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
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
from qutip import *
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)
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))
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)
p_ex, p_ex_ss = sd_qubit_integrate(delta, eps0, A, w, gamma1, gamma2, psi0, tlist)
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');
from qutip.ipynbtools import version_table
version_table()
Software | Version |
---|---|
QuTiP | 4.3.0.dev0+6e5b1d43 |
Numpy | 1.13.1 |
SciPy | 0.19.1 |
matplotlib | 2.0.2 |
Cython | 0.25.2 |
Number of CPUs | 2 |
BLAS Info | INTEL MKL |
IPython | 6.1.0 |
Python | 3.6.2 |Anaconda custom (x86_64)| (default, Jul 20 2017, 13:14:59) [GCC 4.2.1 Compatible Apple LLVM 6.0 (clang-600.0.57)] |
OS | posix [darwin] |
Thu Jul 20 22:25:24 2017 MDT |