J.R. Johansson and P.D. Nation
For more information about QuTiP see http://qutip.org
In [1]:
%pylab inline
In [2]:
from qutip import *
In [3]:
def qubit_integrate(w, theta, gamma1, gamma2, psi0, tlist):
# Hamiltonian
sx = sigmax()
sy = sigmay()
sz = sigmaz()
sm = sigmam()
H = w * (cos(theta) * sz + sin(theta) * sx)
# Lindblad master equation
c_op_list = []
n_th = 0.0 # zero temperature
rate = gamma1 * (n_th + 1)
if rate > 0.0:
c_op_list.append(sqrt(rate) * sm)
rate = gamma1 * n_th
if rate > 0.0:
c_op_list.append(sqrt(rate) * sm.dag())
lme_results = mesolve(H, psi0, tlist, c_op_list, [sx, sy, sz]).expect
# Bloch-Redfield tensor
#ohmic_spectrum = lambda w: gamma1 * w / (2*pi)**2 * (w > 0.0)
def ohmic_spectrum(w):
if w == 0.0:
# dephasing inducing noise
return gamma1/2
else:
# relaxation inducing noise
return gamma1/2 * w / (2*pi) * (w > 0.0)
brme_results = brmesolve(H, psi0, tlist, [sx], [sx, sy, sz], [ohmic_spectrum]).expect
# alternative:
#R, ekets = bloch_redfield_tensor(H, [sx], [ohmic_spectrum])
#brme_results = bloch_redfield_solve(R, ekets, psi0, tlist, [sx, sy, sz])
return lme_results, brme_results
In [4]:
w = 1.0 * 2 * pi # qubit angular frequency
theta = 0.05 * pi # qubit angle from sigma_z axis (toward sigma_x axis)
gamma1 = 0.5 # qubit relaxation rate
gamma2 = 0.0 # qubit dephasing rate
# initial state
a = 0.8
psi0 = (a* basis(2,0) + (1-a)*basis(2,1))/(sqrt(a**2 + (1-a)**2))
tlist = linspace(0,15,5000)
In [5]:
lme_results, brme_results = qubit_integrate(w, theta, gamma1, gamma2, psi0, tlist)
In [6]:
fig = figure(figsize=(12,12))
ax = fig.add_subplot(2,2,1)
title('Lindblad master equation')
ax.plot(tlist, lme_results[0], 'r')
ax.plot(tlist, lme_results[1], 'g')
ax.plot(tlist, lme_results[2], 'b')
ax.legend(("sx", "sy", "sz"))
ax = fig.add_subplot(2,2,2)
title('Bloch-Redfield master equation')
ax.plot(tlist, brme_results[0], 'r')
ax.plot(tlist, brme_results[1], 'g')
ax.plot(tlist, brme_results[2], 'b')
ax.legend(("sx", "sy", "sz"))
sphere=Bloch(axes=fig.add_subplot(2,2,3, projection='3d'))
sphere.add_points([lme_results[0],lme_results[1],lme_results[2]], meth='l')
sphere.vector_color = ['r']
sphere.add_vectors([sin(theta),0,cos(theta)])
sphere.make_sphere()
sphere=Bloch(axes=fig.add_subplot(2,2,4, projection='3d'))
sphere.add_points([brme_results[0],brme_results[1],brme_results[2]], meth='l')
sphere.vector_color = ['r']
sphere.add_vectors([sin(theta),0,cos(theta)])
sphere.make_sphere()
In [7]:
def qubit_integrate(w, theta, g, gamma1, gamma2, psi0, tlist):
#
# Hamiltonian
#
sx1 = tensor(sigmax(),qeye(2))
sy1 = tensor(sigmay(),qeye(2))
sz1 = tensor(sigmaz(),qeye(2))
sm1 = tensor(sigmam(),qeye(2))
sx2 = tensor(qeye(2),sigmax())
sy2 = tensor(qeye(2),sigmay())
sz2 = tensor(qeye(2),sigmaz())
sm2 = tensor(qeye(2),sigmam())
H = w[0] * (cos(theta[0]) * sz1 + sin(theta[0]) * sx1) # qubit 1
H += w[1] * (cos(theta[1]) * sz2 + sin(theta[1]) * sx2) # qubit 2
H += g * sx1 * sx2 # interaction
#
# Lindblad master equation
#
c_op_list = []
n_th = 0.0 # zero temperature
rate = gamma1[0] * (n_th + 1)
if rate > 0.0: c_op_list.append(sqrt(rate) * sm1)
rate = gamma1[1] * (n_th + 1)
if rate > 0.0: c_op_list.append(sqrt(rate) * sm2)
lme_results = mesolve(H, psi0, tlist, c_op_list, [sx1, sy1, sz1]).expect
#
# Bloch-Redfield tensor
#
def ohmic_spectrum1(w):
if w == 0.0:
# dephasing inducing noise
return gamma1[0]/2
else:
# relaxation inducing noise
return gamma1[0] * w / (2*pi) * (w > 0.0)
def ohmic_spectrum2(w):
if w == 0.0:
# dephasing inducing noise
return gamma1[1]/2
else:
# relaxation inducing noise
return gamma1[1] * w / (2*pi) * (w > 0.0)
brme_results = brmesolve(H, psi0, tlist, [sx1, sx2], [sx1, sy1, sz1], \
[ohmic_spectrum1, ohmic_spectrum2]).expect
# alternative:
#R, ekets = bloch_redfield_tensor(H, [sx1, sx2], [ohmic_spectrum1, ohmic_spectrum2])
#brme_results = brmesolve(R, ekets, psi0, tlist, [sx1, sy1, sz1])
return lme_results, brme_results
In [8]:
w = array([1.0, 1.0]) * 2 * pi # qubit angular frequency
theta = array([0.15, 0.45]) * 2 * pi # qubit angle from sigma_z axis (toward sigma_x axis)
gamma1 = [0.25, 0.35] # qubit relaxation rate
gamma2 = [0.0, 0.0] # qubit dephasing rate
g = 0.1 * 2 * pi
# initial state
a = 0.8
psi1 = (a*basis(2,0) + (1-a)*basis(2,1))/(sqrt(a**2 + (1-a)**2))
psi2 = ((1-a)*basis(2,0) + a*basis(2,1))/(sqrt(a**2 + (1-a)**2))
psi0 = tensor(psi1, psi2)
tlist = linspace(0,15,5000)
In [9]:
lme_results, brme_results = qubit_integrate(w, theta, g, gamma1, gamma2, psi0, tlist)
In [10]:
fig = figure(figsize=(12,12))
ax = fig.add_subplot(2,2,1)
title('Lindblad master equation')
ax.plot(tlist, lme_results[0], 'r')
ax.plot(tlist, lme_results[1], 'g')
ax.plot(tlist, lme_results[2], 'b')
ax.legend(("sx", "sy", "sz"))
ax = fig.add_subplot(2,2,2)
title('Bloch-Redfield master equation')
ax.plot(tlist, brme_results[0], 'r')
ax.plot(tlist, brme_results[1], 'g')
ax.plot(tlist, brme_results[2], 'b')
ax.legend(("sx", "sy", "sz"))
sphere=Bloch(axes=fig.add_subplot(2,2,3, projection='3d'))
sphere.add_points([lme_results[0],lme_results[1],lme_results[2]], meth='l')
sphere.vector_color = ['r']
sphere.add_vectors([sin(theta[0]),0,cos(theta[0])])
sphere.make_sphere()
sphere=Bloch(axes=fig.add_subplot(2,2,4, projection='3d'))
sphere.add_points([brme_results[0],brme_results[1],brme_results[2]], meth='l')
sphere.vector_color = ['r']
sphere.add_vectors([sin(theta[0]),0,cos(theta[0])])
sphere.make_sphere()
In [11]:
from qutip.ipynbtools import version_table
version_table()
Out[11]: