J.R. Johansson and P.D. Nation
For more information about QuTiP see http://qutip.org
In [1]:
%matplotlib inline
In [2]:
import matplotlib.pyplot as plt
In [3]:
import numpy as np
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from qutip import *
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def qubit_integrate(epsilon, delta, g1, g2, solver):
H = epsilon / 2.0 * sigmaz() + delta / 2.0 * sigmax()
# collapse operators
c_ops = []
if g1 > 0.0:
c_ops.append(np.sqrt(g1) * sigmam())
if g2 > 0.0:
c_ops.append(np.sqrt(g2) * sigmaz())
e_ops = [sigmax(), sigmay(), sigmaz()]
if solver == "me":
output = mesolve(H, psi0, tlist, c_ops, e_ops)
elif solver == "es":
output = essolve(H, psi0, tlist, c_ops, e_ops)
elif solver == "mc":
ntraj = 250
output = mcsolve(H, psi0, tlist, ntraj, c_ops, [sigmax(), sigmay(), sigmaz()])
else:
raise ValueError("unknown solver")
return output.expect[0], output.expect[1], output.expect[2]
In [6]:
epsilon = 0.0 * 2 * np.pi # cavity frequency
delta = 1.0 * 2 * np.pi # atom frequency
g2 = 0.15
g1 = 0.0
# intial state
psi0 = basis(2,0)
tlist = np.linspace(0,5,200)
# analytics
sx_analytic = np.zeros(shape(tlist))
sy_analytic = -np.sin(2*np.pi*tlist) * np.exp(-tlist * g2)
sz_analytic = np.cos(2*np.pi*tlist) * np.exp(-tlist * g2)
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sx1, sy1, sz1 = qubit_integrate(epsilon, delta, g1, g2, "me")
In [8]:
fig, ax = plt.subplots(figsize=(12,6))
ax.plot(tlist, np.real(sx1), 'r')
ax.plot(tlist, np.real(sy1), 'b')
ax.plot(tlist, np.real(sz1), 'g')
ax.plot(tlist, sx_analytic, 'r*')
ax.plot(tlist, sy_analytic, 'g*')
ax.plot(tlist, sz_analytic, 'g*')
ax.legend(("sx", "sy", "sz"))
ax.set_xlabel('Time')
ax.set_ylabel('expectation value');
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sx2, sy2, sz2 = qubit_integrate(epsilon, delta, 0, 0, "me")
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# analytics
sx_analytic = np.zeros(np.shape(tlist))
sy_analytic = -np.sin(2*np.pi*tlist)
sz_analytic = np.cos(2*np.pi*tlist)
In [11]:
fig, ax = plt.subplots(figsize=(12,6))
ax.plot(tlist, np.real(sx2), 'r')
ax.plot(tlist, np.real(sy2), 'b')
ax.plot(tlist, np.real(sz2), 'g')
ax.plot(tlist, sx_analytic, 'r*')
ax.plot(tlist, sy_analytic, 'g*')
ax.plot(tlist, sz_analytic, 'g*')
ax.legend(("sx", "sy", "sz"))
ax.set_xlabel('Time')
ax.set_ylabel('expectation value');
In [12]:
w = 1.0 * 2 * np.pi # qubit angular frequency
theta = 0.2 * np.pi # qubit angle from sigma_z axis (toward sigma_x axis)
gamma1 = 0.05 # qubit relaxation rate
gamma2 = 0.02 # qubit dephasing rate
# initial state
a = 1.0
psi0 = (a * basis(2,0) + (1-a)*basis(2,1))/(np.sqrt(a**2 + (1-a)**2))
tlist = np.linspace(0,15,1000)
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def qubit_integrate(w, theta, gamma1, gamma2, psi0, tlist):
# Hamiltonian
sx = sigmax()
sy = sigmay()
sz = sigmaz()
sm = sigmam()
H = w * (np.cos(theta) * sz + np.sin(theta) * sx)
# collapse operators
c_op_list = []
n_th = 0.5 # zero temperature
rate = gamma1 * (n_th + 1)
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * sm)
rate = gamma1 * n_th
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * sm.dag())
rate = gamma2
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * sz)
# evolve and calculate expectation values
output = mesolve(H, psi0, tlist, c_op_list, [sx, sy, sz])
return output.expect[0], output.expect[1], output.expect[2]
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sx, sy, sz = qubit_integrate(w, theta, gamma1, gamma2, psi0, tlist)
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sphere=Bloch()
sphere.add_points([sx,sy,sz], meth='l')
sphere.vector_color = ['r']
sphere.add_vectors([np.sin(theta), 0, np.cos(theta)])
sphere.show()
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from qutip.ipynbtools import version_table
version_table()
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