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * sm.dag())
# dephasing
rate = gamma2
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * sz)
# evolve and calculate expectation values
output = mesolve(hamiltonian_t, psi0, tlist, c_op_list, [sm.dag() * sm], H_args)
T = 2 * np.pi / w
U = propagator(hamiltonian_t, T, c_op_list, H_args)
rho_ss = propagator_steadystate(U)
return output.expect[0], expect(sm.dag() * sm, rho_ss) * np.ones(shape(tlist))
In [7]:
delta = 0.3 * 2 * np.pi # qubit sigma_x coefficient
eps0 = 1.0 * 2 * np.pi # qubit sigma_z coefficient
A = 0.05 * 2 * np.pi # driving amplitude (sigma_x coupled)
w = 1.0 * 2 * np.pi # driving frequency
gamma1 = 0.15 # relaxation rate
gamma2 = 0.05 # dephasing rate
# intial state
psi0 = basis(2,0)
tlist = np.linspace(0,50,500)
In [8]:
p_ex, p_ex_ss = sd_qubit_integrate(delta, eps0, A, w, gamma1, gamma2, psi0, tlist)
In [9]:
fig, ax = plt.subplots(figsize=(12,6))
ax.plot(tlist, np.real(p_ex))
ax.plot(tlist, np.real(p_ex_ss))
ax.set_xlabel('Time')
ax.set_ylabel('P_ex')
ax.set_ylim(0,1)
ax.set_title('Excitation probabilty of qubit');
In [10]:
from qutip.ipynbtools import version_table
version_table()
Out[10]: