QuTiP example: Bloch-Redfield Master Equation


In [1]:
%matplotlib inline
import matplotlib.pyplot as plt

In [2]:
import numpy as np

In [3]:
from qutip import *

Two-level system


In [4]:
delta = 0.0 * 2 * np.pi
epsilon = 0.5 * 2 * np.pi
gamma = 0.25
times = np.linspace(0, 10, 100)

In [5]:
H = delta/2 * sigmax() + epsilon/2 * sigmaz()
H


Out[5]:
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}1.571 & 0.0\\0.0 & -1.571\\\end{array}\right)\end{equation*}

In [6]:
psi0 = (2 * basis(2, 0) + basis(2, 1)).unit()

In [7]:
c_ops = [np.sqrt(gamma) * sigmam()]
a_ops = [sigmax()]

In [8]:
e_ops = [sigmax(), sigmay(), sigmaz()]

In [9]:
result_me = mesolve(H, psi0, times, c_ops, e_ops)

In [10]:
result_brme = brmesolve(H, psi0, times, a_ops, e_ops, spectra_cb=[lambda w : gamma * (w > 0)])

In [11]:
plot_expectation_values([result_me, result_brme]);



In [12]:
b = Bloch()
b.add_points(result_me.expect, meth='l')
b.add_points(result_brme.expect, meth='l')
b.make_sphere()


Harmonic oscillator


In [13]:
N = 10

w0 = 1.0 * 2 * np.pi
g = 0.05 * w0
kappa = 0.15

times = np.linspace(0, 25, 1000)

In [14]:
a = destroy(N)

In [15]:
H = w0 * a.dag() * a + g * (a + a.dag())

In [16]:
# start in a superposition state
psi0 = ket2dm((basis(N, 4) + basis(N, 2) + basis(N,0)).unit())

In [19]:
c_ops = [np.sqrt(kappa) * a]
a_ops = [[a + a.dag(),lambda w : kappa * (w > 0)]]

In [20]:
e_ops = [a.dag() * a, a + a.dag()]

Zero temperature


In [21]:
result_me = mesolve(H, psi0, times, c_ops, e_ops)

In [22]:
result_brme = brmesolve(H, psi0, times, a_ops, e_ops)

In [23]:
plot_expectation_values([result_me, result_brme]);


Finite temperature


In [24]:
times = np.linspace(0, 25, 250)

In [27]:
n_th = 1.5
c_ops = [np.sqrt(kappa * (n_th + 1)) * a, np.sqrt(kappa * n_th) * a.dag()]

In [28]:
result_me = mesolve(H, psi0, times, c_ops, e_ops)

In [29]:
w_th = w0/np.log(1 + 1/n_th)

def S_w(w):
    if w >= 0:
        return (n_th + 1) * kappa
    else:
        return (n_th + 1) * kappa * np.exp(w / w_th)
    
a_ops = [[a + a.dag(),S_w]]

In [30]:
result_brme = brmesolve(H, psi0, times, a_ops, e_ops)

In [31]:
plot_expectation_values([result_me, result_brme]);


Storing states instead of expectation values


In [32]:
result_me = mesolve(H, psi0, times, c_ops, [])

In [33]:
result_brme = brmesolve(H, psi0, times, a_ops, [])

In [34]:
n_me = expect(a.dag() * a, result_me.states)

In [35]:
n_brme = expect(a.dag() * a, result_brme.states)

In [36]:
fig, ax = plt.subplots()

ax.plot(times, n_me, label='me')
ax.plot(times, n_brme, label='brme')
ax.legend()
ax.set_xlabel("t");


Atom-Cavity


In [37]:
N = 10
a = tensor(destroy(N), identity(2))
sm = tensor(identity(N), destroy(2))
psi0 = ket2dm(tensor(basis(N, 1), basis(2, 0)))
e_ops = [a.dag() * a, sm.dag() * sm]

Weak coupling


In [38]:
w0 = 1.0 * 2 * np.pi
g = 0.05 * 2 * np.pi
kappa = 0.05
times = np.linspace(0, 5 * 2 * np.pi / g, 1000)

a_ops = [[(a + a.dag()),lambda w : kappa*(w > 0)]]

In [39]:
c_ops = [np.sqrt(kappa) * a]
H = w0 * a.dag() * a + w0 * sm.dag() * sm + g * (a + a.dag()) * (sm + sm.dag())

In [40]:
result_me = mesolve(H, psi0, times, c_ops, e_ops)

In [41]:
result_brme = brmesolve(H, psi0, times, a_ops, e_ops)

In [42]:
plot_expectation_values([result_me, result_brme]);


In the weak coupling regime there is no significant difference between the Lindblad master equation and the Bloch-Redfield master equation.

Strong coupling


In [43]:
w0 = 1.0 * 2 * np.pi
g = 0.75 * 2 * np.pi
kappa = 0.05
times = np.linspace(0, 5 * 2 * np.pi / g, 1000)

In [44]:
c_ops = [np.sqrt(kappa) * a]
H = w0 * a.dag() * a + w0 * sm.dag() * sm + g * (a + a.dag()) * (sm + sm.dag())

In [45]:
result_me = mesolve(H, psi0, times, c_ops, e_ops)

In [48]:
result_brme = brmesolve(H, psi0, times, a_ops, e_ops)

In [49]:
plot_expectation_values([result_me, result_brme]);


In the strong coupling regime there are some corrections to the Lindblad master equation that is due to the fact system eigenstates are hybridized states with both atomic and cavity contributions.

Versions


In [50]:
from qutip.ipynbtools import version_table

version_table()


Out[50]:
SoftwareVersion
QuTiP4.2.0
Numpy1.13.1
SciPy0.19.1
matplotlib2.0.2
Cython0.25.2
Number of CPUs2
BLAS InfoINTEL MKL
IPython6.1.0
Python3.6.1 |Anaconda custom (x86_64)| (default, May 11 2017, 13:04:09) [GCC 4.2.1 Compatible Apple LLVM 6.0 (clang-600.0.57)]
OSposix [darwin]
Wed Jul 19 22:10:15 2017 MDT

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