# QuTiP example: Calculate the quasi-steadystate of a time-dependent (period) quantum system¶

J.R. Johansson and P.D. Nation

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)

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');


## Software version:¶

In [10]:
from qutip.ipynbtools import version_table

version_table()

Out[10]:
SoftwareVersion
QuTiP4.3.0.dev0+6e5b1d43
Numpy1.13.1
SciPy0.19.1
matplotlib2.0.2
Cython0.25.2
Number of CPUs2
BLAS InfoINTEL MKL
IPython6.1.0
Python3.6.2 |Anaconda custom (x86_64)| (default, Jul 20 2017, 13:14:59) [GCC 4.2.1 Compatible Apple LLVM 6.0 (clang-600.0.57)]
OSposix [darwin]
Thu Jul 20 22:25:24 2017 MDT